Optimal. Leaf size=463 \[ \frac{5 d^4 (c x+1) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (c x+1)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.370384, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4673, 669, 671, 641, 216, 4761, 627, 43, 4641} \[ \frac{5 d^4 (c x+1) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (c x+1)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 669
Rule 671
Rule 641
Rule 216
Rule 4761
Rule 627
Rule 43
Rule 4641
Rubi steps
\begin{align*} \int \frac{(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{(f-c f x)^{3/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{3/2} \int \frac{(d+c d x)^4 \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{2 d^4 (1+c x)^3 \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{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (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 \left (\frac{15 d^4}{2 c}+\frac{5 d^4 (1+c x)}{2 c}+\frac{2 d^4 (1+c x)^3}{c \left (1-c^2 x^2\right )}-\frac{15 d^4 \sin ^{-1}(c x)}{2 c \sqrt{1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (1+c x)^3 \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{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{(1+c x)^3}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{\left (15 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (1+c x)^3 \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{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{(1+c x)^2}{1-c x} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (1+c x)^3 \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{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (-3-c x+\frac{4}{1-c x}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac{3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (1+c x)^3 \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{15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac{8 b d^4 \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 = 3.99489, size = 768, normalized size = 1.66 \[ \frac{d^2 \left (\frac{8 a \left (c^2 x^2+7 c x-24\right ) \sqrt{c d x+d} \sqrt{f-c f x}}{c x-1}+120 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{8 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (\left (\sin ^{-1}(c x)-4\right ) \sin ^{-1}(c x)-8 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+4\right )-8 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt{1-c^2 x^2} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac{32 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left (-\sin ^{-1}(c x) \left (\left (\sqrt{1-c^2 x^2}+2\right ) \cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\left (\sqrt{1-c^2 x^2}-2\right ) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+\sin ^{-1}(c x)^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+\left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (c x-4 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt{1-c^2 x^2} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}+\frac{b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left (-20 \sin ^{-1}(c x)^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin ^{-1}(c x) \left (24 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )-7 \sin \left (\frac{3}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{5}{2} \sin ^{-1}(c x)\right )+24 \cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+7 \cos \left (\frac{3}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{5}{2} \sin ^{-1}(c x)\right )\right )+2 \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (-16 c x+\cos \left (2 \sin ^{-1}(c x)\right )+32 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt{1-c^2 x^2} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^2}\right )}{16 c f^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.245, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) ) \left ( cdx+d \right ) ^{{\frac{5}{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 [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} d^{2} x^{2} + 2 \, a c d^{2} x + a d^{2} +{\left (b c^{2} d^{2} x^{2} + 2 \, b c d^{2} x + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c f x + f}}{c^{2} f^{2} x^{2} - 2 \, c f^{2} x + 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{{\left (c d x + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\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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