Optimal. Leaf size=93 \[ \frac {d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)}-\frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} (a d-b c (3 m+4)) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b (3 m+4)} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {388, 246, 245} \[ x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (c-\frac {a d}{3 b m+4 b}\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )+\frac {d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right ) \, dx &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (-c+\frac {a d}{4 b+3 b m}\right ) \int \left (a+b x^3\right )^m \, dx\\ &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (\left (-c+\frac {a d}{4 b+3 b m}\right ) \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac {b x^3}{a}\right )^m \, dx\\ &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}+\left (c-\frac {a d}{4 b+3 b m}\right ) x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.97 \[ \frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left ((b c (3 m+4)-a d) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )+d \left (a+b x^3\right ) \left (\frac {b x^3}{a}+1\right )^m\right )}{b (3 m+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+c \right ) \left (b \,x^{3}+a \right )^{m}\, dx \]
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 \[ \int {\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^m\,\left (d\,x^3+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 100.31, size = 75, normalized size = 0.81 \[ \frac {a^{m} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, - m \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{m} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, - m \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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