Integrand size = 31, antiderivative size = 641 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.52 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {4861, 4847, 4747, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 197, 4749, 4266, 267} \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 i b g \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
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Rule 197
Rule 267
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4745
Rule 4747
Rule 4749
Rule 4765
Rule 4767
Rule 4847
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (\frac {f (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {g x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (f \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (2 f \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 f \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b f \sqrt {1-c^2 x^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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 g}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 f x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b g x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 f x (a+b \arcsin (c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 6.23 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.07 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {f \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}-4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )-b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )\right )}{4 c}-\frac {(c f-g) \left (2 b (a+b \arcsin (c x)) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )+(a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right ) \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )-2 \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}-4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )-b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )\right )\right )}{24 c^2}-\frac {f \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}+4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )+b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{4 c}-\frac {(c f+g) \left (2 b (a+b \arcsin (c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )-4 b^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )-(a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )+2 \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}+4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )+b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )\right )}{24 c^2}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5893 vs. \(2 (610 ) = 1220\).
Time = 1.25 (sec) , antiderivative size = 5894, normalized size of antiderivative = 9.20
method | result | size |
default | \(\text {Expression too large to display}\) | \(5894\) |
parts | \(\text {Expression too large to display}\) | \(5894\) |
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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