Optimal. Leaf size=44 \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.0690032, 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} \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0534622, size = 44, normalized size = 1. \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 38, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,a}\sqrt{-{a}^{2}{x}^{2}+1} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}{\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 [A] time = 1.95064, size = 112, normalized size = 2.55 \begin{align*} \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{3 \,{\left (a^{3} c x^{2} - a c\right )} \arcsin \left (a x\right )^{\frac{3}{2}}} \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{1}{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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