Optimal. Leaf size=271 \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (m+n p+n+1) (a d (m+1)-b c (m+n (p+2)+1))-a (m+1) (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1))))}{b^2 e (m+1) (m+n p+n+1) (m+n (p+2)+1)}-\frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1} (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1)))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}+\frac{d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]
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Rubi [A] time = 0.327428, antiderivative size = 255, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {596, 459, 365, 364} \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1} (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}-\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (\frac{a (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b (m+n p+n+1)}+a A d-\frac{A b c (m+n (p+2)+1)}{m+1}\right )}{b e (m+n (p+2)+1)}+\frac{d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]
Antiderivative was successfully verified.
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Rule 596
Rule 459
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx &=\frac{d (e x)^{1+m} \left (a+b x^n\right )^{1+p} \left (A+B x^n\right )}{b e (1+m+n (2+p))}+\frac{\int (e x)^m \left (a+b x^n\right )^p \left (-A (a d (1+m)-b c (1+m+n (2+p)))+(A b d n-a B d (1+m+n)+b B c (1+m+n (2+p))) x^n\right ) \, dx}{b (1+m+n (2+p))}\\ &=\frac{(A b d n-a B d (1+m+n)+b B c (1+m+n (2+p))) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n+n p) (1+m+n (2+p))}+\frac{d (e x)^{1+m} \left (a+b x^n\right )^{1+p} \left (A+B x^n\right )}{b e (1+m+n (2+p))}-\frac{\left (A (a d (1+m)-b c (1+m+n (2+p)))+\frac{a (1+m) (A b d n-a B d (1+m+n)+b B c (1+m+n (2+p)))}{b (1+m+n+n p)}\right ) \int (e x)^m \left (a+b x^n\right )^p \, dx}{b (1+m+n (2+p))}\\ &=\frac{(A b d n-a B d (1+m+n)+b B c (1+m+n (2+p))) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n+n p) (1+m+n (2+p))}+\frac{d (e x)^{1+m} \left (a+b x^n\right )^{1+p} \left (A+B x^n\right )}{b e (1+m+n (2+p))}-\frac{\left (\left (A (a d (1+m)-b c (1+m+n (2+p)))+\frac{a (1+m) (A b d n-a B d (1+m+n)+b B c (1+m+n (2+p)))}{b (1+m+n+n p)}\right ) \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}{b (1+m+n (2+p))}\\ &=\frac{(A b d n-a B d (1+m+n)+b B c (1+m+n (2+p))) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n+n p) (1+m+n (2+p))}+\frac{d (e x)^{1+m} \left (a+b x^n\right )^{1+p} \left (A+B x^n\right )}{b e (1+m+n (2+p))}-\frac{\left (A (a d (1+m)-b c (1+m+n (2+p)))+\frac{a (1+m) (A b d n-a B d (1+m+n)+b B c (1+m+n (2+p)))}{b (1+m+n+n p)}\right ) (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 )}{b e (1+m) (1+m+n (2+p))}\\ \end{align*}
Mathematica [A] time = 0.207783, size = 164, normalized size = 0.61 \[ x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (x^n \left (\frac{(A d+B c) \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+\frac{B d x^n \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}\right )+\frac{A c \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) \, 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{\left (B x^{n} + A\right )}{\left (d x^{n} + c\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}\,{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 ({\left (B d x^{2 \, n} + A c +{\left (B c + A d\right )} x^{n}\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}, 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{\left (B x^{n} + A\right )}{\left (d x^{n} + c\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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