Optimal. Leaf size=44 \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0752011, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0465079, size = 44, normalized size = 1. \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 38, normalized size = 0.9 \begin{align*}{\frac{2}{3\,a} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt{\operatorname{asin}{\left (a x \right )}}}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{\arcsin \left (a x\right )}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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