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 = 0.12, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 1.32 \[ \frac {4 \left (a x -1\right ) \left (a x +1\right ) \left (a x -2\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{3 a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 28, normalized size = 0.68 \[ -\frac {4 \, {\left (a x + 1\right )}^{\frac {1}{8}} {\left (a x - 2\right )}}{3 \, {\left (-a x + 1\right )}^{\frac {3}{8}} a^{3} c^{\frac {9}{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 52, normalized size = 1.27 \[ \frac {\left (\frac {8}{3\,a^3\,c}-\frac {4\,x}{3\,a^2\,c}\right )\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{{\left (c-a^2\,c\,x^2\right )}^{1/8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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