Optimal. Leaf size=78 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
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Rubi [A] time = 0.0395393, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {459, 364} \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 459
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx &=\frac{B (e x)^{1+m}}{d e (1+m)}-\frac{(B c (1+m)-A d (1+m)) \int \frac{(e x)^m}{c+d x^n} \, dx}{d (1+m)}\\ &=\frac{B (e x)^{1+m}}{d e (1+m)}-\frac{(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 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0645983, size = 57, normalized size = 0.73 \[ \frac{x (e x)^m \left ((A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+B c\right )}{c d (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.36, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \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*} \frac{B e^{m} x x^{m}}{d{\left (m + 1\right )}} -{\left (B c e^{m} - A d e^{m}\right )} \int \frac{x^{m}}{d^{2} x^{n} + c d}\,{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 (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 = 3.17009, size = 284, normalized size = 3.64 \begin{align*} \frac{A e^{m} m x x^{m} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{m}{n} + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + \frac{1}{n}\right )}{c n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} + \frac{A e^{m} x x^{m} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{m}{n} + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + \frac{1}{n}\right )}{c n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} + \frac{B e^{m} m x x^{m} x^{n} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{m}{n} + 1 + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )}{c n^{2} \Gamma \left (\frac{m}{n} + 2 + \frac{1}{n}\right )} + \frac{B e^{m} x x^{m} x^{n} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{m}{n} + 1 + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )}{c n \Gamma \left (\frac{m}{n} + 2 + \frac{1}{n}\right )} + \frac{B e^{m} x x^{m} x^{n} \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{m}{n} + 1 + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )}{c n^{2} \Gamma \left (\frac{m}{n} + 2 + \frac{1}{n}\right )} \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}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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