Optimal. Leaf size=414 \[ \frac{3 f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac{3 f (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}+\frac{f \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac{1}{4} f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^4 f x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c^3 f x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^2 f x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac{5 b c f x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac{b f x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.38232, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4673, 4763, 4649, 4647, 4641, 30, 14, 4677, 194} \[ \frac{3 f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac{3 f (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}+\frac{f \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac{1}{4} f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^4 f x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c^3 f x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^2 f x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac{5 b c f x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac{b f x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4763
Rule 4649
Rule 4647
Rule 4641
Rule 30
Rule 14
Rule 4677
Rule 194
Rubi steps
\begin{align*} \int (d+c d x)^{3/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int (f-c f x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (f \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-c f x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{\left (f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}-\frac{\left (c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac{\left (3 f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{3 f x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac{f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac{\left (3 f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{b f x (d+c d x)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac{5 b c f x^2 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac{2 b c^2 f x^3 (d+c d x)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac{b c^3 f x^4 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c^4 f x^5 (d+c d x)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac{1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{3 f x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac{f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac{3 f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.45541, size = 305, normalized size = 0.74 \[ \frac{d f^2 \left (\sqrt{c d x+d} \sqrt{f-c f x} \left (240 a \sqrt{1-c^2 x^2} \left (8 c^4 x^4-10 c^3 x^3-16 c^2 x^2+25 c x+8\right )-128 b c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+1200 b \cos \left (2 \sin ^{-1}(c x)\right )+75 b \cos \left (4 \sin ^{-1}(c x)\right )\right )-3600 a \sqrt{d} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )+60 b \sqrt{c d x+d} \sqrt{f-c f x} \left (32 \left (1-c^2 x^2\right )^{5/2}+40 \sin \left (2 \sin ^{-1}(c x)\right )+5 \sin \left (4 \sin ^{-1}(c x)\right )\right ) \sin ^{-1}(c x)+1800 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2\right )}{9600 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.224, size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{3}{2}}} \left ( -cfx+f \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) \, 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 ({\left (a c^{3} d f^{2} x^{3} - a c^{2} d f^{2} x^{2} - a c d f^{2} x + a d f^{2} +{\left (b c^{3} d f^{2} x^{3} - b c^{2} d f^{2} x^{2} - b c d f^{2} x + b d f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c f x + f}, 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{\left (c d x + d\right )}^{\frac{3}{2}}{\left (-c f x + f\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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