Optimal. Leaf size=252 \[ -\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b f^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{4 b f^3 \left (1-c^2 x^2\right )^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.436466, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4673, 4775, 637, 4761, 12, 627, 31, 4641, 4677, 8} \[ -\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (1-c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b f^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{4 b f^3 \left (1-c^2 x^2\right )^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4775
Rule 637
Rule 4761
Rule 12
Rule 627
Rule 31
Rule 4641
Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \frac{(f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{3/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{3/2} \int \frac{(f-c f x)^3 \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{\left (1-c^2 x^2\right )^{3/2} \int \left (\frac{4 \left (f^3-c f^3 x\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}}-\frac{3 f^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{c f^3 x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac{\left (4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{\left (f^3-c f^3 x\right ) \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{\left (3 f^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (c f^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{4 f^3 (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{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (4 b c \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{f^3 (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{\left (b f^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int 1 \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac{b f^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (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{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (4 b f^3 \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{b f^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (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{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (4 b f^3 \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{b f^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (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{f^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{3 f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{4 b f^3 \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 = 3.15703, size = 291, normalized size = 1.15 \[ \frac{f \left (6 a \sqrt{d} \sqrt{f} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )-\frac{\sqrt{c d x+d} \sqrt{f-c f x} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (2 \left (a (c x+5) \left (\sqrt{1-c^2 x^2}+c x-1\right )+b c x \left (\sqrt{1-c^2 x^2}-c x-1\right )+8 b \left (\sqrt{1-c^2 x^2}-c x-1\right ) \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-3 b \left (\sqrt{1-c^2 x^2}-c x-1\right ) \sin ^{-1}(c x)^2+2 b (c x+5) \left (\sqrt{1-c^2 x^2}+c x-1\right ) \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2} \left (\cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )+1\right )}\right )}{2 c d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( -cfx+f \right ) ^{{\frac{3}{2}}} \left ( cdx+d \right ) ^{-{\frac{3}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a c f x - a f +{\left (b c f x - b f\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{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{{\left (-c f x + f\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (c d x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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