Integrand size = 19, antiderivative size = 79 \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440} \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right ) \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^4}{a}\right )^p \left (c+d x^4\right )^q \, dx \\ & = \left (\left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q}\right ) \int \left (1+\frac {b x^4}{a}\right )^p \left (1+\frac {d x^4}{c}\right )^q \, dx \\ & = x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} F_1\left (\frac {1}{4};-p,-q;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(79)=158\).
Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.18 \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\frac {5 a c x \left (a+b x^4\right )^p \left (c+d x^4\right )^q \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a c \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+4 x^4 \left (b c p \operatorname {AppellF1}\left (\frac {5}{4},1-p,-q,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+a d q \operatorname {AppellF1}\left (\frac {5}{4},-p,1-q,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )} \]
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\[\int \left (b \,x^{4}+a \right )^{p} \left (d \,x^{4}+c \right )^{q}d x\]
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\[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )}^{p} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\text {Timed out} \]
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\[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )}^{p} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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\[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\int { {\left (b x^{4} + a\right )}^{p} {\left (d x^{4} + c\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\int {\left (b\,x^4+a\right )}^p\,{\left (d\,x^4+c\right )}^q \,d x \]
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