3.2.18 \(\int \frac {(d+e x+f x^2) (a+b \arcsin (c x))^2}{g+h x} \, dx\) [118]

3.2.18.1 Optimal result

Integrand size = 28, antiderivative size = 1067 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \arcsin (c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \arcsin (c x)}{h^2}+\frac {a b f x^2 \arcsin (c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \arcsin (c x)}{2 c h}-\frac {b^2 f \arcsin (c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^2}{h^3}-\frac {b^2 (f g-e h) x \arcsin (c x)^2}{h^2}+\frac {b^2 f x^2 \arcsin (c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \arcsin (c x)^3}{3 h^3}+\frac {2 a b \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \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}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3} \]

output

-a^2*(-e*h+f*g)*x/h^2-1/2*a*b*f*arcsin(c*x)/c^2/h-2*a*b*(-e*h+f*g)*x*arcsi 
n(c*x)/h^2+2*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1 
)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3+2*a*b*(d*h^2-e*g*h+f*g^2)*arcsin 
(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-1 
/2*a*b*(-f*h*x-4*e*h+4*f*g)*(-c^2*x^2+1)^(1/2)/c/h^2-2*b^2*(-e*h+f*g)*arcs 
in(c*x)*(-c^2*x^2+1)^(1/2)/c/h^2-I*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)^2/h 
^3-2*I*a*b*(d*h^2-e*g*h+f*g^2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c 
*g-(c^2*g^2-h^2)^(1/2)))/h^3+a^2*(d*h^2-e*g*h+f*g^2)*ln(h*x+g)/h^3+b^2*(d* 
h^2-e*g*h+f*g^2)*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g-(c 
^2*g^2-h^2)^(1/2)))/h^3+b^2*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)^2*ln(1-I*(I*c* 
x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-1/4*b^2*f*arcsin(c* 
x)^2/c^2/h+1/2*b^2*f*x^2*arcsin(c*x)^2/h-1/3*I*b^2*(d*h^2-e*g*h+f*g^2)*arc 
sin(c*x)^3/h^3-b^2*(-e*h+f*g)*x*arcsin(c*x)^2/h^2+a*b*f*x^2*arcsin(c*x)/h+ 
1/2*b^2*f*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/h+2*b^2*(d*h^2-e*g*h+f*g^2)*p 
olylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3+2*b 
^2*(d*h^2-e*g*h+f*g^2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2* 
g^2-h^2)^(1/2)))/h^3+2*b^2*(-e*h+f*g)*x/h^2+1/2*a^2*f*x^2/h-1/4*b^2*f*x^2/ 
h-2*I*b^2*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^ 
(1/2))*h/(c*g-(c^2*g^2-h^2)^(1/2)))/h^3-2*I*a*b*(d*h^2-e*g*h+f*g^2)*polylo 
g(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*h/(c*g+(c^2*g^2-h^2)^(1/2)))/h^3-2*I*b...
 

3.2.18.2 Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.52 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\frac {12 h (-f g+e h) x (a+b \arcsin (c x))^2+6 f h^2 x^2 (a+b \arcsin (c x))^2-\frac {4 i \left (f g^2+h (-e g+d h)\right ) (a+b \arcsin (c x))^3}{b}+24 b h (f g-e h) \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-3 b f h^2 \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right )+12 \left (f g^2+h (-e g+d h)\right ) (a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )+12 \left (f g^2+h (-e g+d h)\right ) (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-24 b \left (f g^2+h (-e g+d h)\right ) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )-24 b \left (f g^2+h (-e g+d h)\right ) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{12 h^3} \]

input

Integrate[((d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2)/(g + h*x),x]
 

output

(12*h*(-(f*g) + e*h)*x*(a + b*ArcSin[c*x])^2 + 6*f*h^2*x^2*(a + b*ArcSin[c 
*x])^2 - ((4*I)*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^3)/b + 24*b 
*h*(f*g - e*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - 3*b*f*h 
^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[ 
c*x])^2/(b*c^2)) + 12*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log 
[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] + 12*(f*g^2 + 
 h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/( 
c*g + Sqrt[c^2*g^2 - h^2])] - 24*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*Ar 
cSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])] 
 - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])]) - 24 
*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*Ar 
cSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c 
*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/(12*h^3)
 

3.2.18.3 Rubi [A] (verified)

Time = 2.31 (sec) , antiderivative size = 1085, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5258, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx\)

\(\Big \downarrow \) 5258

\(\displaystyle \int \left (\frac {a^2 \left (d+e x+f x^2\right )}{g+h x}+\frac {2 a b \arcsin (c x) \left (d+e x+f x^2\right )}{g+h x}+\frac {b^2 \arcsin (c x)^2 \left (d+e x+f x^2\right )}{g+h x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i b^2 \left (f g^2-e h g+d h^2\right ) \arcsin (c x)^3}{3 h^3}+\frac {b^2 f x^2 \arcsin (c x)^2}{2 h}-\frac {i a b \left (f g^2-e h g+d h^2\right ) \arcsin (c x)^2}{h^3}-\frac {b^2 (f g-e h) x \arcsin (c x)^2}{h^2}+\frac {b^2 \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)^2}{h^3}+\frac {b^2 \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)^2}{h^3}-\frac {b^2 f \arcsin (c x)^2}{4 c^2 h}+\frac {a b f x^2 \arcsin (c x)}{h}-\frac {2 a b (f g-e h) x \arcsin (c x)}{h^2}+\frac {2 a b \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}+\frac {2 a b \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}-\frac {2 i b^2 \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 ) \arcsin (c x)}{h^3}-\frac {2 i b^2 \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 ) \arcsin (c x)}{h^3}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \arcsin (c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \arcsin (c x)}{2 c h}-\frac {a b f \arcsin (c x)}{2 c^2 h}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 \left (f g^2-e h g+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \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}-\frac {2 i a b \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}+\frac {2 b^2 \left (f g^2-e h g+d h^2\right ) \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 \left (f g^2-e h g+d h^2\right ) \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 g-e h) \sqrt {1-c^2 x^2}}{c h^2}+\frac {a b f x \sqrt {1-c^2 x^2}}{2 c h}\)

input

Int[((d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2)/(g + h*x),x]
 

output

-((a^2*(f*g - e*h)*x)/h^2) + (2*b^2*(f*g - e*h)*x)/h^2 + (a^2*f*x^2)/(2*h) 
 - (b^2*f*x^2)/(4*h) - (2*a*b*(f*g - e*h)*Sqrt[1 - c^2*x^2])/(c*h^2) + (a* 
b*f*x*Sqrt[1 - c^2*x^2])/(2*c*h) - (a*b*f*ArcSin[c*x])/(2*c^2*h) - (2*a*b* 
(f*g - e*h)*x*ArcSin[c*x])/h^2 + (a*b*f*x^2*ArcSin[c*x])/h - (2*b^2*(f*g - 
 e*h)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*h^2) + (b^2*f*x*Sqrt[1 - c^2*x^2]* 
ArcSin[c*x])/(2*c*h) - (b^2*f*ArcSin[c*x]^2)/(4*c^2*h) - (I*a*b*(f*g^2 - e 
*g*h + d*h^2)*ArcSin[c*x]^2)/h^3 - (b^2*(f*g - e*h)*x*ArcSin[c*x]^2)/h^2 + 
 (b^2*f*x^2*ArcSin[c*x]^2)/(2*h) - ((I/3)*b^2*(f*g^2 - e*g*h + d*h^2)*ArcS 
in[c*x]^3)/h^3 + (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^( 
I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + (b^2*(f*g^2 - e*g*h 
+ d*h^2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 
 - h^2])])/h^3 + (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^( 
I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (b^2*(f*g^2 - e*g*h 
+ d*h^2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 
 - h^2])])/h^3 + (a^2*(f*g^2 - e*g*h + d*h^2)*Log[g + h*x])/h^3 - ((2*I)*a 
*b*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[ 
c^2*g^2 - h^2])])/h^3 - ((2*I)*b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Pol 
yLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - ((2*I) 
*a*b*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqr 
t[c^2*g^2 - h^2])])/h^3 - ((2*I)*b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x...
 

3.2.18.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5258

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(Px_)*((d_) + (e_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x] 
, x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && In 
tegerQ[m]
 

3.2.18.4 Maple [F]

\[\int \frac {\left (f \,x^{2}+e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{h x +g}d x\]

input

int((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x)
 

output

int((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x)
 

3.2.18.5 Fricas [F]

\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\int { \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{h x + g} \,d x } \]

input

integrate((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x, algorithm="fricas")
 

output

integral((a^2*f*x^2 + a^2*e*x + a^2*d + (b^2*f*x^2 + b^2*e*x + b^2*d)*arcs 
in(c*x)^2 + 2*(a*b*f*x^2 + a*b*e*x + a*b*d)*arcsin(c*x))/(h*x + g), x)
 

3.2.18.6 Sympy [F]

\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \]

input

integrate((f*x**2+e*x+d)*(a+b*asin(c*x))**2/(h*x+g),x)
 

output

Integral((a + b*asin(c*x))**2*(d + e*x + f*x**2)/(g + h*x), x)
 

3.2.18.7 Maxima [F]

\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\int { \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{h x + g} \,d x } \]

input

integrate((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x, algorithm="maxima")
 

output

a^2*e*(x/h - g*log(h*x + g)/h^2) + 1/2*a^2*f*(2*g^2*log(h*x + g)/h^3 + (h* 
x^2 - 2*g*x)/h^2) + a^2*d*log(h*x + g)/h + integrate(((b^2*f*x^2 + b^2*e*x 
 + b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*f*x^2 + a* 
b*e*x + a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(h*x + g), x)
 

3.2.18.8 Giac [F]

\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\int { \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{h x + g} \,d x } \]

input

integrate((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x, algorithm="giac")
 

output

integrate((f*x^2 + e*x + d)*(b*arcsin(c*x) + a)^2/(h*x + g), x)
 

3.2.18.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{g+h x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right )}{g+h\,x} \,d x \]

input

int(((a + b*asin(c*x))^2*(d + e*x + f*x^2))/(g + h*x),x)
 

output

int(((a + b*asin(c*x))^2*(d + e*x + f*x^2))/(g + h*x), x)