Optimal. Leaf size=486 \[ \frac {i e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
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Rubi [A]
time = 0.77, antiderivative size = 486, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4763, 4859,
4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \begin {gather*} \frac {i e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) (a+b \text {ArcSin}(c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3798
Rule 3852
Rule 4269
Rule 4271
Rule 4763
Rule 4857
Rule 4859
Rubi steps
\begin {align*} \int \frac {\sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{5/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(e-c e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \left (\frac {2 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac {\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac {\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 c e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{(c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac {\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int 1 \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 b e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (8 b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 5.77, size = 527, normalized size = 1.08 \begin {gather*} \frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (\frac {a^2 (-1+c x)^2}{(1+c x)^2}-\frac {a b \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {3}{2} \text {ArcSin}(c x)\right ) \left (\text {ArcSin}(c x)+2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )-\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (4+3 \text {ArcSin}(c x)+6 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+2 \left (-2+\left (2+\sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)-2 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}-\frac {b^2 (-1+c x)^2 \left (-i \pi \text {ArcSin}(c x)+(1+i) \text {ArcSin}(c x)^2-4 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )-2 (\pi +2 \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+4 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+4 i \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )+\frac {4 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}-\frac {2 \text {ArcSin}(c x) (2+\text {ArcSin}(c x))}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {2 \left (-4+\text {ArcSin}(c x)^2\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}\right )}{3 c d^3 (-1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c e x +e}}{\left (c d x +d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e-c\,e\,x}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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