Optimal. Leaf size=408 \[ \frac{b c (2-m) m \sqrt{1-c^2 x^2} x^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )}{3 d^2 \left (m^2+3 m+2\right ) \sqrt{d-c^2 d x^2}}-\frac{(2-m) m \sqrt{1-c^2 x^2} x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 (m+1) \sqrt{d-c^2 d x^2}}-\frac{b c (2-m) \sqrt{1-c^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (2,\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}}+\frac{(2-m) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.454834, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {4705, 4713, 4711, 364} \[ \frac{b c (2-m) m \sqrt{1-c^2 x^2} x^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right )}{3 d^2 \left (m^2+3 m+2\right ) \sqrt{d-c^2 d x^2}}-\frac{(2-m) m \sqrt{1-c^2 x^2} x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 (m+1) \sqrt{d-c^2 d x^2}}+\frac{(2-m) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{b c (2-m) \sqrt{1-c^2 x^2} x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} x^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4705
Rule 4713
Rule 4711
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{(2-m) \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{x^{1+m}}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c x^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (2,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt{d-c^2 d x^2}}-\frac{((2-m) m) \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx}{3 d^2}-\frac{\left (b c (2-m) \sqrt{1-c^2 x^2}\right ) \int \frac{x^{1+m}}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c (2-m) x^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt{d-c^2 d x^2}}-\frac{b c x^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (2,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt{d-c^2 d x^2}}-\frac{\left ((2-m) m \sqrt{1-c^2 x^2}\right ) \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{(2-m) m x^{1+m} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};c^2 x^2\right )}{3 d^2 (1+m) \sqrt{d-c^2 d x^2}}-\frac{b c (2-m) x^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt{d-c^2 d x^2}}-\frac{b c x^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (2,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt{d-c^2 d x^2}}+\frac{b c (2-m) m x^{2+m} \sqrt{1-c^2 x^2} \, _3F_2\left (1,1+\frac{m}{2},1+\frac{m}{2};\frac{3}{2}+\frac{m}{2},2+\frac{m}{2};c^2 x^2\right )}{3 d^2 \left (2+3 m+m^2\right ) \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.380634, size = 279, normalized size = 0.68 \[ \frac{x^{m+1} \left ((2-m) \left (d-c^2 d x^2\right ) \left (-m \sqrt{1-c^2 x^2} \left ((m+2) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-b c x \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )\right )-b c (m+1) x \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (1,\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )+(m+1) (m+2) \left (a+b \sin ^{-1}(c x)\right )\right )-b c d (m+1) x \left (1-c^2 x^2\right )^{3/2} \text{Hypergeometric2F1}\left (2,\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )+d (m+1) (m+2) \left (a+b \sin ^{-1}(c x)\right )\right )}{3 d^2 (m+1) (m+2) \left (d-c^2 d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.602, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b\arcsin \left ( cx \right ) \right ) \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, 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 \frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{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 (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, 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 \frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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