Optimal. Leaf size=595 \[ -\frac {d e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.59, antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4725, 4723, 4406, 3308, 2181} \[ -\frac {d e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 4406
Rule 4723
Rule 4725
Rubi steps
\begin {align*} \int x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos ^4(x) \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{8} (a+b x)^n \sin (x)+\frac {3}{16} (a+b x)^n \sin (3 x)+\frac {1}{16} (a+b x)^n \sin (5 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sin (5 x) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sin (3 x) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 i d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {d e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 2.41, size = 464, normalized size = 0.78 \[ -\frac {d^2 15^{-n-1} e^{-\frac {5 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-3 n} \left (\left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (2\ 15^{n+1} e^{\frac {6 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3 \left (5^{n+1} e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{2 n} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5^{n+1} e^{\frac {8 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3^n \left (e^{\frac {10 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{3 n} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )\right )+2\ 15^{n+1} e^{\frac {4 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{32 c^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (c^{2} d x^{3} - d x\right )} \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \]
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 = 0.19, size = 0, normalized size = 0.00 \[ \int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx \]
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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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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