Optimal. Leaf size=99 \[ \frac{b f \left (1-c^2 x^2\right )^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.213356, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4673, 637, 4761, 12, 627, 31} \[ \frac{b f \left (1-c^2 x^2\right )^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 637
Rule 4761
Rule 12
Rule 627
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} \sqrt{f-c f x}} \, dx &=\frac{\left (1-c^2 x^2\right )^{3/2} \int \frac{(f-c f x) \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (b c \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{f (1-c x)}{c \left (1-c^2 x^2\right )} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (b f \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{1-c x}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (b f \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{1}{1+c x} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{f (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{b f \left (1-c^2 x^2\right )^{3/2} \log (1+c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.363893, size = 79, normalized size = 0.8 \[ \frac{\sqrt{c d x+d} \left (a (c x-1)+b \sqrt{1-c^2 x^2} \log (-f (c x+1))+b (c x-1) \sin ^{-1}(c x)\right )}{c d^2 (c x+1) \sqrt{f-c f x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( cdx+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cfx+f}}}}\, 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 = 2.59884, size = 803, normalized size = 8.11 \begin{align*} \left [\frac{{\left (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 \, \sqrt{c d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}}{2 \,{\left (c^{2} d^{2} f x + c d^{2} f\right )}}, \frac{{\left (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 ) - \sqrt{c d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{2} d^{2} f x + c d^{2} f}\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*} \int \frac{a + b \operatorname{asin}{\left (c x \right )}}{\left (d \left (c x + 1\right )\right )^{\frac{3}{2}} \sqrt{- f \left (c x - 1\right )}}\, dx \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}{{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-c f x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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