Optimal. Leaf size=345 \[ \frac{5 f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{3 f^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{11 f^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{b c^2 f^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{3 b c f^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{11 b f^3 x \sqrt{1-c^2 x^2}}{3 \sqrt{c d x+d} \sqrt{f-c f x}} \]
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Rubi [A] time = 0.591949, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {4673, 4763, 4641, 4677, 8, 4707, 30} \[ \frac{5 f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{3 f^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{11 f^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{b c^2 f^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{3 b c f^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{11 b f^3 x \sqrt{1-c^2 x^2}}{3 \sqrt{c d x+d} \sqrt{f-c f x}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4763
Rule 4641
Rule 4677
Rule 8
Rule 4707
Rule 30
Rubi steps
\begin{align*} \int \frac{(f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d+c d x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f-c f x)^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{f^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3 c f^3 x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 c^2 f^3 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{c^3 f^3 x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{\left (f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (3 c f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}+\frac{\left (3 c^2 f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (c^3 f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{3 f^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{3 f^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{\left (3 f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (3 b f^3 \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (2 c f^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{\left (3 b c f^3 \sqrt{1-c^2 x^2}\right ) \int x \, dx}{2 \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (b c^2 f^3 \sqrt{1-c^2 x^2}\right ) \int x^2 \, dx}{3 \sqrt{d+c d x} \sqrt{f-c f x}}\\ &=-\frac{3 b f^3 x \sqrt{1-c^2 x^2}}{\sqrt{d+c d x} \sqrt{f-c f x}}+\frac{3 b c f^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{b c^2 f^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{11 f^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{3 f^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{5 f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{\left (2 b f^3 \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{3 \sqrt{d+c d x} \sqrt{f-c f x}}\\ &=-\frac{11 b f^3 x \sqrt{1-c^2 x^2}}{3 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{3 b c f^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{b c^2 f^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{11 f^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{d+c d x} \sqrt{f-c f x}}-\frac{3 f^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{5 f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{d+c d x} \sqrt{f-c f x}}\\ \end{align*}
Mathematica [A] time = 1.73426, size = 274, normalized size = 0.79 \[ \frac{f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left (12 a \sqrt{1-c^2 x^2} \left (2 c^2 x^2-9 c x+22\right )-270 b c x+2 b \sin \left (3 \sin ^{-1}(c x)\right )-27 b \cos \left (2 \sin ^{-1}(c x)\right )\right )-180 a \sqrt{d} f^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )-6 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left (9 \sqrt{1-c^2 x^2} (2 c x-5)+\cos \left (3 \sin ^{-1}(c x)\right )\right )+90 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{72 c d \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( -cfx+f \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{cdx+d}}}}\, 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^{2} f^{2} x^{2} - 2 \, a c f^{2} x + a f^{2} +{\left (b c^{2} f^{2} x^{2} - 2 \, b c f^{2} x + b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c f x + f}}{\sqrt{c d x + d}}, 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{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{c d x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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