Optimal. Leaf size=25 \[ \frac{x^{m+1} \text{Hypergeometric2F1}(2,m+1,m+2,a x)}{c (m+1)} \]
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Rubi [A] time = 0.0812267, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 64} \[ \frac{x^{m+1} \, _2F_1(2,m+1;m+2;a x)}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 64
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^m}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^m}{(1-a x)^2} \, dx}{c}\\ &=\frac{x^{1+m} \, _2F_1(2,1+m;2+m;a x)}{c (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0128715, size = 25, normalized size = 1. \[ \frac{x^{m+1} \text{Hypergeometric2F1}(2,m+1,m+2,a x)}{c (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.372, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2}{x}^{m}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) \left ( -{a}^{2}c{x}^{2}+c \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 \frac{{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}{\left (a^{2} x^{2} - 1\right )}}\,{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{x^{m}}{a^{2} c x^{2} - 2 \, a c x + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{m}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c} \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 (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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