\(\int (g+h x)^3 (d+e x+f x^2) (a+b \arcsin (c x))^2 \, dx\) [177]

Optimal result

Integrand size = 28, antiderivative size = 1016 \[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d g^3 x-\frac {16 b^2 h^2 (3 f g+e h) x}{75 c^4}-\frac {4 b^2 g \left (f g^2+3 h (e g+d h)\right ) x}{9 c^2}-\frac {5 b^2 f h^3 x^2}{96 c^4}-\frac {1}{4} b^2 g^2 (e g+3 d h) x^2-\frac {3 b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^2}{32 c^2}-\frac {8 b^2 h^2 (3 f g+e h) x^3}{225 c^2}-\frac {2}{27} b^2 g \left (f g^2+3 h (e g+d h)\right ) x^3-\frac {5 b^2 f h^3 x^4}{288 c^2}-\frac {1}{32} b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^4-\frac {2}{125} b^2 h^2 (3 f g+e h) x^5-\frac {1}{108} b^2 f h^3 x^6+\frac {2 b d g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {16 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^5}+\frac {4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {5 b f h^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{48 c^5}+\frac {b g^2 (e g+3 d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {3 b h \left (3 f g^2+h (3 e g+d h)\right ) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {8 b h^2 (3 f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^3}+\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {5 b f h^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{72 c^3}+\frac {b h \left (3 f g^2+h (3 e g+d h)\right ) x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {2 b h^2 (3 f g+e h) x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}+\frac {b f h^3 x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{18 c}-\frac {5 f h^3 (a+b \arcsin (c x))^2}{96 c^6}-\frac {g^2 (e g+3 d h) (a+b \arcsin (c x))^2}{4 c^2}-\frac {3 h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \arcsin (c x))^2}{32 c^4}+d g^3 x (a+b \arcsin (c x))^2+\frac {1}{2} g^2 (e g+3 d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 (a+b \arcsin (c x))^2+\frac {1}{5} h^2 (3 f g+e h) x^5 (a+b \arcsin (c x))^2+\frac {1}{6} f h^3 x^6 (a+b \arcsin (c x))^2 \] Output:

d*g^3*x*(a+b*arcsin(c*x))^2-2*b^2*d*g^3*x-1/4*b^2*g^2*(3*d*h+e*g)*x^2-2/27 
*b^2*g*(f*g^2+3*h*(d*h+e*g))*x^3-1/32*b^2*h*(3*f*g^2+h*(d*h+3*e*g))*x^4-2/ 
125*b^2*h^2*(e*h+3*f*g)*x^5-1/108*b^2*f*h^3*x^6-5/96*f*h^3*(a+b*arcsin(c*x 
))^2/c^6-1/4*g^2*(3*d*h+e*g)*(a+b*arcsin(c*x))^2/c^2-3/32*h*(3*f*g^2+h*(d* 
h+3*e*g))*(a+b*arcsin(c*x))^2/c^4+1/2*g^2*(3*d*h+e*g)*x^2*(a+b*arcsin(c*x) 
)^2+1/3*g*(f*g^2+3*h*(d*h+e*g))*x^3*(a+b*arcsin(c*x))^2+1/4*h*(3*f*g^2+h*( 
d*h+3*e*g))*x^4*(a+b*arcsin(c*x))^2+1/5*h^2*(e*h+3*f*g)*x^5*(a+b*arcsin(c* 
x))^2+1/6*f*h^3*x^6*(a+b*arcsin(c*x))^2+2*b*d*g^3*(-c^2*x^2+1)^(1/2)*(a+b* 
arcsin(c*x))/c+16/75*b*h^2*(e*h+3*f*g)*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x) 
)/c^5+4/9*b*g*(f*g^2+3*h*(d*h+e*g))*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c 
^3+1/8*b*h*(3*f*g^2+h*(d*h+3*e*g))*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x) 
)/c+2/25*b*h^2*(e*h+3*f*g)*x^4*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+1/18 
*b*f*h^3*x^5*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+5/48*b*f*h^3*x*(-c^2*x 
^2+1)^(1/2)*(a+b*arcsin(c*x))/c^5+1/2*b*g^2*(3*d*h+e*g)*x*(-c^2*x^2+1)^(1/ 
2)*(a+b*arcsin(c*x))/c+3/16*b*h*(3*f*g^2+h*(d*h+3*e*g))*x*(-c^2*x^2+1)^(1/ 
2)*(a+b*arcsin(c*x))/c^3+8/75*b*h^2*(e*h+3*f*g)*x^2*(-c^2*x^2+1)^(1/2)*(a+ 
b*arcsin(c*x))/c^3+2/9*b*g*(f*g^2+3*h*(d*h+e*g))*x^2*(-c^2*x^2+1)^(1/2)*(a 
+b*arcsin(c*x))/c+5/72*b*f*h^3*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^ 
3-16/75*b^2*h^2*(e*h+3*f*g)*x/c^4-4/9*b^2*g*(f*g^2+3*h*(d*h+e*g))*x/c^2-5/ 
96*b^2*f*h^3*x^2/c^4-3/32*b^2*h*(3*f*g^2+h*(d*h+3*e*g))*x^2/c^2-8/225*b...
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 734, normalized size of antiderivative = 0.72 \[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=d g^3 x (a+b \arcsin (c x))^2+\frac {1}{2} g^2 (e g+3 d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 (a+b \arcsin (c x))^2+\frac {1}{5} h^2 (3 f g+e h) x^5 (a+b \arcsin (c x))^2+\frac {1}{6} f h^3 x^6 (a+b \arcsin (c x))^2-\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^3}-\frac {2 b h^2 (3 f g+e h) \left (-15 a \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+b c x \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)\right )}{1125 c^5}-\frac {f h^3 \left (45 a^2-6 a b c x \sqrt {1-c^2 x^2} \left (15+10 c^2 x^2+8 c^4 x^4\right )+b^2 c^2 x^2 \left (45+15 c^2 x^2+8 c^4 x^4\right )-6 b \left (-15 a+b c x \sqrt {1-c^2 x^2} \left (15+10 c^2 x^2+8 c^4 x^4\right )\right ) \arcsin (c x)+45 b^2 \arcsin (c x)^2\right )}{864 c^6}-2 b d g^3 \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-\frac {1}{32} b h \left (3 f g^2+h (3 e g+d h)\right ) \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3}-\frac {4 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {3 (a+b \arcsin (c x))^2}{b c^4}\right )-\frac {1}{4} b g^2 (e g+3 d h) \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right ) \] Input:

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

Output:

d*g^3*x*(a + b*ArcSin[c*x])^2 + (g^2*(e*g + 3*d*h)*x^2*(a + b*ArcSin[c*x]) 
^2)/2 + (g*(f*g^2 + 3*h*(e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2)/3 + (h*(3* 
f*g^2 + h*(3*e*g + d*h))*x^4*(a + b*ArcSin[c*x])^2)/4 + (h^2*(3*f*g + e*h) 
*x^5*(a + b*ArcSin[c*x])^2)/5 + (f*h^3*x^6*(a + b*ArcSin[c*x])^2)/6 - (2*b 
*g*(f*g^2 + 3*h*(e*g + d*h))*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x 
*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3 
) - (2*b*h^2*(3*f*g + e*h)*(-15*a*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4 
*x^4) + b*c*x*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*b*Sqrt[1 - c^2*x^2]*(8 + 
 4*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(1125*c^5) - (f*h^3*(45*a^2 - 6*a*b* 
c*x*Sqrt[1 - c^2*x^2]*(15 + 10*c^2*x^2 + 8*c^4*x^4) + b^2*c^2*x^2*(45 + 15 
*c^2*x^2 + 8*c^4*x^4) - 6*b*(-15*a + b*c*x*Sqrt[1 - c^2*x^2]*(15 + 10*c^2* 
x^2 + 8*c^4*x^4))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(864*c^6) - 2*b*d*g 
^3*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (b*h*(3*f*g^2 + h*( 
3*e*g + d*h))*((3*b*x^2)/c^2 + b*x^4 - (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSi 
n[c*x]))/c^3 - (4*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (3*(a + b 
*ArcSin[c*x])^2)/(b*c^4)))/32 - (b*g^2*(e*g + 3*d*h)*(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)))/4
 

Rubi [A] (verified)

Time = 2.08 (sec) , antiderivative size = 1016, 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 = {5250, 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.

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

\(\Big \downarrow \) 5250

\(\displaystyle \int \left (h x^3 (a+b \arcsin (c x))^2 \left (h (d h+3 e g)+3 f g^2\right )+g x^2 (a+b \arcsin (c x))^2 \left (3 h (d h+e g)+f g^2\right )+g^2 x (3 d h+e g) (a+b \arcsin (c x))^2+d g^3 (a+b \arcsin (c x))^2+h^2 x^4 (e h+3 f g) (a+b \arcsin (c x))^2+f h^3 x^5 (a+b \arcsin (c x))^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{108} b^2 f h^3 x^6+\frac {1}{6} f h^3 (a+b \arcsin (c x))^2 x^6+\frac {1}{5} h^2 (3 f g+e h) (a+b \arcsin (c x))^2 x^5-\frac {2}{125} b^2 h^2 (3 f g+e h) x^5+\frac {b f h^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5}{18 c}-\frac {5 b^2 f h^3 x^4}{288 c^2}+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \arcsin (c x))^2 x^4-\frac {1}{32} b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^4+\frac {2 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^4}{25 c}+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) (a+b \arcsin (c x))^2 x^3-\frac {8 b^2 h^2 (3 f g+e h) x^3}{225 c^2}-\frac {2}{27} b^2 g \left (f g^2+3 h (e g+d h)\right ) x^3+\frac {5 b f h^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{72 c^3}+\frac {b h \left (3 f g^2+h (3 e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{8 c}-\frac {5 b^2 f h^3 x^2}{96 c^4}+\frac {1}{2} g^2 (e g+3 d h) (a+b \arcsin (c x))^2 x^2-\frac {1}{4} b^2 g^2 (e g+3 d h) x^2-\frac {3 b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^2}{32 c^2}+\frac {8 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^2}{75 c^3}+\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^2}{9 c}-2 b^2 d g^3 x+d g^3 (a+b \arcsin (c x))^2 x-\frac {16 b^2 h^2 (3 f g+e h) x}{75 c^4}-\frac {4 b^2 g \left (f g^2+3 h (e g+d h)\right ) x}{9 c^2}+\frac {5 b f h^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{48 c^5}+\frac {b g^2 (e g+3 d h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c}+\frac {3 b h \left (3 f g^2+h (3 e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{16 c^3}-\frac {5 f h^3 (a+b \arcsin (c x))^2}{96 c^6}-\frac {g^2 (e g+3 d h) (a+b \arcsin (c x))^2}{4 c^2}-\frac {3 h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \arcsin (c x))^2}{32 c^4}+\frac {2 b d g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {16 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^5}+\frac {4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}\)

Input:

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

Output:

-2*b^2*d*g^3*x - (16*b^2*h^2*(3*f*g + e*h)*x)/(75*c^4) - (4*b^2*g*(f*g^2 + 
 3*h*(e*g + d*h))*x)/(9*c^2) - (5*b^2*f*h^3*x^2)/(96*c^4) - (b^2*g^2*(e*g 
+ 3*d*h)*x^2)/4 - (3*b^2*h*(3*f*g^2 + h*(3*e*g + d*h))*x^2)/(32*c^2) - (8* 
b^2*h^2*(3*f*g + e*h)*x^3)/(225*c^2) - (2*b^2*g*(f*g^2 + 3*h*(e*g + d*h))* 
x^3)/27 - (5*b^2*f*h^3*x^4)/(288*c^2) - (b^2*h*(3*f*g^2 + h*(3*e*g + d*h)) 
*x^4)/32 - (2*b^2*h^2*(3*f*g + e*h)*x^5)/125 - (b^2*f*h^3*x^6)/108 + (2*b* 
d*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (16*b*h^2*(3*f*g + e*h)*S 
qrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*g*(f*g^2 + 3*h*(e*g 
+ d*h))*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (5*b*f*h^3*x*Sqrt 
[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(48*c^5) + (b*g^2*(e*g + 3*d*h)*x*Sqrt[ 
1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*h*(3*f*g^2 + h*(3*e*g + d*h 
))*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (8*b*h^2*(3*f*g + e 
*h)*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*g*(f*g^2 + 
3*h*(e*g + d*h))*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (5*b*f 
*h^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(72*c^3) + (b*h*(3*f*g^2 + 
 h*(3*e*g + d*h))*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (2*b* 
h^2*(3*f*g + e*h)*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (b*f 
*h^3*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (5*f*h^3*(a + b*A 
rcSin[c*x])^2)/(96*c^6) - (g^2*(e*g + 3*d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2 
) - (3*h*(3*f*g^2 + h*(3*e*g + d*h))*(a + b*ArcSin[c*x])^2)/(32*c^4) + ...
 

Defintions of rubi rules used

rule 2009

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

rule 5250

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(Px_), x_Symbol] :> Int[ExpandI 
ntegrand[Px*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && Poly 
nomialQ[Px, x]
 

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 1706, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1706\)
default	\(\text {Expression too large to display}\)	\(1706\)
parts	\(\text {Expression too large to display}\)	\(1903\)
orering	\(\text {Expression too large to display}\)	\(5066\)

Input:

int((h*x+g)^3*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/c*(a^2/c^5*(1/6*h^3*f*c^6*x^6+1/5*(c*e*h^3+3*c*f*g*h^2)*c^5*x^5+1/4*(c^2 
*d*h^3+3*c^2*e*g*h^2+3*c^2*f*g^2*h)*c^4*x^4+1/3*(3*c^3*d*g*h^2+3*c^3*e*g^2 
*h+c^3*f*g^3)*c^3*x^3+1/2*(3*c^4*d*g^2*h+c^4*e*g^3)*c^2*x^2+c^6*g^3*d*x)+b 
^2/c^5*(c^5*d*g^3*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2 
))+1/4*c^4*g^3*e*(2*arcsin(c*x)^2*c^2*x^2+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2) 
*c*x-arcsin(c*x)^2-c^2*x^2)+1/27*c^3*f*g^3*(9*arcsin(c*x)^2*c^3*x^3+6*arcs 
in(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*c^3*x^3+12*arcsin(c*x)*(-c^2*x^2+1)^( 
1/2)-12*c*x)+3/4*c^4*g^2*h*d*(2*arcsin(c*x)^2*c^2*x^2+2*arcsin(c*x)*(-c^2* 
x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/9*c^3*e*g^2*h*(9*arcsin(c*x)^2*c 
^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*c^3*x^3+12*arcsin(c*x)*( 
-c^2*x^2+1)^(1/2)-12*c*x)+3/128*c^2*f*g^2*h*(32*arcsin(c*x)^2*c^4*x^4+16*a 
rcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-4*c^4*x^4+24*arcsin(c*x)*(-c^2*x^2+1 
)^(1/2)*c*x-12*arcsin(c*x)^2-12*c^2*x^2-9)+1/9*c^3*d*g*h^2*(9*arcsin(c*x)^ 
2*c^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*c^3*x^3+12*arcsin(c*x 
)*(-c^2*x^2+1)^(1/2)-12*c*x)+3/128*c^2*e*g*h^2*(32*arcsin(c*x)^2*c^4*x^4+1 
6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-4*c^4*x^4+24*arcsin(c*x)*(-c^2*x^ 
2+1)^(1/2)*c*x-12*arcsin(c*x)^2-12*c^2*x^2-9)+1/375*c*f*g*h^2*(225*arcsin( 
c*x)^2*c^5*x^5+90*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-18*c^5*x^5+120*ar 
csin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-40*c^3*x^3+240*arcsin(c*x)*(-c^2*x^2+ 
1)^(1/2)-240*c*x)+1/128*h^3*d*c^2*(32*arcsin(c*x)^2*c^4*x^4+16*arcsin(c...
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1537, normalized size of antiderivative = 1.51 \[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/108000*(1000*(18*a^2 - b^2)*c^6*f*h^3*x^6 + 864*(3*(25*a^2 - 2*b^2)*c^6* 
f*g*h^2 + (25*a^2 - 2*b^2)*c^6*e*h^3)*x^5 + 375*(27*(8*a^2 - b^2)*c^6*f*g^ 
2*h + 27*(8*a^2 - b^2)*c^6*e*g*h^2 + (9*(8*a^2 - b^2)*c^6*d - 5*b^2*c^4*f) 
*h^3)*x^4 + 160*(25*(9*a^2 - 2*b^2)*c^6*f*g^3 + 75*(9*a^2 - 2*b^2)*c^6*e*g 
^2*h - 24*b^2*c^4*e*h^3 + 3*(25*(9*a^2 - 2*b^2)*c^6*d - 24*b^2*c^4*f)*g*h^ 
2)*x^3 + 1125*(24*(2*a^2 - b^2)*c^6*e*g^3 - 27*b^2*c^4*e*g*h^2 + 9*(8*(2*a 
^2 - b^2)*c^6*d - 3*b^2*c^4*f)*g^2*h - (9*b^2*c^4*d + 5*b^2*c^2*f)*h^3)*x^ 
2 + 225*(80*b^2*c^6*f*h^3*x^6 + 480*b^2*c^6*d*g^3*x - 120*b^2*c^4*e*g^3 - 
135*b^2*c^2*e*g*h^2 + 96*(3*b^2*c^6*f*g*h^2 + b^2*c^6*e*h^3)*x^5 + 120*(3* 
b^2*c^6*f*g^2*h + 3*b^2*c^6*e*g*h^2 + b^2*c^6*d*h^3)*x^4 - 45*(8*b^2*c^4*d 
 + 3*b^2*c^2*f)*g^2*h - 5*(9*b^2*c^2*d + 5*b^2*f)*h^3 + 160*(b^2*c^6*f*g^3 
 + 3*b^2*c^6*e*g^2*h + 3*b^2*c^6*d*g*h^2)*x^3 + 240*(b^2*c^6*e*g^3 + 3*b^2 
*c^6*d*g^2*h)*x^2)*arcsin(c*x)^2 - 480*(300*b^2*c^4*e*g^2*h + 48*b^2*c^2*e 
*h^3 - 25*(9*(a^2 - 2*b^2)*c^6*d - 4*b^2*c^4*f)*g^3 + 12*(25*b^2*c^4*d + 1 
2*b^2*c^2*f)*g*h^2)*x + 450*(80*a*b*c^6*f*h^3*x^6 + 480*a*b*c^6*d*g^3*x - 
120*a*b*c^4*e*g^3 - 135*a*b*c^2*e*g*h^2 + 96*(3*a*b*c^6*f*g*h^2 + a*b*c^6* 
e*h^3)*x^5 + 120*(3*a*b*c^6*f*g^2*h + 3*a*b*c^6*e*g*h^2 + a*b*c^6*d*h^3)*x 
^4 - 45*(8*a*b*c^4*d + 3*a*b*c^2*f)*g^2*h - 5*(9*a*b*c^2*d + 5*a*b*f)*h^3 
+ 160*(a*b*c^6*f*g^3 + 3*a*b*c^6*e*g^2*h + 3*a*b*c^6*d*g*h^2)*x^3 + 240*(a 
*b*c^6*e*g^3 + 3*a*b*c^6*d*g^2*h)*x^2)*arcsin(c*x) + 30*(200*a*b*c^5*f*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2992 vs. \(2 (1006) = 2012\).

Time = 1.08 (sec) , antiderivative size = 2992, normalized size of antiderivative = 2.94 \[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**2*d*g**3*x + 3*a**2*d*g**2*h*x**2/2 + a**2*d*g*h**2*x**3 + a 
**2*d*h**3*x**4/4 + a**2*e*g**3*x**2/2 + a**2*e*g**2*h*x**3 + 3*a**2*e*g*h 
**2*x**4/4 + a**2*e*h**3*x**5/5 + a**2*f*g**3*x**3/3 + 3*a**2*f*g**2*h*x** 
4/4 + 3*a**2*f*g*h**2*x**5/5 + a**2*f*h**3*x**6/6 + 2*a*b*d*g**3*x*asin(c* 
x) + 3*a*b*d*g**2*h*x**2*asin(c*x) + 2*a*b*d*g*h**2*x**3*asin(c*x) + a*b*d 
*h**3*x**4*asin(c*x)/2 + a*b*e*g**3*x**2*asin(c*x) + 2*a*b*e*g**2*h*x**3*a 
sin(c*x) + 3*a*b*e*g*h**2*x**4*asin(c*x)/2 + 2*a*b*e*h**3*x**5*asin(c*x)/5 
 + 2*a*b*f*g**3*x**3*asin(c*x)/3 + 3*a*b*f*g**2*h*x**4*asin(c*x)/2 + 6*a*b 
*f*g*h**2*x**5*asin(c*x)/5 + a*b*f*h**3*x**6*asin(c*x)/3 + 2*a*b*d*g**3*sq 
rt(-c**2*x**2 + 1)/c + 3*a*b*d*g**2*h*x*sqrt(-c**2*x**2 + 1)/(2*c) + 2*a*b 
*d*g*h**2*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + a*b*d*h**3*x**3*sqrt(-c**2*x** 
2 + 1)/(8*c) + a*b*e*g**3*x*sqrt(-c**2*x**2 + 1)/(2*c) + 2*a*b*e*g**2*h*x* 
*2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*a*b*e*g*h**2*x**3*sqrt(-c**2*x**2 + 1)/( 
8*c) + 2*a*b*e*h**3*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 2*a*b*f*g**3*x**2*s 
qrt(-c**2*x**2 + 1)/(9*c) + 3*a*b*f*g**2*h*x**3*sqrt(-c**2*x**2 + 1)/(8*c) 
 + 6*a*b*f*g*h**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + a*b*f*h**3*x**5*sqrt( 
-c**2*x**2 + 1)/(18*c) - 3*a*b*d*g**2*h*asin(c*x)/(2*c**2) - a*b*e*g**3*as 
in(c*x)/(2*c**2) + 4*a*b*d*g*h**2*sqrt(-c**2*x**2 + 1)/(3*c**3) + 3*a*b*d* 
h**3*x*sqrt(-c**2*x**2 + 1)/(16*c**3) + 4*a*b*e*g**2*h*sqrt(-c**2*x**2 + 1 
)/(3*c**3) + 9*a*b*e*g*h**2*x*sqrt(-c**2*x**2 + 1)/(16*c**3) + 8*a*b*e*...
 

Maxima [F]

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

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

Output:

1/6*a^2*f*h^3*x^6 + 3/5*a^2*f*g*h^2*x^5 + 1/5*a^2*e*h^3*x^5 + 3/4*a^2*f*g^ 
2*h*x^4 + 3/4*a^2*e*g*h^2*x^4 + 1/4*a^2*d*h^3*x^4 + 1/3*a^2*f*g^3*x^3 + a^ 
2*e*g^2*h*x^3 + a^2*d*g*h^2*x^3 + b^2*d*g^3*x*arcsin(c*x)^2 + 1/2*a^2*e*g^ 
3*x^2 + 3/2*a^2*d*g^2*h*x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 
1)*x/c^2 - arcsin(c*x)/c^3))*a*b*e*g^3 + 2/9*(3*x^3*arcsin(c*x) + c*(sqrt( 
-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*f*g^3 + 3/2*(2*x^2* 
arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d*g^2*h 
+ 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 
 + 1)/c^4))*a*b*e*g^2*h + 3/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)* 
x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*f*g^2*h + 
 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 
+ 1)/c^4))*a*b*d*g*h^2 + 3/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x 
^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*e*g*h^2 + 
2/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 
 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*f*g*h^2 + 1/16*(8*x^4*arc 
sin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3* 
arcsin(c*x)/c^5)*c)*a*b*d*h^3 + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^ 
2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)* 
c)*a*b*e*h^3 + 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 
 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsi...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3444 vs. \(2 (932) = 1864\).

Time = 0.21 (sec) , antiderivative size = 3444, normalized size of antiderivative = 3.39 \[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*a^2*f*h^3*x^6 + 3/5*a^2*f*g*h^2*x^5 + 1/5*a^2*e*h^3*x^5 + 3/4*a^2*f*g^ 
2*h*x^4 + 3/4*a^2*e*g*h^2*x^4 + 1/4*a^2*d*h^3*x^4 + 1/3*a^2*f*g^3*x^3 + a^ 
2*e*g^2*h*x^3 + a^2*d*g*h^2*x^3 + b^2*d*g^3*x*arcsin(c*x)^2 + 2*a*b*d*g^3* 
x*arcsin(c*x) + 1/3*(c^2*x^2 - 1)*b^2*f*g^3*x*arcsin(c*x)^2/c^2 + (c^2*x^2 
 - 1)*b^2*e*g^2*h*x*arcsin(c*x)^2/c^2 + (c^2*x^2 - 1)*b^2*d*g*h^2*x*arcsin 
(c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*b^2*e*g^3*x*arcsin(c*x)/c + 3/2*sqrt( 
-c^2*x^2 + 1)*b^2*d*g^2*h*x*arcsin(c*x)/c + a^2*d*g^3*x - 2*b^2*d*g^3*x + 
2/3*(c^2*x^2 - 1)*a*b*f*g^3*x*arcsin(c*x)/c^2 + 2*(c^2*x^2 - 1)*a*b*e*g^2* 
h*x*arcsin(c*x)/c^2 + 2*(c^2*x^2 - 1)*a*b*d*g*h^2*x*arcsin(c*x)/c^2 + 1/2* 
(c^2*x^2 - 1)*b^2*e*g^3*arcsin(c*x)^2/c^2 + 3/2*(c^2*x^2 - 1)*b^2*d*g^2*h* 
arcsin(c*x)^2/c^2 + 1/3*b^2*f*g^3*x*arcsin(c*x)^2/c^2 + b^2*e*g^2*h*x*arcs 
in(c*x)^2/c^2 + b^2*d*g*h^2*x*arcsin(c*x)^2/c^2 + 3/5*(c^2*x^2 - 1)^2*b^2* 
f*g*h^2*x*arcsin(c*x)^2/c^4 + 1/5*(c^2*x^2 - 1)^2*b^2*e*h^3*x*arcsin(c*x)^ 
2/c^4 + 1/2*sqrt(-c^2*x^2 + 1)*a*b*e*g^3*x/c + 3/2*sqrt(-c^2*x^2 + 1)*a*b* 
d*g^2*h*x/c + 2*sqrt(-c^2*x^2 + 1)*b^2*d*g^3*arcsin(c*x)/c - 3/8*(-c^2*x^2 
 + 1)^(3/2)*b^2*f*g^2*h*x*arcsin(c*x)/c^3 - 3/8*(-c^2*x^2 + 1)^(3/2)*b^2*e 
*g*h^2*x*arcsin(c*x)/c^3 - 1/8*(-c^2*x^2 + 1)^(3/2)*b^2*d*h^3*x*arcsin(c*x 
)/c^3 - 2/27*(c^2*x^2 - 1)*b^2*f*g^3*x/c^2 - 2/9*(c^2*x^2 - 1)*b^2*e*g^2*h 
*x/c^2 - 2/9*(c^2*x^2 - 1)*b^2*d*g*h^2*x/c^2 + (c^2*x^2 - 1)*a*b*e*g^3*arc 
sin(c*x)/c^2 + 3*(c^2*x^2 - 1)*a*b*d*g^2*h*arcsin(c*x)/c^2 + 2/3*a*b*f*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (g+h x)^3 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {too large to display} \] Input:

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

Output:

(3600*asin(c*x)**2*b**2*c**6*d*g**3*x + 5400*asin(c*x)**2*b**2*c**6*d*g**2 
*h*x**2 + 1800*asin(c*x)**2*b**2*c**6*e*g**3*x**2 - 2700*asin(c*x)**2*b**2 
*c**4*d*g**2*h - 900*asin(c*x)**2*b**2*c**4*e*g**3 + 7200*sqrt( - c**2*x** 
2 + 1)*asin(c*x)*b**2*c**5*d*g**3 + 5400*sqrt( - c**2*x**2 + 1)*asin(c*x)* 
b**2*c**5*d*g**2*h*x + 1800*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c**5*e*g 
**3*x + 7200*asin(c*x)*a*b*c**6*d*g**3*x + 10800*asin(c*x)*a*b*c**6*d*g**2 
*h*x**2 + 7200*asin(c*x)*a*b*c**6*d*g*h**2*x**3 + 1800*asin(c*x)*a*b*c**6* 
d*h**3*x**4 + 3600*asin(c*x)*a*b*c**6*e*g**3*x**2 + 7200*asin(c*x)*a*b*c** 
6*e*g**2*h*x**3 + 5400*asin(c*x)*a*b*c**6*e*g*h**2*x**4 + 1440*asin(c*x)*a 
*b*c**6*e*h**3*x**5 + 2400*asin(c*x)*a*b*c**6*f*g**3*x**3 + 5400*asin(c*x) 
*a*b*c**6*f*g**2*h*x**4 + 4320*asin(c*x)*a*b*c**6*f*g*h**2*x**5 + 1200*asi 
n(c*x)*a*b*c**6*f*h**3*x**6 - 5400*asin(c*x)*a*b*c**4*d*g**2*h - 1800*asin 
(c*x)*a*b*c**4*e*g**3 - 675*asin(c*x)*a*b*c**2*d*h**3 - 2025*asin(c*x)*a*b 
*c**2*e*g*h**2 - 2025*asin(c*x)*a*b*c**2*f*g**2*h - 375*asin(c*x)*a*b*f*h* 
*3 + 7200*sqrt( - c**2*x**2 + 1)*a*b*c**5*d*g**3 + 5400*sqrt( - c**2*x**2 
+ 1)*a*b*c**5*d*g**2*h*x + 2400*sqrt( - c**2*x**2 + 1)*a*b*c**5*d*g*h**2*x 
**2 + 450*sqrt( - c**2*x**2 + 1)*a*b*c**5*d*h**3*x**3 + 1800*sqrt( - c**2* 
x**2 + 1)*a*b*c**5*e*g**3*x + 2400*sqrt( - c**2*x**2 + 1)*a*b*c**5*e*g**2* 
h*x**2 + 1350*sqrt( - c**2*x**2 + 1)*a*b*c**5*e*g*h**2*x**3 + 288*sqrt( - 
c**2*x**2 + 1)*a*b*c**5*e*h**3*x**4 + 800*sqrt( - c**2*x**2 + 1)*a*b*c*...