Optimal. Leaf size=41 \[ \frac{x^{m+1} F_1\left (-m-1;\frac{3}{4},-\frac{3}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{m+1} \]
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Rubi [A] time = 0.0381142, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6173, 133} \[ \frac{x^{m+1} F_1\left (-m-1;\frac{3}{4},-\frac{3}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6173
Rule 133
Rubi steps
\begin{align*} \int e^{\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m} \left (1+\frac{x}{a}\right )^{3/4}}{\left (1-\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{x^{1+m} F_1\left (-1-m;\frac{3}{4},-\frac{3}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.328393, size = 0, normalized size = 0. \[ \int e^{\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{4}}}}\, 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{x^{m}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}\,{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 x + 1\right )} x^{m} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x - 1}, 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{x^{m}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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