Optimal. Leaf size=130 \[ -\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.117625, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4647, 4641, 4635, 4406, 12, 3305, 3351} \[ -\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4647
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{2 \sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{4 \sqrt{1-a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a \sqrt{1-a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a \sqrt{1-a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt{1-a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a \sqrt{1-a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0671401, size = 138, normalized size = 1.06 \[ \frac{\sqrt{c-a^2 c x^2} \left (3 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+3 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+16 \sin ^{-1}(a x) \left (3 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right )\right )}{96 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{\arcsin \left ( ax \right ) }\, dx \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 \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{\operatorname{asin}{\left (a x \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 \sqrt{-a^{2} c x^{2} + c} \sqrt{\arcsin \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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