Optimal. Leaf size=391 \[ -\frac{e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.440236, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4725
Rule 4723
Rule 4406
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac{\sqrt{d-c^2 d x^2} \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \cos ^2(x) \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int \left (\frac{1}{4} (a+b x)^n \sin (x)+\frac{1}{4} (a+b x)^n \sin (3 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 \sqrt{1-c^2 x^2}}+\frac{\sqrt{d-c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \sin (3 x) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (i \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 )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (i \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 )}{8 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (i \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 )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (i \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 )}{8 c^2 \sqrt{1-c^2 x^2}}\\ &=-\frac{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 )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{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 )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-1-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 (1+n,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-1-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 (1+n,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.840741, size = 272, normalized size = 0.7 \[ \frac{d e^{-\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (3 e^{\frac{2 i a}{b}} \left (\left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \left (-\text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )-e^{\frac{2 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )-3^{-n} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (e^{\frac{6 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{24 c^2 \sqrt{d \left (1-c^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.225, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{-{c}^{2}d{x}^{2}+d} \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 \sqrt{-c^{2} d x^{2} + d}{\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 (\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}^{n} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{n}\, dx \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 \sqrt{-c^{2} d x^{2} + d}{\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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