Optimal. Leaf size=76 \[ \frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.224563, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6192, 6193, 43} \[ \frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^3 \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x^2 (-1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-x^2+a x^3\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0269459, size = 45, normalized size = 0.59 \[ \frac{x^2 (3 a x-4) \sqrt{c-a^2 c x^2}}{12 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 47, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 3\,ax-4 \right ) }{12\,ax-12}\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{{\frac{ax-1}{ax+1}}}} \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 \sqrt{-a^{2} c x^{2} + c} x^{2} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58257, size = 53, normalized size = 0.7 \begin{align*} \frac{{\left (3 \, a x^{4} - 4 \, x^{3}\right )} \sqrt{-a^{2} c}}{12 \, a} \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 \sqrt{-a^{2} c x^{2} + c} x^{2} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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