Optimal. Leaf size=112 \[ \frac {24 e^{\text {ArcSin}(a x)}}{85 a}+\frac {24}{85} e^{\text {ArcSin}(a x)} x \sqrt {1-a^2 x^2}+\frac {12 e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )}{85 a}+\frac {4}{17} e^{\text {ArcSin}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )^2}{17 a} \]
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Rubi [A]
time = 0.21, antiderivative size = 112, normalized size of antiderivative = 1.00, 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 = {4920, 6820,
6852, 4520, 2225} \begin {gather*} \frac {\left (1-a^2 x^2\right )^2 e^{\text {ArcSin}(a x)}}{17 a}+\frac {4}{17} x \left (1-a^2 x^2\right )^{3/2} e^{\text {ArcSin}(a x)}+\frac {12 \left (1-a^2 x^2\right ) e^{\text {ArcSin}(a x)}}{85 a}+\frac {24}{85} x \sqrt {1-a^2 x^2} e^{\text {ArcSin}(a x)}+\frac {24 e^{\text {ArcSin}(a x)}}{85 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4520
Rule 4920
Rule 6820
Rule 6852
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \cos (x) \cos ^2(x)^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {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 \text {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 \text {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*}
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Mathematica [A]
time = 0.12, size = 51, normalized size = 0.46 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} (255+68 \cos (2 \text {ArcSin}(a x))+5 \cos (4 \text {ArcSin}(a x))+136 \sin (2 \text {ArcSin}(a x))+20 \sin (4 \text {ArcSin}(a x)))}{680 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 55, normalized size = 0.49 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.74, size = 95, normalized size = 0.85 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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