Optimal. Leaf size=66 \[ \frac {B x^{1+m}}{b (1+m)}+\frac {(A b-a B) x^{1+m} \, _2F_1\left (1,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a b (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, 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 = {470, 371}
\begin {gather*} \frac {x^{m+1} (A b-a B) \, _2F_1\left (1,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a b (m+1)}+\frac {B x^{m+1}}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 470
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {B x^{1+m}}{b (1+m)}-\frac {(-A b (1+m)+a B (1+m)) \int \frac {x^m}{a+b x^3} \, dx}{b (1+m)}\\ &=\frac {B x^{1+m}}{b (1+m)}+\frac {(A b-a B) x^{1+m} \, _2F_1\left (1,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 55, normalized size = 0.83 \begin {gather*} \frac {x^{1+m} \left (a B+(A b-a B) \, _2F_1\left (1,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )\right )}{a b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (B \,x^{3}+A \right )}{b \,x^{3}+a}\, dx\]
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 = 8.35, size = 190, normalized size = 2.88 \begin {gather*} \frac {A m x x^{m} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{9 a \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {A x x^{m} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {1}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {1}{3}\right )}{9 a \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {B m x^{4} x^{m} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {4}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )}{9 a \Gamma \left (\frac {m}{3} + \frac {7}{3}\right )} + \frac {4 B x^{4} x^{m} \Phi \left (\frac {b x^{3} e^{i \pi }}{a}, 1, \frac {m}{3} + \frac {4}{3}\right ) \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )}{9 a \Gamma \left (\frac {m}{3} + \frac {7}{3}\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 = -0.02 \begin {gather*} \int \frac {x^m\,\left (B\,x^3+A\right )}{b\,x^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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