Optimal. Leaf size=93 \[ \frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)}-\frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} (b c-a d (4 q+5)) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d (4 q+5)} \]
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Rubi [A] time = 0.0407276, 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} \[ x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a-\frac{b c}{4 d q+5 d}\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+\frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)} \]
Antiderivative was successfully verified.
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Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx &=\frac{b x \left (c+d x^4\right )^{1+q}}{d (5+4 q)}-\left (-a+\frac{b c}{5 d+4 d q}\right ) \int \left (c+d x^4\right )^q \, dx\\ &=\frac{b x \left (c+d x^4\right )^{1+q}}{d (5+4 q)}-\left (\left (-a+\frac{b c}{5 d+4 d q}\right ) \left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q}\right ) \int \left (1+\frac{d x^4}{c}\right )^q \, dx\\ &=\frac{b x \left (c+d x^4\right )^{1+q}}{d (5+4 q)}+\left (a-\frac{b c}{5 d+4 d q}\right ) x \left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q} \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0299453, size = 90, normalized size = 0.97 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left ((a d (4 q+5)-b c) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+b \left (c+d x^4\right ) \left (\frac{d x^4}{c}+1\right )^q\right )}{d (4 q+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.212, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{4}+a \right ) \left ( d{x}^{4}+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^{4} + a\right )}{\left (d x^{4} + 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^{4} + a\right )}{\left (d x^{4} + c\right )}^{q}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 92.1663, size = 75, normalized size = 0.81 \begin{align*} \frac{a c^{q} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - q \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{b c^{q} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, - q \\ \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \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^{4} + a\right )}{\left (d x^{4} + c\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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