Optimal. Leaf size=164 \[ \frac{d^3 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^3 \left (1-c^2 x^2\right )^{5/2} \log (1-c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A] time = 0.256099, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4673, 651, 4761, 12, 627, 43} \[ \frac{d^3 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^3 \left (1-c^2 x^2\right )^{5/2} \log (1-c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 651
Rule 4761
Rule 12
Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c d x} \left (a+b \sin ^{-1}(c x)\right )}{(f-c f x)^{5/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(d+c d x)^3 \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{d^3 (1+c x)^3 \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{\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{d^3 (1+c x)^3}{3 c \left (1-c^2 x^2\right )^2} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{d^3 (1+c x)^3 \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{\left (b d^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{(1+c x)^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{d^3 (1+c x)^3 \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{\left (b d^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1+c x}{(1-c x)^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac{d^3 (1+c x)^3 \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{\left (b d^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac{2}{(-1+c x)^2}+\frac{1}{-1+c x}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac{2 b d^3 \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{d^3 (1+c x)^3 \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{b d^3 \left (1-c^2 x^2\right )^{5/2} \log (1-c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.486667, size = 126, normalized size = 0.77 \[ \frac{\sqrt{c d x+d} \sqrt{f-c f x} \left ((c x+1) \left (a \sqrt{1-c^2 x^2}+b c x-b\right )+b (c x+1) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-b (c x-1)^2 \log (f-c f x)\right )}{3 c f^3 (c x-1)^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.242, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) )\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 = 3.24564, size = 1137, normalized size = 6.93 \begin{align*} \left [\frac{{\left (b c^{3} f x^{3} - b c^{2} f x^{2} - b c f x + b f\right )} \sqrt{\frac{d}{f}} \log \left (\frac{c^{6} d x^{6} - 4 \, c^{5} d x^{5} + 5 \, c^{4} d x^{4} - 4 \, c^{2} d x^{2} + 4 \, c d 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{\frac{d}{f}} - 2 \, d}{c^{4} x^{4} - 2 \, c^{3} x^{3} + 2 \, c x - 1}\right ) + 2 \,{\left (a c^{2} x^{2} - 2 \, \sqrt{-c^{2} x^{2} + 1} b c x + 2 \, a c x +{\left (b c^{2} x^{2} + 2 \, b c x + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{6 \,{\left (c^{4} f^{3} x^{3} - c^{3} f^{3} x^{2} - c^{2} f^{3} x + c f^{3}\right )}}, -\frac{{\left (b c^{3} f x^{3} - b c^{2} f x^{2} - b c f x + b f\right )} \sqrt{-\frac{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{-\frac{d}{f}}}{c^{4} d x^{4} - 2 \, c^{3} d x^{3} - c^{2} d x^{2} + 2 \, c d x}\right ) -{\left (a c^{2} x^{2} - 2 \, \sqrt{-c^{2} x^{2} + 1} b c x + 2 \, a c x +{\left (b c^{2} x^{2} + 2 \, b c x + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{3 \,{\left (c^{4} f^{3} x^{3} - c^{3} f^{3} x^{2} - c^{2} f^{3} x + c 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{\sqrt{c d x + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\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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