Optimal. Leaf size=96 \[ \frac{x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.174714, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4673, 4651, 260} \[ \frac{x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4651
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{3/2} \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac{x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(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{x}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac{x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.454177, size = 105, normalized size = 1.09 \[ \frac{\sqrt{c d x+d} \left (2 a c x+b \sqrt{1-c^2 x^2} \log (-f (c x+1))+b \sqrt{1-c^2 x^2} \log (f-c f x)+2 b c x \sin ^{-1}(c x)\right )}{2 c d^2 f (c x+1) \sqrt{f-c f x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( cdx+d \right ) ^{-{\frac{3}{2}}} \left ( -cfx+f \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52171, size = 117, normalized size = 1.22 \begin{align*} -\frac{b c \sqrt{\frac{1}{c^{4} d f}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{2 \, d f} + \frac{b x \arcsin \left (c x\right )}{\sqrt{-c^{2} d f x^{2} + d f} d f} + \frac{a x}{\sqrt{-c^{2} d f x^{2} + d f} d f} \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 d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} f^{2} x^{4} - 2 \, c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}}, 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{b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac{3}{2}}{\left (-c f x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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