Optimal. Leaf size=80 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{c (m+1)}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{c (m+2)} \]
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Rubi [A] time = 0.10641, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6148, 808, 364} \[ \frac{x^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{c (m+1)}+\frac{a x^{m+2} \, _2F_1\left (\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{c (m+2)} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 808
Rule 364
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^m}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^m (1+a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{\int \frac{x^m}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}+\frac{a \int \frac{x^{1+m}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{x^{1+m} \, _2F_1\left (\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{c (1+m)}+\frac{a x^{2+m} \, _2F_1\left (\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{c (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0296975, size = 82, normalized size = 1.02 \[ \frac{\frac{x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+1}{2},\frac{m+1}{2}+1,a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+2}{2}+1,a^2 x^2\right )}{m+2}}{c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ){x}^{m}}{-{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, 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 )} x^{m}}{{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} x^{2} + 1}}\,{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{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{3} c x^{3} - a^{2} c x^{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} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x x^{m}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 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 )} x^{m}}{{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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