Integrand size = 28, antiderivative size = 1323 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3} \]
[Out]
Time = 1.68 (sec) , antiderivative size = 1323, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.893, Rules used = {4843, 712, 4837, 12, 6874, 267, 739, 210, 222, 2451, 4825, 4615, 2221, 2317, 2438, 4715, 4767, 8, 4827, 4857, 3404, 2296, 2611, 2320, 6724} \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)^2}{h^3}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)^2}{h^3}-\frac {b^2 \left (f g^2-e h g+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3}-\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3}+\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \arcsin (c x)}{h^3}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}-\frac {2 a b \left (f g^2-e h g+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}+\frac {2 a b c \left (f g^2-e h g+d h^2\right ) \arctan \left (\frac {g x c^2+h}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}-\frac {a^2 \left (f g^2-e h g+d h^2\right )}{h^3 (g+h x)} \]
[In]
[Out]
Rule 8
Rule 12
Rule 210
Rule 222
Rule 267
Rule 712
Rule 739
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2451
Rule 2611
Rule 3404
Rule 4615
Rule 4715
Rule 4767
Rule 4825
Rule 4827
Rule 4837
Rule 4843
Rule 4857
Rule 6724
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 \left (d+e x+f x^2\right )}{(g+h x)^2}+\frac {2 a b \left (d+e x+f x^2\right ) \arcsin (c x)}{(g+h x)^2}+\frac {b^2 \left (d+e x+f x^2\right ) \arcsin (c x)^2}{(g+h x)^2}\right ) \, dx \\ & = a^2 \int \frac {d+e x+f x^2}{(g+h x)^2} \, dx+(2 a b) \int \frac {\left (d+e x+f x^2\right ) \arcsin (c x)}{(g+h x)^2} \, dx+b^2 \int \frac {\left (d+e x+f x^2\right ) \arcsin (c x)^2}{(g+h x)^2} \, dx \\ & = \frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log (g+h x)}{h^3}+a^2 \int \left (\frac {f}{h^2}+\frac {f g^2-e g h+d h^2}{h^2 (g+h x)^2}+\frac {-2 f g+e h}{h^2 (g+h x)}\right ) \, dx+b^2 \int \left (\frac {f \arcsin (c x)^2}{h^2}+\frac {\left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^2 (g+h x)^2}+\frac {(-2 f g+e h) \arcsin (c x)^2}{h^2 (g+h x)}\right ) \, dx-(2 a b c) \int \frac {f h x-\frac {f g^2-e g h+d h^2}{g+h x}-(2 f g-e h) \log (g+h x)}{h^3 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log (g+h x)}{h^3}-\frac {(2 a b c) \int \frac {f h x-\frac {f g^2-e g h+d h^2}{g+h x}-(2 f g-e h) \log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 f\right ) \int \arcsin (c x)^2 \, dx}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \int \frac {\arcsin (c x)^2}{g+h x} \, dx}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\arcsin (c x)^2}{(g+h x)^2} \, dx}{h^2} \\ & = \frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log (g+h x)}{h^3}-\frac {(2 a b c) \int \left (\frac {f h x}{\sqrt {1-c^2 x^2}}+\frac {-f g^2+e g h-d h^2}{(g+h x) \sqrt {1-c^2 x^2}}-\frac {(2 f g-e h) \log (g+h x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{h^3}-\frac {\left (2 b^2 c f\right ) \int \frac {x \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \frac {x^2 \cos (x)}{c g+h \sin (x)} \, dx,x,\arcsin (c x)\right )}{h^2}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\arcsin (c x)}{(g+h x) \sqrt {1-c^2 x^2}} \, dx}{h^3} \\ & = \frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log (g+h x)}{h^3}-\frac {\left (2 b^2 f\right ) \int 1 \, dx}{h^2}-\frac {(2 a b c f) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{h^2}+\frac {(2 a b c (2 f g-e h)) \int \frac {\log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}-\frac {\left (b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h-\sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h+\sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {x}{c g+h \sin (x)} \, dx,x,\arcsin (c x)\right )}{h^3} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {\left (2 b^2 (2 f g-e h)\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}+\frac {\left (2 b^2 (2 f g-e h)\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \int \frac {\arcsin (c x)}{c g+c h x} \, dx}{h^2}-\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 g^2+h^2-x^2} \, dx,x,\frac {h+c^2 g x}{\sqrt {1-c^2 x^2}}\right )}{h^3}+\frac {\left (4 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} g+i h-i e^{2 i x} h} \, dx,x,\arcsin (c x)\right )}{h^3} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {\left (2 i b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}-\frac {\left (2 i b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 g+c h \sin (x)} \, dx,x,\arcsin (c x)\right )}{h^2}-\frac {\left (4 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c g-2 i e^{i x} h-2 \sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2 \sqrt {c^2 g^2-h^2}}+\frac {\left (4 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c g-2 i e^{i x} h+2 \sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2 \sqrt {c^2 g^2-h^2}} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {\left (2 b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i h x}{c g-\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3}-\frac {\left (2 b^2 (2 f g-e h)\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i h x}{c g+\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h-c \sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2}-\frac {(2 a b c (2 f g-e h)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h+c \sqrt {c^2 g^2-h^2}} \, dx,x,\arcsin (c x)\right )}{h^2}+\frac {\left (2 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} h}{2 c g-2 \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {\left (2 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} h}{2 c g+2 \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3 \sqrt {c^2 g^2-h^2}} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {(2 a b (2 f g-e h)) \text {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}+\frac {(2 a b (2 f g-e h)) \text {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\arcsin (c x)\right )}{h^3}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i h x}{2 c g-2 \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i h x}{2 c g+2 \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3 \sqrt {c^2 g^2-h^2}} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {(2 i a b (2 f g-e h)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3}-\frac {(2 i a b (2 f g-e h)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{h^3} \\ & = \frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \arcsin (c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {i a b (2 f g-e h) \arcsin (c x)^2}{h^3}+\frac {b^2 f x \arcsin (c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \arcsin (c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \arctan \left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i a b (2 f g-e h) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3} \\ \end{align*}
Time = 1.35 (sec) , antiderivative size = 688, normalized size of antiderivative = 0.52 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\frac {3 f h x (a+b \arcsin (c x))^2-\frac {3 \left (f g^2+h (-e g+d h)\right ) (a+b \arcsin (c x))^2}{g+h x}+\frac {i (2 f g-e h) (a+b \arcsin (c x))^3}{b}-6 b f h \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-3 (2 f g-e h) (a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )-3 (2 f g-e h) (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )+\frac {6 b c \left (f g^2+h (-e g+d h)\right ) \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{\sqrt {c^2 g^2-h^2}}+6 b (2 f g-e h) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )+6 b (2 f g-e h) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{3 h^3} \]
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\[\int \frac {\left (f \,x^{2}+e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (h x +g \right )^{2}}d x\]
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\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\int { \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\int { \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]
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