Optimal. Leaf size=164 \[ \frac{B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
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Rubi [A] time = 0.179496, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {597, 365, 364, 511, 510} \[ \frac{B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 597
Rule 365
Rule 364
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{c+d x^n} \, dx &=\int \left (\frac{B (e x)^m \left (a+b x^n\right )^p}{d}+\frac{(-B c+A d) (e x)^m \left (a+b x^n\right )^p}{d \left (c+d x^n\right )}\right ) \, dx\\ &=\frac{B \int (e x)^m \left (a+b x^n\right )^p \, dx}{d}+\frac{(-B c+A d) \int \frac{(e x)^m \left (a+b x^n\right )^p}{c+d x^n} \, dx}{d}\\ &=\frac{\left (B \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^n}{a}\right )^p \, dx}{d}+\frac{\left ((-B c+A d) \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int \frac{(e x)^m \left (1+\frac{b x^n}{a}\right )^p}{c+d x^n} \, dx}{d}\\ &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} F_1\left (\frac{1+m}{n};-p,1;\frac{1+m+n}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c d e (1+m)}+\frac{B (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{n},-p;\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{d e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.252439, size = 138, normalized size = 0.84 \[ \frac{x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (A (m+n+1) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac{m+n+1}{n};-p,1;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}{c (m+1) (m+n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.676, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) }{c+d{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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