Optimal. Leaf size=41 \[ \frac{4 (2-a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{3 a^3 c \sqrt [8]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.119413, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {6146} \[ \frac{4 (2-a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{3 a^3 c \sqrt [8]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6146
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{9/8}} \, dx &=\frac{4 e^{\frac{1}{2} \tanh ^{-1}(a x)} (2-a x)}{3 a^3 c \sqrt [8]{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0329122, size = 63, normalized size = 1.54 \[ -\frac{4 (a x-2) \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{3 a^3 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.028, size = 54, normalized size = 1.3 \begin{align*}{\frac{ \left ( 4\,ax-4 \right ) \left ( ax+1 \right ) \left ( ax-2 \right ) }{3\,{a}^{3}}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{8}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.53288, size = 38, normalized size = 0.93 \begin{align*} -\frac{4 \,{\left (a x + 1\right )}^{\frac{1}{8}}{\left (a x - 2\right )}}{3 \,{\left (-a x + 1\right )}^{\frac{3}{8}} a^{3} c^{\frac{9}{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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