Optimal. Leaf size=112 \[ \frac{\left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{17 a}+\frac{4}{17} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{12 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{24 e^{\sin ^{-1}(a x)}}{85 a} \]
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Rubi [A] time = 0.303462, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4435, 2194} \[ \frac{\left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{17 a}+\frac{4}{17} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{12 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{24 e^{\sin ^{-1}(a x)}}{85 a} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 6688
Rule 6720
Rule 4435
Rule 2194
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \cos ^2(x)^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}+\frac{12 \operatorname{Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{17 a}\\ &=\frac{24}{85} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{12 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{85 a}+\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}+\frac{24 \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{85 a}\\ &=\frac{24 e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{12 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{85 a}+\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}\\ \end{align*}
Mathematica [A] time = 0.150005, size = 51, normalized size = 0.46 \[ \frac{e^{\sin ^{-1}(a x)} \left (136 \sin \left (2 \sin ^{-1}(a x)\right )+20 \sin \left (4 \sin ^{-1}(a x)\right )+68 \cos \left (2 \sin ^{-1}(a x)\right )+5 \cos \left (4 \sin ^{-1}(a x)\right )+255\right )}{680 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}\, dx \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{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01153, size = 132, normalized size = 1.18 \begin{align*} \frac{{\left (5 \, a^{4} x^{4} - 22 \, a^{2} x^{2} - 4 \,{\left (5 \, a^{3} x^{3} - 11 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} + 41\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.3924, size = 95, normalized size = 0.85 \begin{align*} \begin{cases} \frac{a^{3} x^{4} e^{\operatorname{asin}{\left (a x \right )}}}{17} - \frac{4 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{17} - \frac{22 a x^{2} e^{\operatorname{asin}{\left (a x \right )}}}{85} + \frac{44 x \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{85} + \frac{41 e^{\operatorname{asin}{\left (a x \right )}}}{85 a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26209, size = 123, normalized size = 1.1 \begin{align*} \frac{4}{17} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{24}{85} \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{{\left (a^{2} x^{2} - 1\right )}^{2} e^{\left (\arcsin \left (a x\right )\right )}}{17 \, a} - \frac{12 \,{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} + \frac{24 \, e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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