Optimal. Leaf size=84 \[ \frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)}-\frac{x \left (c+d x^3\right )^{q+1} (b c-a d (3 q+4)) \, _2F_1\left (1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right )}{c d (3 q+4)} \]
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Rubi [A] time = 0.0388256, antiderivative size = 85, normalized size of antiderivative = 1.01, 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} \[ x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (a-\frac{b c}{3 d q+4 d}\right ) \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+\frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)} \]
Antiderivative was successfully verified.
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Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^3\right ) \left (c+d x^3\right )^q \, dx &=\frac{b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}-\left (-a+\frac{b c}{4 d+3 d q}\right ) \int \left (c+d x^3\right )^q \, dx\\ &=\frac{b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}-\left (\left (-a+\frac{b c}{4 d+3 d q}\right ) \left (c+d x^3\right )^q \left (1+\frac{d x^3}{c}\right )^{-q}\right ) \int \left (1+\frac{d x^3}{c}\right )^q \, dx\\ &=\frac{b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}+\left (a-\frac{b c}{4 d+3 d q}\right ) x \left (c+d x^3\right )^q \left (1+\frac{d x^3}{c}\right )^{-q} \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0292039, size = 90, normalized size = 1.07 \[ \frac{x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left ((a d (3 q+4)-b c) \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+b \left (c+d x^3\right ) \left (\frac{d x^3}{c}+1\right )^q\right )}{d (3 q+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.239, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) \left ( d{x}^{3}+c \right ) ^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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