Optimal. Leaf size=162 \[ \frac {\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac {6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac {30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac {120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac {72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac {144}{629} x \sqrt {1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac {144 e^{\sin ^{-1}(a x)}}{629 a} \]
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Rubi [A] time = 0.43, 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 = {4836, 6688, 6720, 4435, 2194} \[ \frac {\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac {6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac {30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac {120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac {72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac {144}{629} x \sqrt {1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac {144 e^{\sin ^{-1}(a x)}}{629 a} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4435
Rule 4836
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \cos ^2(x)^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 = 0.40, size = 69, normalized size = 0.43 \[ \frac {e^{\sin ^{-1}(a x)} \left (3774 \sin \left (2 \sin ^{-1}(a x)\right )+888 \sin \left (4 \sin ^{-1}(a x)\right )+102 \sin \left (6 \sin ^{-1}(a x)\right )+1887 \cos \left (2 \sin ^{-1}(a x)\right )+222 \cos \left (4 \sin ^{-1}(a x)\right )+17 \cos \left (6 \sin ^{-1}(a x)\right )+6290\right )}{20128 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 71, normalized size = 0.44 \[ -\frac {{\left (17 \, a^{6} x^{6} - 81 \, a^{4} x^{4} + 183 \, a^{2} x^{2} - 6 \, {\left (17 \, a^{5} x^{5} - 54 \, a^{3} x^{3} + 61 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} - 263\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, 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 \[ \int {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,{\left (1-a^2\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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