Optimal. Leaf size=486 \[ -\frac{4 i b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{-i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{-i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
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Rubi [A] time = 1.06978, antiderivative size = 486, normalized size of antiderivative = 1., 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} \[ -\frac{4 i b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{-i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{-i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b^2 d^3 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4775
Rule 4773
Rule 3318
Rule 4186
Rule 3767
Rule 8
Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c d x} \left (a+b \sin ^{-1}(c x)\right )^2}{(e-c e x)^{5/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(d+c d 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 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x)^2 \sqrt{1-c^2 x^2}}+\frac{d^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 (d^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 d^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 (d^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 d^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 (d^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 (d^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{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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 (d^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 d^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 d^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 d^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{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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 d^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 d^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 d^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 d^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 d^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{4 b d^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{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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 (8 b d^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 d^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 d^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 d^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{4 b d^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{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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 i b^2 d^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 d^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 d^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 d^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{4 b d^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 d^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{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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 (8 i b^2 d^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 d^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 d^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{4 b d^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 d^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}}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \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{d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \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}}\\ \end{align*}
Mathematica [A] time = 8.23552, size = 683, normalized size = 1.41 \[ \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{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-\frac{2 \sin ^{-1}(c x)^2}{c x-1}-(1-i) \sin ^{-1}(c x)^2+\frac{4 \sin ^{-1}(c x)}{c x-1}-3 i \pi \sin ^{-1}(c x)-4 \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-4 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+2 \pi \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+\frac{2 \left (\sin ^{-1}(c x)^2+c x \left (\sin ^{-1}(c x)^2-4\right )+4\right ) \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )}{\left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}+4 \pi \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2 \pi \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )}{3 c e^3 \sqrt{1-c^2 x^2} \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 )^2}+\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left (\frac{a^2}{3 e^3 (c x-1)}+\frac{2 a^2}{3 e^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 (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 (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin ^{-1}(c x)+4 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (3 \sin ^{-1}(c x)-6 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-4\right )-\cos \left (\frac{3}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x)-2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{3 c e^3 \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 )^4 \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.273, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{cdx+d} \left ( -cex+e \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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} e^{3} x^{3} - 3 \, c^{2} e^{3} x^{2} + 3 \, c e^{3} x - e^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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