def replacement1883(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1883) return -Dist( (-a * v + b * u) / a, Int(S(1) / u, x), x) + Simp(b * x / a, x)
def replacement1890(u, v, x): a = D(u, x) b = D(v, x) return Simp( -S(2) * atanh(sqrt(v) / Rt(-(-a * v + b * u) / a, S(2))) / (a * Rt(-(-a * v + b * u) / a, S(2))), x)
def replacement1885(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1885) return -Dist(a / (-a * v + b * u), Int(S(1) / u, x), x) + Dist( b / (-a * v + b * u), Int(S(1) / v, x), x)
def replacement1892(v, u, m, n, x): a = D(u, x) b = D(v, x) rubi.append(1892) return -Simp( u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) * (-a * v + b * u)), x)
def replacement1896(m, n, u, v, x): a = D(u, x) b = D(v, x) return -Dist(b * n / (a * (m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x), x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
def replacement1894(u, v, x): a = D(u, x) b = D(v, x) return Simp( S(2) * ArcTan(sqrt(u) * Rt(-a * b, S(2)) / (a * sqrt(v))) / Rt(-a * b, S(2)), x)
def replacement1893(u, v, x): a = D(u, x) b = D(v, x) return Simp( S(2) * atanh(sqrt(u) * Rt(a * b, S(2)) / (a * sqrt(v))) / Rt(a * b, S(2)), x)
def replacement1889(u, v, x): a = D(u, x) b = D(v, x) return Simp( S(2) * ArcTan(sqrt(v) / Rt((-a * v + b * u) / a, S(2))) / (a * Rt( (-a * v + b * u) / a, S(2))), x)
def replacement1901(m, n, u, v, x): a = D(u, x) b = D(v, x) return Simp( u**m * v**(n + S(1)) * (b * u / (-a * v + b * u))**(-m) * Hypergeometric2F1(-m, n + S(1), n + S(2), -a * v / (-a * v + b * u)) / (b * (n + S(1))), x)
def replacement1898(m, n, u, v, x): a = D(u, x) b = D(v, x) return -Dist(n * (-a * v + b * u) / (a * (m + n + S(1))), Int(u**m * v**(n + S(-1)), x), x) + Simp( u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
def replacement1892(n, u, v, x): a = D(u, x) b = D(v, x) return Simp( v**(n + S(1)) * Hypergeometric2F1(S(1), n + S(1), n + S(2), -a * v / (-a * v + b * u)) / ((n + S(1)) * (-a * v + b * u)), x)
def replacement1884(v, x, n, u): a = D(u, x) b = D(v, x) rubi.append(1884) return -Dist( (-a * v + b * u) / a, Int(v**(n + S(-1)) / u, x), x) + Simp( v**n / (a * n), x)
def replacement1886(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1886) return Simp( S(2) * ArcTan(sqrt(v) / Rt((-a * v + b * u) / a, S(2))) / (a * Rt( (-a * v + b * u) / a, S(2))), x)
def With1898(v, u, m, n, x): if isinstance(x, (int, Integer, float, Float)): return False a = D(u, x) b = D(v, x) if NonzeroQ(-a * v + b * u): return True return False
def replacement1895(v, u, m, n, x): a = D(u, x) b = D(v, x) rubi.append(1895) return -Dist(n * (-a * v + b * u) / (a * (m + n + S(1))), Int(u**m * v**(n + S(-1)), x), x) + Simp(u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
def With1891(v, x, u): if isinstance(x, (int, Integer, float, Float)): return False a = D(u, x) b = D(v, x) if And(NonzeroQ(-a * v + b * u), NegQ(a * b)): return True return False
def replacement1893(v, u, m, n, x): a = D(u, x) b = D(v, x) rubi.append(1893) return -Dist(b * n / (a * (m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x), x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
def With1886(v, x, u): if isinstance(x, (int, Integer, float, Float)): return False a = D(u, x) b = D(v, x) if And(NonzeroQ(-a * v + b * u), PosQ((-a * v + b * u) / a)): return True return False
def replacement1888(v, x, n, u): a = D(u, x) b = D(v, x) rubi.append(1888) return -Dist(a / (-a * v + b * u), Int(v**(n + S(1)) / u, x), x) + Simp( v**(n + S(1)) / ((n + S(1)) * (-a * v + b * u)), x)
def replacement1890(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1890) return Simp( S(2) * atanh(sqrt(u) * Rt(a * b, S(2)) / (a * sqrt(v))) / Rt(a * b, S(2)), x)
def replacement1887(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1887) return Simp( -S(2) * atanh(sqrt(v) / Rt(-(-a * v + b * u) / a, S(2))) / (a * Rt(-(-a * v + b * u) / a, S(2))), x)
def replacement1891(v, x, u): a = D(u, x) b = D(v, x) rubi.append(1891) return Simp( S(2) * ArcTan(sqrt(u) * Rt(-a * b, S(2)) / (a * sqrt(v))) / Rt(-a * b, S(2)), x)
def replacement1898(v, u, m, n, x): a = D(u, x) b = D(v, x) rubi.append(1898) return Simp( u**m * v**(n + S(1)) * (b * u / (-a * v + b * u))**(-m) * Hypergeometric2F1(-m, n + S(1), n + S(2), -a * v / (-a * v + b * u)) / (b * (n + S(1))), x)
def replacement1900(m, n, u, v, x): a = D(u, x) b = D(v, x) return Dist(b * (m + n + S(2)) / ((m + S(1)) * (-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp( u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) * (-a * v + b * u)), x)
def replacement1897(v, u, m, n, x): a = D(u, x) b = D(v, x) rubi.append(1897) return Dist(b * (m + n + S(2)) / ( (m + S(1)) * (-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp( u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) * (-a * v + b * u)), x)
def replacement1889(v, x, n, u): a = D(u, x) b = D(v, x) rubi.append(1889) return Simp( v**(n + S(1)) * Hypergeometric2F1(S(1), n + S(1), n + S(2), -a * v / (-a * v + b * u)) / ((n + S(1)) * (-a * v + b * u)), x)
def With1899(u, b, a, n, x): c = D(u, x) rubi.append(1899) return -Dist(c * n / b, Int(u**(n + S(-1)) * (a + b * x) * log(a + b * x), x), x) - Int( u**n, x) + Simp( u**n * (a + b * x) * log(a + b * x) / b, x)
def With1903(a, b, m, n, u, x): c = D(u, x) return -Dist( c * n / (b * (m + S(1))), Int(u**(n + S(-1)) * (a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist( S(1) / (m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp( u**n * (a + b * x)**(m + S(1)) * log(a + b * x) / (b * (m + S(1))), x)
def With34(p, Pm, b, n2, c, a, n, Qm, x): if isinstance(x, (int, Integer, float, Float)): return False m = Expon(Pm, x) if And( Equal(Expon(Qm, x), m + S(-1)), ZeroQ(-Qm * m * Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1)) * D(Pm, x))): return True return False
def With1900(u, m, b, a, n, x): c = D(u, x) rubi.append(1900) return -Dist( c * n / (b * (m + S(1))), Int(u**(n + S(-1)) * (a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist( S(1) / (m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp( u**n * (a + b * x)**(m + S(1)) * log(a + b * x) / (b * (m + S(1))), x)