Optimal. Leaf size=162 \[ \frac {144 e^{\text {ArcSin}(a x)}}{629 a}+\frac {144}{629} e^{\text {ArcSin}(a x)} x \sqrt {1-a^2 x^2}+\frac {72 e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac {120}{629} e^{\text {ArcSin}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {30 e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac {6}{37} e^{\text {ArcSin}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )^3}{37 a} \]
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Rubi [A]
time = 0.30, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 )^3 e^{\text {ArcSin}(a x)}}{37 a}+\frac {6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\text {ArcSin}(a x)}+\frac {30 \left (1-a^2 x^2\right )^2 e^{\text {ArcSin}(a x)}}{629 a}+\frac {120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\text {ArcSin}(a x)}+\frac {72 \left (1-a^2 x^2\right ) e^{\text {ArcSin}(a x)}}{629 a}+\frac {144}{629} x \sqrt {1-a^2 x^2} e^{\text {ArcSin}(a x)}+\frac {144 e^{\text {ArcSin}(a x)}}{629 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 )^{5/2} \, dx &=\frac {\text {Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \cos (x) \cos ^2(x)^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \cos ^6(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac {30 \text {Subst}\left (\int e^x \cos ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{37 a}\\ &=\frac {120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac {6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac {360 \text {Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac {144}{629} e^{\sin ^{-1}(a x)} x \sqrt {1-a^2 x^2}+\frac {72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac {120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac {6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac {144 \text {Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac {144 e^{\sin ^{-1}(a x)}}{629 a}+\frac {144}{629} e^{\sin ^{-1}(a x)} x \sqrt {1-a^2 x^2}+\frac {72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac {120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac {6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 69, normalized size = 0.43 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} (6290+1887 \cos (2 \text {ArcSin}(a x))+222 \cos (4 \text {ArcSin}(a x))+17 \cos (6 \text {ArcSin}(a x))+3774 \sin (2 \text {ArcSin}(a x))+888 \sin (4 \text {ArcSin}(a x))+102 \sin (6 \text {ArcSin}(a x)))}{20128 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}\, dx\]
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 = 2.09, size = 71, normalized size = 0.44 \begin {gather*} -\frac {{\left (17 \, a^{6} x^{6} - 81 \, a^{4} x^{4} + 183 \, a^{2} x^{2} - 6 \, {\left (17 \, a^{5} x^{5} - 54 \, a^{3} x^{3} + 61 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} - 263\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 17.88, size = 141, normalized size = 0.87 \begin {gather*} \begin {cases} - \frac {a^{5} x^{6} e^{\operatorname {asin}{\left (a x \right )}}}{37} + \frac {6 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{37} + \frac {81 a^{3} x^{4} e^{\operatorname {asin}{\left (a x \right )}}}{629} - \frac {324 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{629} - \frac {183 a x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{629} + \frac {366 x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{629} + \frac {263 e^{\operatorname {asin}{\left (a x \right )}}}{629 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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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