Integrand size = 21, antiderivative size = 167 \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+n}{2 n},-p,1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)} \]
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Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1452, 441, 440, 525, 524} \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {n+1}{2 n},-p,1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \]
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Rule 440
Rule 441
Rule 524
Rule 525
Rule 1452
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}}+\frac {e x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}}\right ) \, dx \\ & = d \int \frac {\left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}} \, dx+e \int \frac {x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}} \, dx \\ & = \left (d \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^{2 n}}{a}\right )^p}{d^2-e^2 x^{2 n}} \, dx+\left (e \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^n \left (1+\frac {c x^{2 n}}{a}\right )^p}{-d^2+e^2 x^{2 n}} \, dx \\ & = \frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2 n};-p,1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+n}{2 n};-p,1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)} \\ \end{align*}
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \]
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\[\int \frac {\left (a +c \,x^{2 n}\right )^{p}}{d +e \,x^{n}}d x\]
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\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \]
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\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {{\left (a+c\,x^{2\,n}\right )}^p}{d+e\,x^n} \,d x \]
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