Integrand size = 32, antiderivative size = 486 \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=-\frac {i d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 i b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
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Time = 0.73 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4763, 4859, 4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=-\frac {i d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{-i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2} \sec ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) \sec ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 i b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
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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*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(d+c d x)^3 (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 {2 d^3 (a+b \arcsin (c x))^2}{(-1+c x)^2 \sqrt {1-c^2 x^2}}+\frac {d^3 (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 (d^3 \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 (2 d^3 \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 (d^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,\arcsin (c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 c d^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,\arcsin (c x)\right )}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {\left (d^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,\arcsin (c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (d^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,\arcsin (c x)\right )}{2 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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 (d^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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b d^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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 d^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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {i d^3 \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 {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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 b d^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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b d^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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b^2 d^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} \arcsin (c x)\right )\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {i d^3 \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 {4 b^2 d^3 \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 {4 b d^3 \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 {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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 d^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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (4 b^2 d^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,\arcsin (c x)\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {i d^3 \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 {4 b^2 d^3 \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 {4 b d^3 \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 {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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 i b^2 d^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 \arcsin (c x)}\right )}{c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (8 b^2 d^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,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {i d^3 \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 {4 b^2 d^3 \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 {4 b d^3 \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 {4 i b^2 d^3 \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 {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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 i b^2 d^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 \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {i d^3 \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 {4 b^2 d^3 \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 {4 b d^3 \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 {4 i b^2 d^3 \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}}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \sec ^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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \tan \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 {d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \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}} \\ \end{align*}
Time = 10.90 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\frac {\sqrt {-e (-1+c x)} \sqrt {d (1+c x)} \left (\frac {2 a^2}{3 e^3 (-1+c x)^2}+\frac {a^2}{3 e^3 (-1+c x)}\right )}{c}+\frac {a b \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 ) \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 )-\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 )+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 e^3 \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 \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )}+\frac {b^2 (1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (-3 i \pi \arcsin (c x)+\frac {4 \arcsin (c x)}{-1+c x}-(1-i) \arcsin (c x)^2-\frac {2 \arcsin (c x)^2}{-1+c x}-4 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+2 \pi \log \left (1+i e^{i \arcsin (c x)}\right )-4 \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 (-\cos \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 {2 \left (4+\arcsin (c x)^2+c x \left (-4+\arcsin (c x)^2\right )\right ) \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}\right )}{3 c e^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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\[\int \frac {\sqrt {c d x +d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c e x +e \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\int { \frac {\sqrt {c d x + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\int \frac {\sqrt {d \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- e \left (c x - 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\int { \frac {\sqrt {c d x + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{(e-c e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}}{{\left (e-c\,e\,x\right )}^{5/2}} \,d x \]
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