Optimal. Leaf size=309 \[ -\frac {a^3 (a+b x) e^{\sin ^{-1}(a+b x)}}{2 b^4}-\frac {a^3 \sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b^4}+\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}-\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{5 b^4}-\frac {3 a (a+b x) e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac {9 a e^{\sin ^{-1}(a+b x)} \sin \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}-\frac {3 a \sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{20 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \sin \left (4 \sin ^{-1}(a+b x)\right )}{136 b^4}+\frac {3 a e^{\sin ^{-1}(a+b x)} \cos \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \cos \left (4 \sin ^{-1}(a+b x)\right )}{34 b^4} \]
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Rubi [A] time = 0.53, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4836, 6741, 12, 6742, 4433, 4469, 4432} \[ -\frac {a^3 (a+b x) e^{\sin ^{-1}(a+b x)}}{2 b^4}-\frac {a^3 \sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b^4}+\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}-\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{5 b^4}-\frac {3 a (a+b x) e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac {9 a e^{\sin ^{-1}(a+b x)} \sin \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}-\frac {3 a \sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{20 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \sin \left (4 \sin ^{-1}(a+b x)\right )}{136 b^4}+\frac {3 a e^{\sin ^{-1}(a+b x)} \cos \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \cos \left (4 \sin ^{-1}(a+b x)\right )}{34 b^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 4432
Rule 4433
Rule 4469
Rule 4836
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a+b x)} x^3 \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^3 \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {e^x \cos (x) (-a+\sin (x))^3}{b^3} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) (-a+\sin (x))^3 \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^3 e^x \cos (x)+3 a^2 e^x \cos (x) \sin (x)-3 a e^x \cos (x) \sin ^2(x)+e^x \cos (x) \sin ^3(x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^x \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac {a^3 \operatorname {Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac {a^3 e^{\sin ^{-1}(a+b x)} (a+b x)}{2 b^4}-\frac {a^3 e^{\sin ^{-1}(a+b x)} \sqrt {1-(a+b x)^2}}{2 b^4}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{4} e^x \sin (2 x)-\frac {1}{8} e^x \sin (4 x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac {(3 a) \operatorname {Subst}\left (\int \left (\frac {1}{4} e^x \cos (x)-\frac {1}{4} e^x \cos (3 x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2} e^x \sin (2 x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac {a^3 e^{\sin ^{-1}(a+b x)} (a+b x)}{2 b^4}-\frac {a^3 e^{\sin ^{-1}(a+b x)} \sqrt {1-(a+b x)^2}}{2 b^4}-\frac {\operatorname {Subst}\left (\int e^x \sin (4 x) \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac {\operatorname {Subst}\left (\int e^x \sin (2 x) \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}+\frac {(3 a) \operatorname {Subst}\left (\int e^x \cos (3 x) \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^x \sin (2 x) \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^4}\\ &=-\frac {3 a e^{\sin ^{-1}(a+b x)} (a+b x)}{8 b^4}-\frac {a^3 e^{\sin ^{-1}(a+b x)} (a+b x)}{2 b^4}-\frac {3 a e^{\sin ^{-1}(a+b x)} \sqrt {1-(a+b x)^2}}{8 b^4}-\frac {a^3 e^{\sin ^{-1}(a+b x)} \sqrt {1-(a+b x)^2}}{2 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}-\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \cos \left (2 \sin ^{-1}(a+b x)\right )}{5 b^4}+\frac {3 a e^{\sin ^{-1}(a+b x)} \cos \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \cos \left (4 \sin ^{-1}(a+b x)\right )}{34 b^4}+\frac {e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{20 b^4}+\frac {3 a^2 e^{\sin ^{-1}(a+b x)} \sin \left (2 \sin ^{-1}(a+b x)\right )}{10 b^4}+\frac {9 a e^{\sin ^{-1}(a+b x)} \sin \left (3 \sin ^{-1}(a+b x)\right )}{40 b^4}-\frac {e^{\sin ^{-1}(a+b x)} \sin \left (4 \sin ^{-1}(a+b x)\right )}{136 b^4}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 148, normalized size = 0.48 \[ \frac {e^{\sin ^{-1}(a+b x)} \left (-340 a^3 (a+b x)-85 \left (4 a^2+3\right ) a \sqrt {1-(a+b x)^2}+204 a^2 \sin \left (2 \sin ^{-1}(a+b x)\right )-68 \left (6 a^2+1\right ) \cos \left (2 \sin ^{-1}(a+b x)\right )-255 a (a+b x)+153 a \sin \left (3 \sin ^{-1}(a+b x)\right )+34 \sin \left (2 \sin ^{-1}(a+b x)\right )-5 \sin \left (4 \sin ^{-1}(a+b x)\right )+51 a \cos \left (3 \sin ^{-1}(a+b x)\right )+20 \cos \left (4 \sin ^{-1}(a+b x)\right )\right )}{680 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 129, normalized size = 0.42 \[ \frac {{\left (40 \, b^{4} x^{4} + 7 \, a b^{3} x^{3} - 3 \, {\left (5 \, a^{2} + 2\right )} b^{2} x^{2} + 6 \, a^{4} + 3 \, {\left (8 \, a^{3} + 13 \, a\right )} b x - 57 \, a^{2} + {\left (10 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} - 24 \, a^{3} + 6 \, {\left (5 \, a^{2} + 2\right )} b x - 39 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} - 12\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{170 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.85, size = 334, normalized size = 1.08 \[ -\frac {{\left (b x + a\right )} a^{3} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b^{4}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{3} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b^{4}} - \frac {9 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} a e^{\left (\arcsin \left (b x + a\right )\right )}}{10 \, b^{4}} + \frac {6 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{17 \, b^{4}} + \frac {3 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} a e^{\left (\arcsin \left (b x + a\right )\right )}}{10 \, b^{4}} + \frac {4 \, {\left ({\left (b x + a\right )}^{2} - 1\right )}^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{17 \, b^{4}} - \frac {3 \, {\left (b x + a\right )} a e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {3 \, a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {11 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {37 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} + \frac {11 \, e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (b x +a \right )} x^{3}\, dx \]
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 \[ \int x^{3} e^{\left (\arcsin \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {e}}^{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.81, size = 416, normalized size = 1.35 \[ \begin {cases} \frac {3 a^{4} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} + \frac {12 a^{3} x e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{3}} - \frac {12 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} - \frac {3 a^{2} x^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{34 b^{2}} + \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{17 b^{3}} - \frac {57 a^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{4}} + \frac {7 a x^{3} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b} - \frac {21 a x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{2}} + \frac {39 a x e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{3}} - \frac {39 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{4}} + \frac {4 x^{4} e^{\operatorname {asin}{\left (a + b x \right )}}}{17} + \frac {x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{17 b} - \frac {3 x^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{2}} + \frac {6 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{3}} - \frac {6 e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} e^{\operatorname {asin}{\relax (a )}}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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