3.2.14 \(\int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx\) [114]

3.2.14.1 Optimal result

Integrand size = 25, antiderivative size = 1678 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx =\text {Too large to display} \]

output

a^2*g^2*ln(e*x+d)/e^3+4*I*b^2*c*g*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+( 
-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(1/2)-2*b^ 
2*g*(-d*g+e*f)*arcsin(c*x)^2/e^3/(e*x+d)+2*a*b*g^2*arcsin(c*x)*ln(1-I*e*(I 
*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^3+2*a*b*g^2*arcsin(c 
*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^3-I*a 
*b*g^2*arcsin(c*x)^2/e^3-2*I*a*b*g^2*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/ 
2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^3-2*I*b^2*g^2*arcsin(c*x)*polylog(2,I*e*( 
I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^3-2*I*a*b*g^2*polyl 
og(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^3-2*I*b^2 
*g^2*arcsin(c*x)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^ 
2)^(1/2)))/e^3-I*b^2*c^3*d*(-d*g+e*f)^2*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2* 
x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(3/2)-4*I*b^2*c 
*g*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^ 
2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(1/2)-a*b*(-d*g+e*f)^2*arcsin(c*x)/e^3/(e 
*x+d)^2-b^2*c^3*d*(-d*g+e*f)^2*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c 
*d-(c^2*d^2-e^2)^(1/2)))/e^3/(c^2*d^2-e^2)^(3/2)+b^2*c^3*d*(-d*g+e*f)^2*po 
lylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^3/(c^2 
*d^2-e^2)^(3/2)+b^2*c*(-d*g+e*f)^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/e^2/(c^2 
*d^2-e^2)/(e*x+d)-a*b*c*(-d*g+e*f)*(4*e^2*g-c^2*d*(3*d*g+e*f))*arctan((c^2 
*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^3/(c^2*d^2-e^2)^(3/2)...
 

3.2.14.2 Mathematica [A] (verified)

Time = 4.54 (sec) , antiderivative size = 903, normalized size of antiderivative = 0.54 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\frac {-\frac {3 (e f-d g)^2 (a+b \arcsin (c x))^2}{(d+e x)^2}+\frac {12 g (-e f+d g) (a+b \arcsin (c x))^2}{d+e x}-\frac {2 i g^2 (a+b \arcsin (c x))^3}{b}+6 g^2 (a+b \arcsin (c x))^2 \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 g^2 (a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+\frac {24 b c g (-e f+d g) \left (i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}+\frac {6 b c^2 (e f-d g)^2 \left (\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c d+c e x}-b \log (d+e x)+\frac {c d \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-12 b g^2 \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )\right )-12 b g^2 \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{6 e^3} \]

input

Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]
 

output

((-3*(e*f - d*g)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^2 + (12*g*(-(e*f) + d* 
g)*(a + b*ArcSin[c*x])^2)/(d + e*x) - ((2*I)*g^2*(a + b*ArcSin[c*x])^3)/b 
+ 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sq 
rt[c^2*d^2 - e^2])] + 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin 
[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + (24*b*c*g*(-(e*f) + d*g)*(I*(a + b* 
ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2 
])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + b*Po 
lyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - b*PolyLog[ 
2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e 
^2] + (6*b*c^2*(e*f - d*g)^2*((e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c 
*d + c*e*x) - b*Log[d + e*x] + (c*d*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I* 
e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*A 
rcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin 
[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]) 
)/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2]))/(c^2*d^2 - e^2) - 1 
2*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - S 
qrt[c^2*d^2 - e^2])] - b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^ 
2*d^2 - e^2])]) - 12*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*Arc 
Sin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - b*PolyLog[3, (I*e*E^(I*ArcSin[c* 
x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(6*e^3)
 

3.2.14.3 Rubi [A] (verified)

Time = 3.94 (sec) , antiderivative size = 1678, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

\(\displaystyle \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 5258

\(\displaystyle \int \left (\frac {a^2 (f+g x)^2}{(d+e x)^3}+\frac {2 a b \arcsin (c x) (f+g x)^2}{(d+e x)^3}+\frac {b^2 \arcsin (c x)^2 (f+g x)^2}{(d+e x)^3}\right )dx\)

\(\Big \downarrow \) 2009

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

input

Int[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]
 

output

-1/2*(a^2*(e*f - d*g)^2)/(e^3*(d + e*x)^2) - (2*a^2*g*(e*f - d*g))/(e^3*(d 
 + e*x)) + (a*b*c*(e*f - d*g)^2*Sqrt[1 - c^2*x^2])/(e^2*(c^2*d^2 - e^2)*(d 
 + e*x)) - (a*b*(e*f - d*g)^2*ArcSin[c*x])/(e^3*(d + e*x)^2) - (4*a*b*g*(e 
*f - d*g)*ArcSin[c*x])/(e^3*(d + e*x)) + (b^2*c*(e*f - d*g)^2*Sqrt[1 - c^2 
*x^2]*ArcSin[c*x])/(e^2*(c^2*d^2 - e^2)*(d + e*x)) - (I*a*b*g^2*ArcSin[c*x 
]^2)/e^3 - (b^2*(e*f - d*g)^2*ArcSin[c*x]^2)/(2*e^3*(d + e*x)^2) - (2*b^2* 
g*(e*f - d*g)*ArcSin[c*x]^2)/(e^3*(d + e*x)) - ((I/3)*b^2*g^2*ArcSin[c*x]^ 
3)/e^3 - (a*b*c*(e*f - d*g)*(4*e^2*g - c^2*d*(e*f + 3*d*g))*ArcTan[(e + c^ 
2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*(c^2*d^2 - e^2)^(3/2 
)) + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^ 
2*d^2 - e^2])])/e^3 - ((4*I)*b^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e* 
E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) 
 - (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/ 
(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSi 
n[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 
 + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2* 
d^2 - e^2])])/e^3 + ((4*I)*b^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^ 
(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + 
 (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c 
*d + Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcS...
 

3.2.14.3.1 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]
 

3.2.14.4 Maple [F]

\[\int \frac {\left (g x +f \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (e x +d \right )^{3}}d x\]

input

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x)
 

output

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x)
 

3.2.14.5 Fricas [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="fricas")
 

output

integral((a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x 
 + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin 
(c*x))/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
 

3.2.14.6 Sympy [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]

input

integrate((g*x+f)**2*(a+b*asin(c*x))**2/(e*x+d)**3,x)
 

output

Integral((a + b*asin(c*x))**2*(f + g*x)**2/(d + e*x)**3, x)
 

3.2.14.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]

input

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,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((e-c*d)*(e+c*d)>0)', see `assume 
?` for mor
 

3.2.14.8 Giac [F]

\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="giac")
 

output

integrate((g*x + f)^2*(b*arcsin(c*x) + a)^2/(e*x + d)^3, x)
 

3.2.14.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]

input

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d + e*x)^3,x)
 

output

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d + e*x)^3, x)