Optimal. Leaf size=427 \[ d^2 \text {Int}\left (\frac {\left (a+b \sin ^{-1}(c x)\right )^n}{x \sqrt {d-c^2 d x^2}},x\right )+\frac {5 d^2 e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {d^2 3^{-n-1} e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {5 d^2 e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {d^2 3^{-n-1} e^{\frac {3 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx &=\int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x}, x\right ) \]
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 [A] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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