Optimal. Leaf size=729 \[ \frac{112 i b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{5 e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 i e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{112 b e^5 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 b e^5 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{16 b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[Out]
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Rubi [A] time = 1.30321, antiderivative size = 729, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 16, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4673, 4775, 4641, 4677, 4619, 261, 4773, 3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ \frac{112 i b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{5 e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 i e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{112 b e^5 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 b e^5 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{16 b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4673
Rule 4775
Rule 4641
Rule 4677
Rule 4619
Rule 261
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{(e-c e x)^{5/2} \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)^5 \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{5 e^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c e^5 x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{8 e^5 \left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt{1-c^2 x^2}}-\frac{12 e^5 \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 (5 e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^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{\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 (12 e^5 \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 (c e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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 ) \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 b e^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, 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 ) \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{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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(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 ) \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 )}{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 ) \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 )}{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} \sin ^{-1}(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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \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{8 b e^5 \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{4 e^5 \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 e^5 \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 (24 b e^5 \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 (8 b^2 e^5 \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{\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} \sin ^{-1}(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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \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{8 b e^5 \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{4 e^5 \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 (16 b e^5 \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 (48 b e^5 \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 (16 b^2 e^5 \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{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} \sin ^{-1}(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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \sin ^{-1}(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} \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{8 b e^5 \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{4 e^5 \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{48 b e^5 \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 (32 b e^5 \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 (48 b^2 e^5 \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{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} \sin ^{-1}(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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \sin ^{-1}(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} \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{8 b e^5 \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{4 e^5 \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{112 b e^5 \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 (48 i b^2 e^5 \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 (32 b^2 e^5 \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{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} \sin ^{-1}(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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \sin ^{-1}(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} \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{8 b e^5 \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{4 e^5 \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{112 b e^5 \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{48 i b^2 e^5 \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 (32 i b^2 e^5 \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{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} \sin ^{-1}(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} \left (a+b \sin ^{-1}(c x)\right )^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 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right )^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} \sin ^{-1}(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} \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{8 b e^5 \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{4 e^5 \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{112 b e^5 \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{112 i b^2 e^5 \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 [B] time = 12.4498, size = 2326, normalized size = 3.19 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( -cex+e \right ) ^{{\frac{5}{2}}} \left ( cdx+d \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 (a^{2} c^{2} e^{2} x^{2} - 2 \, a^{2} c e^{2} x + a^{2} e^{2} +{\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\right )} \arcsin \left (c x\right )\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 ) \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{{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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