Optimal. Leaf size=595 \[ -\frac{d e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt{1-c^2 x^2}}-\frac{d 3^{-n} e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d 5^{-n-1} e^{-\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt{1-c^2 x^2}}-\frac{d 3^{-n} e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d 5^{-n-1} e^{\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.586051, antiderivative size = 595, normalized size of antiderivative = 1., 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 4725
Rule 4723
Rule 4406
Rule 3308
Rule 2181
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*}
Mathematica [A] time = 2.09228, size = 464, normalized size = 0.78 \[ -\frac{d^2 15^{-n-1} e^{-\frac{5 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-3 n} \left (\left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (2\ 15^{n+1} e^{\frac{6 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3 \left (5^{n+1} e^{\frac{2 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{2 n} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5^{n+1} e^{\frac{8 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3^n \left (e^{\frac{10 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{3 n} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )\right )+2\ 15^{n+1} e^{\frac{4 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{32 c^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int x \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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