Integrand size = 32, antiderivative size = 544 \[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {32 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
[Out]
Time = 0.79 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {4763, 4859, 4737, 4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {32 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[In]
[Out]
Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3798
Rule 3852
Rule 4269
Rule 4271
Rule 4737
Rule 4763
Rule 4857
Rule 4859
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(e-c e x)^4 (a+b \arcsin (c x))^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 {e^4 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {4 e^4 (a+b \arcsin (c x))^2}{(1+c x)^2 \sqrt {1-c^2 x^2}}-\frac {4 e^4 (a+b \arcsin (c x))^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^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^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 (4 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 e^4 \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,\arcsin (c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 c e^4 \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,\arcsin (c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (e^4 \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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 e^4 \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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 e^4 \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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (8 b e^4 \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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b^2 e^4 \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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {4 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 b e^4 \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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (16 b e^4 \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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (8 b^2 e^4 \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} \arcsin (c x)\right )\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {16 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (16 b e^4 \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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (16 b^2 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (16 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (16 b^2 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {16 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (16 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {32 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1438\) vs. \(2(544)=1088\).
Time = 18.46 (sec) , antiderivative size = 1438, normalized size of antiderivative = 2.64 \[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {\sqrt {-e (-1+c x)} \sqrt {d (1+c x)} \left (-\frac {4 a^2 e}{3 d^3 (1+c x)^2}+\frac {8 a^2 e}{3 d^3 (1+c x)}\right )}{c}-\frac {a^2 e^{3/2} \arctan \left (\frac {c x \sqrt {-e (-1+c x)} \sqrt {d (1+c x)}}{\sqrt {d} \sqrt {e} (-1+c x) (1+c x)}\right )}{c d^{5/2}}-\frac {a b e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right ) \left (\cos \left (\frac {1}{2} \arcsin (c x)\right ) \left (-8+6 \arcsin (c x)+9 \arcsin (c x)^2-84 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+\cos \left (\frac {3}{2} \arcsin (c x)\right ) \left ((14-3 \arcsin (c x)) \arcsin (c x)+28 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \left (-4+4 \arcsin (c x)+6 \arcsin (c x)^2+\sqrt {1-c^2 x^2} \left (\arcsin (c x) (14+3 \arcsin (c x))-28 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )-56 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )\right )}{6 c d^3 (-1+c x) \sqrt {(-d-c d x) (e-c e x)} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^4}-\frac {a b e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right ) \left (\cos \left (\frac {3}{2} \arcsin (c x)\right ) \left (\arcsin (c x)+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )-\cos \left (\frac {1}{2} \arcsin (c x)\right ) \left (4+3 \arcsin (c x)+6 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \left (-2+2 \arcsin (c x)+\sqrt {1-c^2 x^2} \arcsin (c x)-4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )\right )}{3 c d^3 (-1+c x) \sqrt {(-d-c d x) (e-c e x)} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^4}-\frac {b^2 e (-1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (-i \pi \arcsin (c x)+(1+i) \arcsin (c x)^2-4 \pi \log \left (1+e^{-i \arcsin (c x)}\right )-2 (\pi +2 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )+4 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+4 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+\frac {4 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}-\frac {2 \arcsin (c x) (2+\arcsin (c x))}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}-\frac {2 \left (-4+\arcsin (c x)^2\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}\right )}{3 c d^3 \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}+\frac {b^2 e (-1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (7 i \pi \arcsin (c x)-(7+7 i) \arcsin (c x)^2-\arcsin (c x)^3+28 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+14 (\pi +2 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )-28 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-14 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-28 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-\frac {4 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {2 \arcsin (c x) (2+\arcsin (c x))}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}+\frac {2 \left (-4+7 \arcsin (c x)^2\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}\right )}{3 c d^3 \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2} \]
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\[\int \frac {\left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int \frac {\left (- e \left (c x - 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e-c e x)^{3/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{3/2}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \]
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