\(\int \frac {(d+e x+f x^2) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx\) [181]

Optimal result

Integrand size = 28, antiderivative size = 1323 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx =\text {Too large to display} \] Output:

2*a*b*f*x*arcsin(c*x)/h^2-2*a*b*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)/h^3/(h*x+g 
)-2*b^2*(-e*h+2*f*g)*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+2*f*g)*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+1/3*I*b^2*(-e*h+2*f*g)*arcsin(c*x)^3/h 
^3+a^2*f*x/h^2-a^2*(-e*h+2*f*g)*ln(h*x+g)/h^3-a^2*(d*h^2-e*g*h+f*g^2)/h^3/ 
(h*x+g)+2*I*a*b*(-e*h+2*f*g)*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*(-e*h+2*f*g)*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*b^2*(-e*h+2*f*g)*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*b^2*(-e*h+2*f*g)*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*b^2*c*(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/(c^2*g^2-h^2)^ 
(1/2)-2*b^2*c*(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/(c^2*g^2-h^2)^(1/2)+2*a*b*f*(-c^2*x^2+1)^( 
1/2)/c/h^2-2*a*b*(-e*h+2*f*g)*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*(-e*h+2*f*g)*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*b^2*f*(-c^2*x 
^2+1)^(1/2)*arcsin(c*x)/c/h^2-2*b^2*f*x/h^2-b^2*(d*h^2-e*g*h+f*g^2)*arcsin 
(c*x)^2/h^3/(h*x+g)+b^2*f*x*arcsin(c*x)^2/h^2+I*a*b*(-e*h+2*f*g)*arcsin(c* 
x)^2/h^3-2*I*b^2*c*(d*h^2-e*g*h+f*g^2)*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*...
 

Mathematica [A] (warning: unable to verify)

Time = 0.79 (sec) , antiderivative size = 688, normalized size of antiderivative = 0.52 \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\frac {3 f h x (a+b \arcsin (c x))^2-\frac {3 \left (f g^2+h (-e g+d h)\right ) (a+b \arcsin (c x))^2}{g+h x}+\frac {i (2 f g-e h) (a+b \arcsin (c x))^3}{b}-6 b f h \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-3 (2 f g-e h) (a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )-3 (2 f g-e h) (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )+\frac {6 b c \left (f g^2+h (-e g+d h)\right ) \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} h}{-c g+\sqrt {c^2 g^2-h^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{\sqrt {c^2 g^2-h^2}}+6 b (2 f g-e h) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )+6 b (2 f g-e h) \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{3 h^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 3.09 (sec) , antiderivative size = 1323, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 5258

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

\(\Big \downarrow \) 2009

\(\displaystyle \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)}\)

Input:

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

Output:

(a^2*f*x)/h^2 - (2*b^2*f*x)/h^2 - (a^2*(f*g^2 - e*g*h + d*h^2))/(h^3*(g + 
h*x)) + (2*a*b*f*Sqrt[1 - c^2*x^2])/(c*h^2) + (2*a*b*f*x*ArcSin[c*x])/h^2 
- (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x])/(h^3*(g + h*x)) + (2*b^2*f*S 
qrt[1 - c^2*x^2]*ArcSin[c*x])/(c*h^2) + (I*a*b*(2*f*g - e*h)*ArcSin[c*x]^2 
)/h^3 + (b^2*f*x*ArcSin[c*x]^2)/h^2 - (b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[ 
c*x]^2)/(h^3*(g + h*x)) + ((I/3)*b^2*(2*f*g - e*h)*ArcSin[c*x]^3)/h^3 + (2 
*a*b*c*(f*g^2 - e*g*h + d*h^2)*ArcTan[(h + c^2*g*x)/(Sqrt[c^2*g^2 - h^2]*S 
qrt[1 - c^2*x^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (2*a*b*(2*f*g - e*h)*ArcSi 
n[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - 
 ((2*I)*b^2*c*(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*Sqrt[c^2*g^2 - h^2]) - (b^2*(2* 
f*g - e*h)*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*(2*f*g - e*h)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin 
[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + ((2*I)*b^2*c*(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*Sqrt[c^2*g^2 - h^2]) - (b^2*(2*f*g - e*h)*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*(2*f* 
g - e*h)*Log[g + h*x])/h^3 + ((2*I)*a*b*(2*f*g - e*h)*PolyLog[2, (I*E^(I*A 
rcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*c*(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]...
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(h-c*g>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2}{(g+h x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*asin(c*x)*a*b*c*f*g**2*h*x + 2*asin(c*x)*a*b*c*f*g*h**2*x**2 + 2*sqrt( 
- c**2*x**2 + 1)*a*b*f*g**2*h + 2*sqrt( - c**2*x**2 + 1)*a*b*f*g*h**2*x + 
2*int(asin(c*x)/(g**2 + 2*g*h*x + h**2*x**2),x)*a*b*c*d*g**2*h**3 + 2*int( 
asin(c*x)/(g**2 + 2*g*h*x + h**2*x**2),x)*a*b*c*d*g*h**4*x - 2*int(asin(c* 
x)/(g**2 + 2*g*h*x + h**2*x**2),x)*a*b*c*f*g**4*h - 2*int(asin(c*x)/(g**2 
+ 2*g*h*x + h**2*x**2),x)*a*b*c*f*g**3*h**2*x + int(asin(c*x)**2/(g**2 + 2 
*g*h*x + h**2*x**2),x)*b**2*c*d*g**2*h**3 + int(asin(c*x)**2/(g**2 + 2*g*h 
*x + h**2*x**2),x)*b**2*c*d*g*h**4*x + 2*int((asin(c*x)*x)/(g**2 + 2*g*h*x 
 + h**2*x**2),x)*a*b*c*e*g**2*h**3 + 2*int((asin(c*x)*x)/(g**2 + 2*g*h*x + 
 h**2*x**2),x)*a*b*c*e*g*h**4*x - 4*int((asin(c*x)*x)/(g**2 + 2*g*h*x + h* 
*2*x**2),x)*a*b*c*f*g**3*h**2 - 4*int((asin(c*x)*x)/(g**2 + 2*g*h*x + h**2 
*x**2),x)*a*b*c*f*g**2*h**3*x + int((asin(c*x)**2*x**2)/(g**2 + 2*g*h*x + 
h**2*x**2),x)*b**2*c*f*g**2*h**3 + int((asin(c*x)**2*x**2)/(g**2 + 2*g*h*x 
 + h**2*x**2),x)*b**2*c*f*g*h**4*x + int((asin(c*x)**2*x)/(g**2 + 2*g*h*x 
+ h**2*x**2),x)*b**2*c*e*g**2*h**3 + int((asin(c*x)**2*x)/(g**2 + 2*g*h*x 
+ h**2*x**2),x)*b**2*c*e*g*h**4*x + log(g + h*x)*a**2*c*e*g**2*h + log(g + 
 h*x)*a**2*c*e*g*h**2*x - 2*log(g + h*x)*a**2*c*f*g**3 - 2*log(g + h*x)*a* 
*2*c*f*g**2*h*x + a**2*c*d*h**3*x - a**2*c*e*g*h**2*x + 2*a**2*c*f*g**2*h* 
x + a**2*c*f*g*h**2*x**2)/(c*g*h**3*(g + h*x))