Optimal. Leaf size=386 \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+n+1))-3 a b c d (A d (m+n+1)-B c (m+2 n+1))+b^2 c^2 (A d (m+2 n+1)-B c (m+3 n+1))\right )}{c d^4 e (m+1) n}-\frac{b^2 x^{n+1} (e x)^m (3 a d (A d (m+n+1)-B c (m+2 n+1))-b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c d^3 n (m+n+1)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c^2 d^4 e (m+1) n}-\frac{(e x)^{m+1} \left (a+b x^n\right )^3 (B c-A d)}{c d e n \left (c+d x^n\right )}-\frac{b^3 x^{2 n+1} (e x)^m (A d (m+2 n+1)-B c (m+3 n+1))}{c d^2 n (m+2 n+1)} \]
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Rubi [A] time = 1.13472, antiderivative size = 381, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {594, 570, 20, 30, 364} \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+n+1))-3 a b c d (A d (m+n+1)-B c (m+2 n+1))+b^2 c^2 (A d (m+2 n+1)-B c (m+3 n+1))\right )}{c d^4 e (m+1) n}-\frac{b^2 x^{n+1} (e x)^m (3 a d (A d (m+n+1)-B c (m+2 n+1))-b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c d^3 n (m+n+1)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c^2 d^4 e (m+1) n}-\frac{(e x)^{m+1} \left (a+b x^n\right )^3 (B c-A d)}{c d e n \left (c+d x^n\right )}-\frac{b^3 x^{2 n+1} (e x)^m \left (A-\frac{B c (m+3 n+1)}{d (m+2 n+1)}\right )}{c d n} \]
Antiderivative was successfully verified.
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Rule 594
Rule 570
Rule 20
Rule 30
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^3}{c d e n \left (c+d x^n\right )}-\frac{\int \frac{(e x)^m \left (a+b x^n\right )^2 \left (-a (B c (1+m)-A d (1+m-n))+b (A d (1+m+2 n)-B c (1+m+3 n)) x^n\right )}{c+d x^n} \, dx}{c d n}\\ &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^3}{c d e n \left (c+d x^n\right )}-\frac{\int \left (\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (1+m+n))-3 a b c d (A d (1+m+n)-B c (1+m+2 n))+b^2 c^2 (A d (1+m+2 n)-B c (1+m+3 n))\right ) (e x)^m}{d^3}+\frac{b^2 (3 a d (A d (1+m+n)-B c (1+m+2 n))-b c (A d (1+m+2 n)-B c (1+m+3 n))) x^n (e x)^m}{d^2}+\frac{b^3 (A d (1+m+2 n)-B c (1+m+3 n)) x^{2 n} (e x)^m}{d}+\frac{(b c-a d)^2 (-a d (B c (1+m)-A d (1+m-n))-b c (A d (1+m+2 n)-B c (1+m+3 n))) (e x)^m}{d^3 \left (c+d x^n\right )}\right ) \, dx}{c d n}\\ &=-\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (1+m+n))-3 a b c d (A d (1+m+n)-B c (1+m+2 n))+b^2 c^2 (A d (1+m+2 n)-B c (1+m+3 n))\right ) (e x)^{1+m}}{c d^4 e (1+m) n}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^3}{c d e n \left (c+d x^n\right )}-\frac{\left (b^3 (A d (1+m+2 n)-B c (1+m+3 n))\right ) \int x^{2 n} (e x)^m \, dx}{c d^2 n}-\frac{\left (b^2 (3 a d (A d (1+m+n)-B c (1+m+2 n))-b c (A d (1+m+2 n)-B c (1+m+3 n)))\right ) \int x^n (e x)^m \, dx}{c d^3 n}+\frac{\left ((b c-a d)^2 (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+2 n)-B c (1+m+3 n)))\right ) \int \frac{(e x)^m}{c+d x^n} \, dx}{c d^4 n}\\ &=-\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (1+m+n))-3 a b c d (A d (1+m+n)-B c (1+m+2 n))+b^2 c^2 (A d (1+m+2 n)-B c (1+m+3 n))\right ) (e x)^{1+m}}{c d^4 e (1+m) n}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^3}{c d e n \left (c+d x^n\right )}+\frac{(b c-a d)^2 (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+2 n)-B c (1+m+3 n))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c^2 d^4 e (1+m) n}-\frac{\left (b^3 (A d (1+m+2 n)-B c (1+m+3 n)) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx}{c d^2 n}-\frac{\left (b^2 (3 a d (A d (1+m+n)-B c (1+m+2 n))-b c (A d (1+m+2 n)-B c (1+m+3 n))) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{c d^3 n}\\ &=-\frac{b^2 (3 a d (A d (1+m+n)-B c (1+m+2 n))-b c (A d (1+m+2 n)-B c (1+m+3 n))) x^{1+n} (e x)^m}{c d^3 n (1+m+n)}-\frac{b^3 (A d (1+m+2 n)-B c (1+m+3 n)) x^{1+2 n} (e x)^m}{c d^2 n (1+m+2 n)}-\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (1+m+n))-3 a b c d (A d (1+m+n)-B c (1+m+2 n))+b^2 c^2 (A d (1+m+2 n)-B c (1+m+3 n))\right ) (e x)^{1+m}}{c d^4 e (1+m) n}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^3}{c d e n \left (c+d x^n\right )}+\frac{(b c-a d)^2 (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+2 n)-B c (1+m+3 n))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c^2 d^4 e (1+m) n}\\ \end{align*}
Mathematica [A] time = 0.52231, size = 220, normalized size = 0.57 \[ \frac{x (e x)^m \left (\frac{b \left (3 a^2 B d^2+3 a b d (A d-2 B c)+b^2 c (3 B c-2 A d)\right )}{m+1}+\frac{b^2 d x^n (3 a B d+A b d-2 b B c)}{m+n+1}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c^2 (m+1)}-\frac{(b c-a d)^2 (-a B d-3 A b d+4 b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c (m+1)}+\frac{b^3 B d^2 x^{2 n}}{m+2 n+1}\right )}{d^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.509, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{3} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{2}}}\, 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*} \text{result too large to display} \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 b^{3} x^{4 \, n} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3 \, n} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2 \, n} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{n}\right )} \left (e x\right )^{m}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, 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 )}^{3} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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