Integrand size = 19, antiderivative size = 57 \[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\frac {x \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},1,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440} \[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\frac {x \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},1,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a} \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \left (\left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q}\right ) \int \frac {\left (1+\frac {d x^4}{c}\right )^q}{a+b x^4} \, dx \\ & = \frac {x \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} F_1\left (\frac {1}{4};1,-q;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).
Time = 0.41 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.84 \[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\frac {5 a c x \left (c+d x^4\right )^q \operatorname {AppellF1}\left (\frac {1}{4},-q,1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{\left (a+b x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},-q,1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+4 x^4 \left (a d q \operatorname {AppellF1}\left (\frac {5}{4},1-q,1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )-b c \operatorname {AppellF1}\left (\frac {5}{4},-q,2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )} \]
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\[\int \frac {\left (d \,x^{4}+c \right )^{q}}{b \,x^{4}+a}d x\]
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\[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\int { \frac {{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\int { \frac {{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a} \,d x } \]
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\[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\int { \frac {{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, dx=\int \frac {{\left (d\,x^4+c\right )}^q}{b\,x^4+a} \,d x \]
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