Optimal. Leaf size=84 \[ \frac{1}{3} x^3 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2}-p,\frac{5}{2},a^2 x^2\right )+\frac{\left (1-a^2 x^2\right )^{p+\frac{1}{2}}}{a^3 (2 p+1)}-\frac{\left (1-a^2 x^2\right )^{p+\frac{3}{2}}}{a^3 (2 p+3)} \]
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Rubi [A] time = 0.120778, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6149, 764, 364, 266, 43} \[ \frac{1}{3} x^3 \, _2F_1\left (\frac{3}{2},\frac{1}{2}-p;\frac{5}{2};a^2 x^2\right )+\frac{\left (1-a^2 x^2\right )^{p+\frac{1}{2}}}{a^3 (2 p+1)}-\frac{\left (1-a^2 x^2\right )^{p+\frac{3}{2}}}{a^3 (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 6149
Rule 764
Rule 364
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx &=\int x^2 (1-a x) \left (1-a^2 x^2\right )^{-\frac{1}{2}+p} \, dx\\ &=-\left (a \int x^3 \left (1-a^2 x^2\right )^{-\frac{1}{2}+p} \, dx\right )+\int x^2 \left (1-a^2 x^2\right )^{-\frac{1}{2}+p} \, dx\\ &=\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{2},\frac{1}{2}-p;\frac{5}{2};a^2 x^2\right )-\frac{1}{2} a \operatorname{Subst}\left (\int x \left (1-a^2 x\right )^{-\frac{1}{2}+p} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{2},\frac{1}{2}-p;\frac{5}{2};a^2 x^2\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \left (\frac{\left (1-a^2 x\right )^{-\frac{1}{2}+p}}{a^2}-\frac{\left (1-a^2 x\right )^{\frac{1}{2}+p}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (1-a^2 x^2\right )^{\frac{1}{2}+p}}{a^3 (1+2 p)}-\frac{\left (1-a^2 x^2\right )^{\frac{3}{2}+p}}{a^3 (3+2 p)}+\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{2},\frac{1}{2}-p;\frac{5}{2};a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0555171, size = 75, normalized size = 0.89 \[ \frac{1}{3} x^3 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2}-p,\frac{5}{2},a^2 x^2\right )+\frac{\left (a^2 (2 p+1) x^2+2\right ) \left (1-a^2 x^2\right )^{p+\frac{1}{2}}}{a^3 \left (4 p^2+8 p+3\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.424, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{p}}{ax+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^{2} x^{2} + 1\right )}^{p + \frac{1}{2}} x^{2}}{a x + 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}{\left (-a^{2} x^{2} + 1\right )}^{p} x^{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^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{a x + 1}\, 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{\sqrt{-a^{2} x^{2} + 1}{\left (-a^{2} x^{2} + 1\right )}^{p} x^{2}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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