Optimal. Leaf size=486 \[ \frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.12, 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 = {4673, 4775, 4773, 3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ \frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3717
Rule 3767
Rule 4184
Rule 4186
Rule 4673
Rule 4773
Rule 4775
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 8.23, size = 698, normalized size = 1.44 \[ \frac {\sqrt {d (c x+1)} \sqrt {-e (c x-1)} \left (\frac {a^2}{3 d^3 (c x+1)}-\frac {2 a^2}{3 d^3 (c x+1)^2}\right )}{c}-\frac {a b \sqrt {c d x+d} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (2 \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (\sqrt {1-c^2 x^2} \sin ^{-1}(c x)-2 \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin ^{-1}(c x)-4 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-2\right )+\cos \left (\frac {3}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x)+2 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )-\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (3 \sin ^{-1}(c x)+6 \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+4\right )\right )}{3 c d^3 (c x-1) \sqrt {(-c d x-d) (e-c e x)} \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^4}-\frac {b^2 (c x-1) \sqrt {c d x+d} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (4 i \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )+(1+i) \sin ^{-1}(c x)^2-i \pi \sin ^{-1}(c x)-4 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )-2 \left (2 \sin ^{-1}(c x)+\pi \right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 \pi \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+\frac {4 \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2}{\left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^3}-\frac {2 \left (\sin ^{-1}(c x)+2\right ) \sin ^{-1}(c x)}{\left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac {2 \left (\sin ^{-1}(c x)^2-4\right ) \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )}{\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )}+4 \pi \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )}{3 c d^3 \sqrt {1-c^2 x^2} \sqrt {(-c d x-d) (e-c e x)} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.35, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e-c\,e\,x}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________