Optimal. Leaf size=376 \[ \frac{1}{4} c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{5 f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}+\frac{2 f^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{3}{8} f^2 x \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^3 f^2 x^4 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{3 b c f^2 x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}-\frac{2 b f^2 x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.539114, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4673, 4763, 4647, 4641, 30, 4677, 4697, 4707} \[ \frac{1}{4} c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{5 f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}+\frac{2 f^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{3}{8} f^2 x \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^3 f^2 x^4 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{3 b c f^2 x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}-\frac{2 b f^2 x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4763
Rule 4647
Rule 4641
Rule 30
Rule 4677
Rule 4697
Rule 4707
Rubi steps
\begin{align*} \int \sqrt{d+c d x} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\left (\sqrt{d+c d x} \sqrt{f-c f x}\right ) \int (f-c f x)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \left (f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c f^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+c^2 f^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (2 c f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} f^2 x \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} c^2 f^2 x^3 \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{2 f^2 \sqrt{d+c d x} \sqrt{f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{\left (f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (b c f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b c^3 f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int x^3 \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b f^2 x \sqrt{d+c d x} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}-\frac{b c f^2 x^2 \sqrt{d+c d x} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f^2 x^3 \sqrt{d+c d x} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 f^2 x^4 \sqrt{d+c d x} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{3}{8} f^2 x \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} c^2 f^2 x^3 \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{2 f^2 \sqrt{d+c d x} \sqrt{f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{f^2 \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{\left (f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (b c f^2 \sqrt{d+c d x} \sqrt{f-c f x}\right ) \int x \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b f^2 x \sqrt{d+c d x} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}-\frac{3 b c f^2 x^2 \sqrt{d+c d x} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f^2 x^3 \sqrt{d+c d x} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 f^2 x^4 \sqrt{d+c d x} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{3}{8} f^2 x \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} c^2 f^2 x^3 \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac{2 f^2 \sqrt{d+c d x} \sqrt{f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{5 f^2 \sqrt{d+c d x} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.2032, size = 293, normalized size = 0.78 \[ \frac{f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left (48 a \sqrt{1-c^2 x^2} \left (6 c^3 x^3-16 c^2 x^2+9 c x+16\right )+256 b c x \left (c^2 x^2-3\right )+144 b \cos \left (2 \sin ^{-1}(c x)\right )-9 b \cos \left (4 \sin ^{-1}(c x)\right )\right )-720 a \sqrt{d} f^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )-12 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left (-64 \left (1-c^2 x^2\right )^{3/2}-24 \sin \left (2 \sin ^{-1}(c x)\right )+3 \sin \left (4 \sin ^{-1}(c x)\right )\right )+360 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{1152 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.235, size = 0, normalized size = 0. \begin{align*} \int \sqrt{cdx+d} \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^{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 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 \sqrt{c d x + d}{\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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