Integrand size = 19, antiderivative size = 189 \[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\frac {x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,2,\frac {5}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4} \]
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Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1254, 441, 440, 525, 524} \[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\frac {x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,2,\frac {5}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3} \]
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Rule 440
Rule 441
Rule 524
Rule 525
Rule 1254
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2}-\frac {2 c e x^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2}+\frac {e^2 x^4 \left (a+b x^4\right )^p}{\left (-c^2+e^2 x^4\right )^2}\right ) \, dx \\ & = c^2 \int \frac {\left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx-(2 c e) \int \frac {x^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx+e^2 \int \frac {x^4 \left (a+b x^4\right )^p}{\left (-c^2+e^2 x^4\right )^2} \, dx \\ & = \left (c^2 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^4}{a}\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx-\left (2 c e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {b x^4}{a}\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx+\left (e^2 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {x^4 \left (1+\frac {b x^4}{a}\right )^p}{\left (-c^2+e^2 x^4\right )^2} \, dx \\ & = \frac {x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {1}{4};-p,2;\frac {5}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {3}{4};-p,2;\frac {7}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {5}{4};-p,2;\frac {9}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4} \\ \end{align*}
\[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \]
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\[\int \frac {\left (b \,x^{4}+a \right )^{p}}{\left (e \,x^{2}+c \right )^{2}}d x\]
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\[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int \frac {{\left (b\,x^4+a\right )}^p}{{\left (e\,x^2+c\right )}^2} \,d x \]
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