Optimal. Leaf size=371 \[ \frac{4 b^2 c^2 d x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{(m+3)^2 \left (m^2+3 m+2\right )}+\frac{2 b^2 c^2 d x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{(m+2) (m+3)^3}-\frac{4 b c d x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{m^3+6 m^2+11 m+6}-\frac{2 b c d x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+2) (m+3)^2}+\frac{d \left (1-c^2 x^2\right ) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m+3}-\frac{2 b c d \sqrt{1-c^2 x^2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+3)^2}+\frac{2 d x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m^2+4 m+3}+\frac{2 b^2 c^2 d x^{m+3}}{(m+3)^3} \]
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Rubi [F] time = 0.0489392, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end{align*}
Mathematica [F] time = 0.0924532, size = 0, normalized size = 0. \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 2.915, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} x^{m}, 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*} - d \left (\int - a^{2} x^{m}\, dx + \int - b^{2} x^{m} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int - 2 a b x^{m} \operatorname{asin}{\left (c x \right )}\, dx + \int a^{2} c^{2} x^{2} x^{m}\, dx + \int b^{2} c^{2} x^{2} x^{m} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{2} x^{m} \operatorname{asin}{\left (c x \right )}\, dx\right ) \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 )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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