Optimal. Leaf size=137 \[ \frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 f \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} (f x)^{7/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )}{35 f^2 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.216347, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {4713, 4711} \[ \frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 f \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} (f x)^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right )}{35 f^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4713
Rule 4711
Rubi steps
\begin{align*} \int \frac{(f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{2 (f x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right )}{5 f \sqrt{d-c^2 d x^2}}-\frac{4 b c (f x)^{7/2} \sqrt{1-c^2 x^2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right )}{35 f^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0419652, size = 97, normalized size = 0.71 \[ -\frac{2 x \sqrt{1-c^2 x^2} (f x)^{3/2} \left (2 b c x \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )-7 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )\right )}{35 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.529, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( fx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}\, 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 (f x\right )^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{-c^{2} d x^{2} + d}}\,{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 f x \arcsin \left (c x\right ) + a f x\right )} \sqrt{f x}}{c^{2} d x^{2} - d}, 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 (f x\right )^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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