Optimal. Leaf size=265 \[ \frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^2 (c x+1) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A] time = 0.286408, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4673, 653, 191, 4761, 627, 44, 207, 260} \[ \frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^2 (c x+1) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 653
Rule 191
Rule 4761
Rule 627
Rule 44
Rule 207
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+c d x} (f-c f x)^{5/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(d+c d x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac{2 d^2 (1+c x)}{3 c \left (1-c^2 x^2\right )^2}+\frac{d^2 x}{3 \left (1-c^2 x^2\right )}\right ) \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1+c x}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{\left (b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1}{(1-c x)^2 (1+c x)} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac{1}{2 (-1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{\left (b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^2 (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.462334, size = 130, normalized size = 0.49 \[ \frac{\sqrt{c d x+d} \sqrt{f-c f x} \left (-(c x-2) \left (a \sqrt{1-c^2 x^2}+b c x-b\right )-b (c x-2) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+b (c x-1)^2 \log (f-c f x)\right )}{3 c d f^3 (c x-1)^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.229, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ){\frac{1}{\sqrt{cdx+d}}} \left ( -cfx+f \right ) ^{-{\frac{5}{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 [A] time = 3.43468, size = 1157, normalized size = 4.37 \begin{align*} \left [\frac{{\left (b c^{3} x^{3} - b c^{2} x^{2} - b c x + b\right )} \sqrt{d f} \log \left (\frac{c^{6} d f x^{6} - 4 \, c^{5} d f x^{5} + 5 \, c^{4} d f x^{4} - 4 \, c^{2} d f x^{2} + 4 \, c d f x -{\left (c^{4} x^{4} - 4 \, c^{3} x^{3} + 6 \, c^{2} x^{2} - 4 \, c x\right )} \sqrt{-c^{2} x^{2} + 1} \sqrt{c d x + d} \sqrt{-c f x + f} \sqrt{d f} - 2 \, d f}{c^{4} x^{4} - 2 \, c^{3} x^{3} + 2 \, c x - 1}\right ) - 2 \,{\left (a c^{2} x^{2} + \sqrt{-c^{2} x^{2} + 1} b c x - a c x +{\left (b c^{2} x^{2} - b c x - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{6 \,{\left (c^{4} d f^{3} x^{3} - c^{3} d f^{3} x^{2} - c^{2} d f^{3} x + c d f^{3}\right )}}, \frac{{\left (b c^{3} x^{3} - b c^{2} x^{2} - b c x + b\right )} \sqrt{-d f} \arctan \left (\frac{{\left (c^{2} x^{2} - 2 \, c x + 2\right )} \sqrt{-c^{2} x^{2} + 1} \sqrt{c d x + d} \sqrt{-c f x + f} \sqrt{-d f}}{c^{4} d f x^{4} - 2 \, c^{3} d f x^{3} - c^{2} d f x^{2} + 2 \, c d f x}\right ) -{\left (a c^{2} x^{2} + \sqrt{-c^{2} x^{2} + 1} b c x - a c x +{\left (b c^{2} x^{2} - b c x - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{3 \,{\left (c^{4} d f^{3} x^{3} - c^{3} d f^{3} x^{2} - c^{2} d f^{3} x + c d f^{3}\right )}}\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{b \arcsin \left (c x\right ) + a}{\sqrt{c d x + d}{\left (-c f x + f\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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