Integrand size = 31, antiderivative size = 617 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]
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Time = 1.08 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {1864, 4837, 12, 6874, 267, 327, 222, 739, 210, 2451, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^4}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \arcsin (c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4}-\frac {b \arcsin (c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}+\frac {b c \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2} \]
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Rule 12
Rule 210
Rule 222
Rule 267
Rule 327
Rule 739
Rule 1864
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 4615
Rule 4825
Rule 4837
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-(b c) \int \frac {2 e (e h-2 d i) x+e^2 i x^2+\frac {2 \left (-e^3 f+d e^2 g-d^2 e h+d^3 i\right )}{d+e x}+2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{2 e^4 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {2 e (e h-2 d i) x+e^2 i x^2+\frac {2 \left (-e^3 f+d e^2 g-d^2 e h+d^3 i\right )}{d+e x}+2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{2 e^4} \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {2 e (e h-2 d i) x}{\sqrt {1-c^2 x^2}}+\frac {e^2 i x^2}{\sqrt {1-c^2 x^2}}-\frac {2 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {2 \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{2 e^4} \\ & = \frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c i) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 e^2}-\frac {(b c (e h-2 d i)) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e^3}-\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}+\frac {\left (b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b i) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e^2}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \int \frac {\arcsin (c x)}{c d+c e x} \, dx}{e^3}-\frac {\left (b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^3} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e^3}+\frac {\left (b c \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e^3} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {\left (b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^4}-\frac {\left (b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (i b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^4}+\frac {\left (i b \left (e^2 g-2 d e h+3 d^2 i\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^4} \\ & = \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.96 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\frac {2 b e (e h-2 d i) \sqrt {1-c^2 x^2}}{c}+\frac {b e^2 i x \sqrt {1-c^2 x^2}}{2 c}-\frac {b e^2 i \arcsin (c x)}{2 c^2}-i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2+2 e (e h-2 d i) x (a+b \arcsin (c x))+e^2 i x^2 (a+b \arcsin (c x))-\frac {2 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{d+e x}+\frac {2 b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)+2 \left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{2 e^4} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2872 vs. \(2 (616 ) = 1232\).
Time = 3.09 (sec) , antiderivative size = 2873, normalized size of antiderivative = 4.66
method | result | size |
parts | \(\text {Expression too large to display}\) | \(2873\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2934\) |
default | \(\text {Expression too large to display}\) | \(2934\) |
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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