Optimal. Leaf size=45 \[ -4 x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)+\frac{4 x^{m+1}}{1-a x}+\frac{x^{m+1}}{m+1} \]
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Rubi [A] time = 0.0570997, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6167, 6126, 89, 80, 64} \[ -4 x^{m+1} \, _2F_1(1,m+1;m+2;a x)+\frac{4 x^{m+1}}{1-a x}+\frac{x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 89
Rule 80
Rule 64
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} x^m \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^m \, dx\\ &=\int \frac{x^m (1+a x)^2}{(1-a x)^2} \, dx\\ &=\frac{4 x^{1+m}}{1-a x}-\frac{\int \frac{x^m \left (a^2 (3+4 m)+a^3 x\right )}{1-a x} \, dx}{a^2}\\ &=\frac{x^{1+m}}{1+m}+\frac{4 x^{1+m}}{1-a x}-(4 (1+m)) \int \frac{x^m}{1-a x} \, dx\\ &=\frac{x^{1+m}}{1+m}+\frac{4 x^{1+m}}{1-a x}-4 x^{1+m} \, _2F_1(1,1+m;2+m;a x)\\ \end{align*}
Mathematica [A] time = 0.0224629, size = 47, normalized size = 1.04 \[ \frac{x^{m+1} (-4 (m+1) (a x-1) \text{Hypergeometric2F1}(1,m+1,m+2,a x)+a x-4 m-5)}{(m+1) (a x-1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.606, size = 201, normalized size = 4.5 \begin{align*} -{\frac{ \left ( -a \right ) ^{-m}}{a} \left ({\frac{{x}^{m} \left ( -a \right ) ^{m} \left ({a}^{2}m{x}^{2}+amx+2\,ax-{m}^{2}-3\,m-2 \right ) }{m \left ( 1+m \right ) \left ( -ax+1 \right ) }}+{x}^{m} \left ( -a \right ) ^{m} \left ( 2+m \right ){\it LerchPhi} \left ( ax,1,m \right ) \right ) }+2\,{\frac{ \left ( -a \right ) ^{-m}}{a} \left ( -{\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( ax-m-1 \right ) }{m \left ( -ax+1 \right ) }}-{x}^{m} \left ( -a \right ) ^{m} \left ( 1+m \right ){\it LerchPhi} \left ( ax,1,m \right ) \right ) }-{\frac{ \left ( -a \right ) ^{-m}}{a} \left ({\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( -1-m \right ) }{ \left ( 1+m \right ) \left ( -ax+1 \right ) }}+{x}^{m} \left ( -a \right ) ^{m}m{\it LerchPhi} \left ( ax,1,m \right ) \right ) } \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 x - 1\right )}^{2}}\,{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 (a^{2} x^{2} + 2 \, a x + 1\right )} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, 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*} \int \frac{x^{m} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \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 x - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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