Integrand size = 19, antiderivative size = 81 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 81, 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} \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \]
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Rule 440
Rule 441
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^n}{a}\right )^p \left (c+d x^n\right )^q \, dx \\ & = \left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q}\right ) \int \left (1+\frac {b x^n}{a}\right )^p \left (1+\frac {d x^n}{c}\right )^q \, dx \\ & = x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} F_1\left (\frac {1}{n};-p,-q;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(81)=162\).
Time = 0.45 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.35 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {a c (1+n) x \left (a+b x^n\right )^p \left (c+d x^n\right )^q \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a d n q x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,1-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )} \]
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\[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q}d x\]
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\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \,d x } \]
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Exception generated. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \,d x } \]
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\[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int {\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q \,d x \]
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