Integrand size = 32, antiderivative size = 896 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \]
[Out]
Time = 0.82 (sec) , antiderivative size = 896, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4763, 4847, 4747, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 197, 4749, 4266, 267, 4771, 4791, 294, 222} \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {c^2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}} \]
[In]
[Out]
Rule 197
Rule 222
Rule 267
Rule 294
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4745
Rule 4747
Rule 4749
Rule 4763
Rule 4765
Rule 4767
Rule 4771
Rule 4791
Rule 4847
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(d+c d x)^2 (a+b \arcsin (c x))^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 {d^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {2 c d^2 x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {c^2 d^2 x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {\left (d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^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 (2 c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))^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 (c^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b c^3 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = -\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (b^2 c^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 i b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (8 i b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (4 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {\left (i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ & = \frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \\ \end{align*}
Time = 6.54 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.43 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (-\frac {2 a^2 (-2+c x)}{(-1+c x)^2}+\frac {2 a b \left (\cos \left (\frac {3}{2} \arcsin (c x)\right ) \left (\arcsin (c x)-2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+\cos \left (\frac {1}{2} \arcsin (c x)\right ) \left (-2+3 \arcsin (c x)+6 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \left (1-\left (-1+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)-2 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {b^2 \left (-8 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+\arcsin (c x) \left (8 \log \left (1+i e^{i \arcsin (c x)}\right )-2 \sec ^2\left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+4 \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+\arcsin (c x)^2 \left (-2 i+\left (2+\sec ^2\left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right ) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{\sqrt {1-c^2 x^2}}\right )}{6 c d e^3} \]
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\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \left (-c e x +e \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {c d x + d} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \left (- e \left (c x - 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {c d x + d} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,{\left (e-c\,e\,x\right )}^{5/2}} \,d x \]
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