Integrand size = 32, antiderivative size = 729 \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=-\frac {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {16 b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {112 b e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {112 i b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
[Out]
Time = 0.89 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4763, 4859, 4737, 4767, 4715, 267, 4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {5 e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {112 b e^5 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {28 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {8 b e^5 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {112 i b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {16 b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[In]
[Out]
Rule 8
Rule 267
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3798
Rule 3852
Rule 4269
Rule 4271
Rule 4715
Rule 4737
Rule 4763
Rule 4767
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)^5 (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 {5 e^5 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c e^5 x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {8 e^5 (a+b \arcsin (c x))^2}{(1+c x)^2 \sqrt {1-c^2 x^2}}-\frac {12 e^5 (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 (5 e^5 \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 (8 e^5 \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 (12 e^5 \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 {\left (c e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (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 {e^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 (12 e^5 \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 (2 b e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int (a+b \arcsin (c x)) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 c e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 (2 b^2 e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \arcsin (c x) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 e^5 \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 (6 e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {12 e^5 \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 {8 b e^5 \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 {4 e^5 \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 (4 e^5 \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 (24 b e^5 \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 (8 b^2 e^5 \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 {\left (2 b^2 c e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {12 i e^5 \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^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {28 e^5 \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 {8 b e^5 \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 {4 e^5 \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 (16 b e^5 \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 (48 b e^5 \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 (16 b^2 e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \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^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {16 b^2 e^5 \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 {28 e^5 \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 {8 b e^5 \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 {4 e^5 \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 {48 b e^5 \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 (32 b e^5 \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 (48 b^2 e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \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^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {16 b^2 e^5 \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 {28 e^5 \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 {8 b e^5 \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 {4 e^5 \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 {112 b e^5 \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 (48 i b^2 e^5 \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 (32 b^2 e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \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^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {16 b^2 e^5 \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 {28 e^5 \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 {8 b e^5 \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 {4 e^5 \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 {112 b e^5 \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 {48 i b^2 e^5 \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 (32 i b^2 e^5 \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 {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \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^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \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 {16 b^2 e^5 \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 {28 e^5 \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 {8 b e^5 \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 {4 e^5 \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 {112 b e^5 \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 {112 i b^2 e^5 \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. \(2338\) vs. \(2(729)=1458\).
Time = 24.67 (sec) , antiderivative size = 2338, normalized size of antiderivative = 3.21 \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e-c e x)^{5/2} (a+b \arcsin (c x))^2}{(d+c d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{5/2}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \]
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