Optimal. Leaf size=729 \[ -\frac {2 a b e^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {5 e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {e^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {28 i e^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {112 b e^5 \left (1-c^2 x^2\right )^{5/2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {28 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {8 b e^5 \left (1-c^2 x^2\right )^{5/2} \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {4 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac {112 i b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 \left (1-c^2 x^2\right )^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b^2 e^5 x \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac {16 b^2 e^5 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}} \]
[Out]
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Rubi [A] time = 1.30, antiderivative size = 729, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8
Rule 261
Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3717
Rule 3767
Rule 4184
Rule 4186
Rule 4619
Rule 4641
Rule 4673
Rule 4677
Rule 4773
Rule 4775
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*}
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Mathematica [B] time = 13.80, size = 2338, normalized size = 3.21 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} e^{2} x^{2} - 2 \, a^{2} c e^{2} x + a^{2} e^{2} + {\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c e x + e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (\frac {3 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {5}{2}}}{c^{5} d^{5} x^{4} + 4 \, c^{4} d^{5} x^{3} + 6 \, c^{3} d^{5} x^{2} + 4 \, c^{2} d^{5} x + c d^{5}} - \frac {5 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} e}{c^{4} d^{4} x^{3} + 3 \, c^{3} d^{4} x^{2} + 3 \, c^{2} d^{4} x + c d^{4}} - \frac {10 \, \sqrt {-c^{2} d e x^{2} + d e} e^{2}}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}} + \frac {35 \, \sqrt {-c^{2} d e x^{2} + d e} e^{2}}{c^{2} d^{3} x + c d^{3}} + \frac {15 \, e^{3} \arcsin \left (c x\right )}{c d^{3} \sqrt {\frac {e}{d}}}\right )} a^{2} + \sqrt {d} \sqrt {e} \int \frac {{\left ({\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{5/2}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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