Integrand size = 32, antiderivative size = 918 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=-\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \]
[Out]
Time = 0.84 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {4763, 4859, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266, 4737, 4715, 267, 4795, 4723, 327, 222} \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=-\frac {5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3 d^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 x \left (1-c^2 x^2\right )^2 d^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 \left (1-c^2 x^2\right )^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {8 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 a b x \left (1-c^2 x^2\right )^{3/2} d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x) d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x) d^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {b c x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {16 b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}} \]
[In]
[Out]
Rule 222
Rule 267
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4715
Rule 4723
Rule 4737
Rule 4745
Rule 4749
Rule 4763
Rule 4765
Rule 4767
Rule 4795
Rule 4847
Rule 4859
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(d+c d x)^4 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = \frac {\left (1-c^2 x^2\right )^{3/2} \int \left (\frac {8 \left (d^4+c d^4 x\right ) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac {7 d^4 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {4 c d^4 x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c^2 d^4 x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = \frac {\left (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\left (d^4+c d^4 x\right ) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (7 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (4 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (c^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = \frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {7 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac {d^4 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {c d^4 x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int (a+b \arcsin (c x)) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (b c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int x (a+b \arcsin (c x)) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (8 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \arcsin (c x) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (b^2 c^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 b^2 c d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (32 i b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {\left (16 b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {\left (8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ & = -\frac {8 a b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 d^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 b^2 d^4 x \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 d^4 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {32 i b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b d^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 d^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2041\) vs. \(2(918)=1836\).
Time = 20.33 (sec) , antiderivative size = 2041, normalized size of antiderivative = 2.22 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c e x +e \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{(e-c e x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}}{{\left (e-c\,e\,x\right )}^{3/2}} \,d x \]
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