Optimal. Leaf size=93 \[ \frac {(A b-a B) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a B (1+m)+A (b-b m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {468, 371}
\begin {gather*} \frac {x^{m+1} (a B (m+1)+A (b-b m)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 468
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a B (1+m)+A (b-b m)) \int \frac {x^m}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a B (1+m)+A (b-b m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 80, normalized size = 0.86 \begin {gather*} \frac {x^{1+m} \left (a B \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )+(A b-a B) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )\right )}{a^2 b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (B \,x^{2}+A \right )}{\left (b \,x^{2}+a \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 15.33, size = 906, normalized size = 9.74 \begin {gather*} A \left (- \frac {a m^{2} x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a m x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {b m^{2} x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}\right ) + B \left (- \frac {a m^{2} x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 a m x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {2 a m x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 a x^{3} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {6 a x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b m^{2} x^{5} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 b m x^{5} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 b x^{5} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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