Optimal. Leaf size=366 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-2 n+1)-A d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )\right )+b^2 c^2 (m-2 n+1) (A d (m-3 n+1)-B c (m-n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (B c (m+1)-A d (m-2 n+1))+b c (A d (m-4 n+1)-B c (m-2 n+1)))}{2 c^2 e n^2 (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (B c-A d)}{2 c e n (b c-a d) \left (c+d x^n\right )^2} \]
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Rubi [A] time = 1.2169, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {595, 597, 364} \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-2 n+1)-A d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )\right )+b^2 c^2 (m-2 n+1) (A d (m-3 n+1)-B c (m-n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (B c (m+1)-A d (m-2 n+1))+b c (A d (m-4 n+1)-B c (m-2 n+1)))}{2 c^2 e n^2 (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (B c-A d)}{2 c e n (b c-a d) \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 595
Rule 597
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx &=\frac{(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e n \left (c+d x^n\right )^2}+\frac{\int \frac{(e x)^m \left (-a B c (1+m)+a A d (1+m-2 n)+2 A b c n-b (B c-A d) (1+m-2 n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{2 c (b c-a d) n}\\ &=\frac{(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e n \left (c+d x^n\right )^2}+\frac{(b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n))) (e x)^{1+m}}{2 c^2 (b c-a d)^2 e n^2 \left (c+d x^n\right )}+\frac{\int \frac{(e x)^m \left (-a (1+m) (b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n)))-(b c-a d) n (a B c (1+m)-a A d (1+m-2 n)-2 A b c n)-b (b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n))) (1+m-n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{2 c^2 (b c-a d)^2 n^2}\\ &=\frac{(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e n \left (c+d x^n\right )^2}+\frac{(b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n))) (e x)^{1+m}}{2 c^2 (b c-a d)^2 e n^2 \left (c+d x^n\right )}+\frac{\int \left (\frac{2 b^2 (A b-a B) c^2 n^2 (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac{\left (-b^2 c^2 (A d (1+m-3 n)-B c (1+m-n)) (1+m-2 n)+a^2 d^2 (B c (1+m)-A d (1+m-2 n)) (1+m-n)-2 a b c d \left (B c (1+m) (1+m-2 n)-A d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )\right )\right ) (e x)^m}{(b c-a d) \left (c+d x^n\right )}\right ) \, dx}{2 c^2 (b c-a d)^2 n^2}\\ &=\frac{(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e n \left (c+d x^n\right )^2}+\frac{(b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n))) (e x)^{1+m}}{2 c^2 (b c-a d)^2 e n^2 \left (c+d x^n\right )}+\frac{\left (b^2 (A b-a B)\right ) \int \frac{(e x)^m}{a+b x^n} \, dx}{(b c-a d)^3}-\frac{\left (b^2 c^2 (A d (1+m-3 n)-B c (1+m-n)) (1+m-2 n)-a^2 d^2 (B c (1+m)-A d (1+m-2 n)) (1+m-n)+2 a b c d \left (B c (1+m) (1+m-2 n)-A d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )\right )\right ) \int \frac{(e x)^m}{c+d x^n} \, dx}{2 c^2 (b c-a d)^3 n^2}\\ &=\frac{(B c-A d) (e x)^{1+m}}{2 c (b c-a d) e n \left (c+d x^n\right )^2}+\frac{(b c (A d (1+m-4 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-2 n))) (e x)^{1+m}}{2 c^2 (b c-a d)^2 e n^2 \left (c+d x^n\right )}+\frac{b^2 (A b-a B) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3 e (1+m)}-\frac{\left (b^2 c^2 (A d (1+m-3 n)-B c (1+m-n)) (1+m-2 n)-a^2 d^2 (B c (1+m)-A d (1+m-2 n)) (1+m-n)+2 a b c d \left (B c (1+m) (1+m-2 n)-A d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{2 c^3 (b c-a d)^3 e (1+m) n^2}\\ \end{align*}
Mathematica [A] time = 0.266575, size = 201, normalized size = 0.55 \[ \frac{x (e x)^m \left (\frac{b^2 (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a}-\frac{d (A b-a B) (b c-a d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c^2}+\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c^3}-\frac{b d (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c}\right )}{(m+1) (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.698, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{3}}}\, 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*}{\left ({\left ({\left (m^{2} - m{\left (5 \, n - 2\right )} + 6 \, n^{2} - 5 \, n + 1\right )} b^{2} c^{2} d e^{m} - 2 \,{\left (m^{2} - 2 \, m{\left (2 \, n - 1\right )} + 3 \, n^{2} - 4 \, n + 1\right )} a b c d^{2} e^{m} +{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} a^{2} d^{3} e^{m}\right )} A -{\left ({\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{2} c^{3} e^{m} - 2 \,{\left (m^{2} - 2 \, m{\left (n - 1\right )} - 2 \, n + 1\right )} a b c^{2} d e^{m} +{\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a^{2} c d^{2} e^{m}\right )} B\right )} \int -\frac{x^{m}}{2 \,{\left (b^{3} c^{6} n^{2} - 3 \, a b^{2} c^{5} d n^{2} + 3 \, a^{2} b c^{4} d^{2} n^{2} - a^{3} c^{3} d^{3} n^{2} +{\left (b^{3} c^{5} d n^{2} - 3 \, a b^{2} c^{4} d^{2} n^{2} + 3 \, a^{2} b c^{3} d^{3} n^{2} - a^{3} c^{2} d^{4} n^{2}\right )} x^{n}\right )}}\,{d x} +{\left (B a b^{2} e^{m} - A b^{3} e^{m}\right )} \int -\frac{x^{m}}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{n}}\,{d x} - \frac{{\left ({\left (a c d^{2} e^{m}{\left (m - 3 \, n + 1\right )} - b c^{2} d e^{m}{\left (m - 5 \, n + 1\right )}\right )} A -{\left (a c^{2} d e^{m}{\left (m - n + 1\right )} - b c^{3} e^{m}{\left (m - 3 \, n + 1\right )}\right )} B\right )} x x^{m} +{\left ({\left (a d^{3} e^{m}{\left (m - 2 \, n + 1\right )} - b c d^{2} e^{m}{\left (m - 4 \, n + 1\right )}\right )} A +{\left (b c^{2} d e^{m}{\left (m - 2 \, n + 1\right )} - a c d^{2} e^{m}{\left (m + 1\right )}\right )} B\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (b^{2} c^{6} n^{2} - 2 \, a b c^{5} d n^{2} + a^{2} c^{4} d^{2} n^{2} +{\left (b^{2} c^{4} d^{2} n^{2} - 2 \, a b c^{3} d^{3} n^{2} + a^{2} c^{2} d^{4} n^{2}\right )} x^{2 \, n} + 2 \,{\left (b^{2} c^{5} d n^{2} - 2 \, a b c^{4} d^{2} n^{2} + a^{2} c^{3} d^{3} n^{2}\right )} x^{n}\right )}} \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 (e x\right )^{m}}{b d^{3} x^{4 \, n} + a c^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} x^{3 \, n} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} x^{2 \, n} +{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{n}}, 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 (e x\right )^{m}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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