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 = 0.04, 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} \[ x \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \left (a-\frac {b c}{4 d q+5 d}\right ) \, _2F_1\left (\frac {1}{4},-q;\frac {5}{4};-\frac {d x^4}{c}\right )+\frac {b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 1.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q}, x\right ) \]
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 \[ \int {\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{4}+a \right ) \left (d \,x^{4}+c \right )^{q}\, dx \]
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 \[ \int {\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q}\,{d x} \]
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 \[ \int \left (b\,x^4+a\right )\,{\left (d\,x^4+c\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 61.01, size = 75, normalized size = 0.81 \[ \frac {a c^{q} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - q \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {b c^{q} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - q \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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