Optimal. Leaf size=187 \[ \frac{(e x)^{m+1} \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d^3 e (m+1)}-\frac{b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)} \]
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Rubi [A] time = 0.254068, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {570, 20, 30, 364} \[ \frac{(e x)^{m+1} \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d^3 e (m+1)}-\frac{b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)} \]
Antiderivative was successfully verified.
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Rule 570
Rule 20
Rule 30
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx &=\int \left (\frac{\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^m}{d^3}+\frac{b (-b B c+A b d+2 a B d) x^n (e x)^m}{d^2}+\frac{b^2 B x^{2 n} (e x)^m}{d}+\frac{(-b c+a d)^2 (-B c+A d) (e x)^m}{d^3 \left (c+d x^n\right )}\right ) \, dx\\ &=\frac{\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}+\frac{\left (b^2 B\right ) \int x^{2 n} (e x)^m \, dx}{d}-\frac{\left ((b c-a d)^2 (B c-A d)\right ) \int \frac{(e x)^m}{c+d x^n} \, dx}{d^3}-\frac{(b (b B c-A b d-2 a B d)) \int x^n (e x)^m \, dx}{d^2}\\ &=\frac{\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac{(b c-a d)^2 (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c d^3 e (1+m)}+\frac{\left (b^2 B x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx}{d}-\frac{\left (b (b B c-A b d-2 a B d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{d^2}\\ &=-\frac{b (b B c-A b d-2 a B d) x^{1+n} (e x)^m}{d^2 (1+m+n)}+\frac{b^2 B x^{1+2 n} (e x)^m}{d (1+m+2 n)}+\frac{\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac{(b c-a d)^2 (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c d^3 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.288652, size = 154, normalized size = 0.82 \[ \frac{x (e x)^m \left (\frac{a^2 B d^2+2 a b d (A d-B c)+b^2 c (B c-A d)}{m+1}-\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c (m+1)}+\frac{b d x^n (2 a B d+A b d-b B c)}{m+n+1}+\frac{b^2 B d^2 x^{2 n}}{m+2 n+1}\right )}{d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.491, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{2} \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*}{\left ({\left (b^{2} c^{2} d e^{m} - 2 \, a b c d^{2} e^{m} + a^{2} d^{3} e^{m}\right )} A -{\left (b^{2} c^{3} e^{m} - 2 \, a b c^{2} d e^{m} + a^{2} c d^{2} e^{m}\right )} B\right )} \int \frac{x^{m}}{d^{4} x^{n} + c d^{3}}\,{d x} + \frac{{\left (m^{2} + m{\left (n + 2\right )} + n + 1\right )} B b^{2} d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} -{\left ({\left ({\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c d e^{m} - 2 \,{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b d^{2} e^{m}\right )} A -{\left ({\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c^{2} e^{m} - 2 \,{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b c d e^{m} +{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{2} d^{2} e^{m}\right )} B\right )} x x^{m} +{\left ({\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} A b^{2} d^{2} e^{m} -{\left ({\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} b^{2} c d e^{m} - 2 \,{\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} a b d^{2} e^{m}\right )} B\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{{\left (m^{3} + 3 \, m^{2}{\left (n + 1\right )} +{\left (2 \, n^{2} + 6 \, n + 3\right )} m + 2 \, n^{2} + 3 \, n + 1\right )} d^{3}} \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^{2} x^{3 \, n} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} x^{2 \, n} +{\left (B a^{2} + 2 \, A a b\right )} x^{n}\right )} \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 [C] time = 26.4276, size = 1085, normalized size = 5.8 \begin{align*} \text{result too large to display} \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 )}^{2} \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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