Integrand size = 17, antiderivative size = 93 \[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\frac {b x \left (c+d x^4\right )^{1+q}}{d (5+4 q)}-\frac {(b c-a d (5+4 q)) x \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-q,\frac {5}{4},-\frac {d x^4}{c}\right )}{d (5+4 q)} \]
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Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {396, 252, 251} \[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=x \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \left (a-\frac {b c}{4 d q+5 d}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-q,\frac {5}{4},-\frac {d x^4}{c}\right )+\frac {b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)} \]
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Rule 251
Rule 252
Rule 396
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\frac {x \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} \left (b \left (c+d x^4\right ) \left (1+\frac {d x^4}{c}\right )^q+(-b c+a d (5+4 q)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-q,\frac {5}{4},-\frac {d x^4}{c}\right )\right )}{d (5+4 q)} \]
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\[\int \left (b \,x^{4}+a \right ) \left (d \,x^{4}+c \right )^{q}d x\]
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\[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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Result contains complex when optimal does not.
Time = 27.99 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.81 \[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\frac {a c^{q} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - q \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {b c^{q} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - q \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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\[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+b x^4\right ) \left (c+d x^4\right )^q \, dx=\int \left (b\,x^4+a\right )\,{\left (d\,x^4+c\right )}^q \,d x \]
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