\(\int (g+h x)^2 (d+e x+f x^2) (a+b \arcsin (c x))^2 \, dx\) [178]

Optimal result

Integrand size = 28, antiderivative size = 701 \[ \int (g+h x)^2 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d g^2 x-\frac {16 b^2 f h^2 x}{75 c^4}-\frac {4 b^2 \left (f g^2+h (2 e g+d h)\right ) x}{9 c^2}-\frac {1}{4} b^2 g (e g+2 d h) x^2-\frac {3 b^2 h (2 f g+e h) x^2}{32 c^2}-\frac {8 b^2 f h^2 x^3}{225 c^2}-\frac {2}{27} b^2 \left (f g^2+h (2 e g+d h)\right ) x^3-\frac {1}{32} b^2 h (2 f g+e h) x^4-\frac {2}{125} b^2 f h^2 x^5+\frac {2 b d g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {16 b f h^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^5}+\frac {4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {b g (e g+2 d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {3 b h (2 f g+e h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{75 c^3}+\frac {2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {b h (2 f g+e h) x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c}-\frac {g (e g+2 d h) (a+b \arcsin (c x))^2}{4 c^2}-\frac {3 h (2 f g+e h) (a+b \arcsin (c x))^2}{32 c^4}+d g^2 x (a+b \arcsin (c x))^2+\frac {1}{2} g (e g+2 d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} h (2 f g+e h) x^4 (a+b \arcsin (c x))^2+\frac {1}{5} f h^2 x^5 (a+b \arcsin (c x))^2 \] Output:

d*g^2*x*(a+b*arcsin(c*x))^2-1/4*g*(2*d*h+e*g)*(a+b*arcsin(c*x))^2/c^2-3/32 
*h*(e*h+2*f*g)*(a+b*arcsin(c*x))^2/c^4+1/2*g*(2*d*h+e*g)*x^2*(a+b*arcsin(c 
*x))^2+1/4*h*(e*h+2*f*g)*x^4*(a+b*arcsin(c*x))^2+1/5*f*h^2*x^5*(a+b*arcsin 
(c*x))^2-2*b^2*d*g^2*x-4/9*b^2*(f*g^2+h*(d*h+2*e*g))*x/c^2-1/4*b^2*g*(2*d* 
h+e*g)*x^2-1/32*b^2*h*(e*h+2*f*g)*x^4-2/125*b^2*f*h^2*x^5+16/75*b*f*h^2*(- 
c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^5+2/9*b*(f*g^2+h*(d*h+2*e*g))*x^2*(-c 
^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+2*b*d*g^2*(-c^2*x^2+1)^(1/2)*(a+b*arcs 
in(c*x))/c+1/2*b*g*(2*d*h+e*g)*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+3/ 
16*b*h*(e*h+2*f*g)*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3+8/75*b*f*h^2 
*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3+1/8*b*h*(e*h+2*f*g)*x^3*(-c^ 
2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c-2/27*b^2*(f*g^2+h*(d*h+2*e*g))*x^3+1/3* 
(f*g^2+h*(d*h+2*e*g))*x^3*(a+b*arcsin(c*x))^2-16/75*b^2*f*h^2*x/c^4-3/32*b 
^2*h*(e*h+2*f*g)*x^2/c^2-8/225*b^2*f*h^2*x^3/c^2+4/9*b*(f*g^2+h*(d*h+2*e*g 
))*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3+2/25*b*f*h^2*x^4*(-c^2*x^2+1)^ 
(1/2)*(a+b*arcsin(c*x))/c
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.76 \[ \int (g+h x)^2 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=d g^2 x (a+b \arcsin (c x))^2+\frac {1}{2} g (e g+2 d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} h (2 f g+e h) x^4 (a+b \arcsin (c x))^2+\frac {1}{5} f h^2 x^5 (a+b \arcsin (c x))^2-\frac {2 b \left (f g^2+h (2 e g+d h)\right ) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^3}-\frac {2 b f h^2 \left (-15 a \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+b c x \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)\right )}{1125 c^5}-2 b d g^2 \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-\frac {1}{32} b h (2 f g+e h) \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3}-\frac {4 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {3 (a+b \arcsin (c x))^2}{b c^4}\right )-\frac {1}{4} b g (e g+2 d h) \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right ) \] Input:

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

Output:

d*g^2*x*(a + b*ArcSin[c*x])^2 + (g*(e*g + 2*d*h)*x^2*(a + b*ArcSin[c*x])^2 
)/2 + ((f*g^2 + h*(2*e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2)/3 + (h*(2*f*g 
+ e*h)*x^4*(a + b*ArcSin[c*x])^2)/4 + (f*h^2*x^5*(a + b*ArcSin[c*x])^2)/5 
- (2*b*(f*g^2 + h*(2*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*f*h^2*(-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) - 2*b*d*g^2*(b*x - (Sqrt[1 - c^2* 
x^2]*(a + b*ArcSin[c*x]))/c) - (b*h*(2*f*g + e*h)*((3*b*x^2)/c^2 + b*x^4 - 
 (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[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*( 
e*g + 2*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 = 1.47 (sec) , antiderivative size = 701, 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.

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

\(\Big \downarrow \) 5250

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-2*b^2*d*g^2*x - (16*b^2*f*h^2*x)/(75*c^4) - (4*b^2*(f*g^2 + h*(2*e*g + d* 
h))*x)/(9*c^2) - (b^2*g*(e*g + 2*d*h)*x^2)/4 - (3*b^2*h*(2*f*g + e*h)*x^2) 
/(32*c^2) - (8*b^2*f*h^2*x^3)/(225*c^2) - (2*b^2*(f*g^2 + h*(2*e*g + d*h)) 
*x^3)/27 - (b^2*h*(2*f*g + e*h)*x^4)/32 - (2*b^2*f*h^2*x^5)/125 + (2*b*d*g 
^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (16*b*f*h^2*Sqrt[1 - c^2*x^2 
]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*(f*g^2 + h*(2*e*g + d*h))*Sqrt[1 - 
c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (b*g*(e*g + 2*d*h)*x*Sqrt[1 - c^2* 
x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*h*(2*f*g + e*h)*x*Sqrt[1 - c^2*x^2] 
*(a + b*ArcSin[c*x]))/(16*c^3) + (8*b*f*h^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*A 
rcSin[c*x]))/(75*c^3) + (2*b*(f*g^2 + h*(2*e*g + d*h))*x^2*Sqrt[1 - c^2*x^ 
2]*(a + b*ArcSin[c*x]))/(9*c) + (b*h*(2*f*g + e*h)*x^3*Sqrt[1 - c^2*x^2]*( 
a + b*ArcSin[c*x]))/(8*c) + (2*b*f*h^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin 
[c*x]))/(25*c) - (g*(e*g + 2*d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2) - (3*h*(2 
*f*g + e*h)*(a + b*ArcSin[c*x])^2)/(32*c^4) + d*g^2*x*(a + b*ArcSin[c*x])^ 
2 + (g*(e*g + 2*d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + ((f*g^2 + h*(2*e*g + d 
*h))*x^3*(a + b*ArcSin[c*x])^2)/3 + (h*(2*f*g + e*h)*x^4*(a + b*ArcSin[c*x 
])^2)/4 + (f*h^2*x^5*(a + b*ArcSin[c*x])^2)/5
 

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 = 0.82 (sec) , antiderivative size = 1176, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1176\)
default	\(\text {Expression too large to display}\)	\(1176\)
parts	\(\text {Expression too large to display}\)	\(1263\)
orering	\(\text {Expression too large to display}\)	\(3875\)

Input:

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

Output:

1/c*(a^2/c^4*(1/5*f*h^2*c^5*x^5+1/4*(c*e*h^2+2*c*f*g*h)*c^4*x^4+1/3*(c^2*d 
*h^2+2*c^2*e*g*h+c^2*f*g^2)*c^3*x^3+1/2*(2*c^3*d*g*h+c^3*e*g^2)*c^2*x^2+c^ 
5*g^2*d*x)+b^2/c^4*(c^4*d*g^2*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2 
*x^2+1)^(1/2))+1/4*c^3*g^2*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^2*f*g^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)+1/2*c^3*g*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)+2/27*c^2*e*g*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*arcs 
in(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x)+1/64*c*f*g*h*(32*arcsin(c*x)^2*c^4*x^4+ 
16*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)+1/27*c^2*d*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)+1/128*c*e*h^2*(32*arcsin(c*x)^2*c^4*x^4+16 
*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)+1/1125*f*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*arcsin 
(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))+2*a*b/c^4*(1/5*arcsin(c*x)*f*h^2*c^5*x^5+1/4*arcsin(c*x)*c^ 
5*e*h^2*x^4+1/2*arcsin(c*x)*c^5*f*g*h*x^4+1/3*arcsin(c*x)*c^5*d*h^2*x^3...
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1029, normalized size of antiderivative = 1.47 \[ \int (g+h x)^2 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/108000*(864*(25*a^2 - 2*b^2)*c^5*f*h^2*x^5 + 3375*(2*(8*a^2 - b^2)*c^5*f 
*g*h + (8*a^2 - b^2)*c^5*e*h^2)*x^4 + 160*(25*(9*a^2 - 2*b^2)*c^5*f*g^2 + 
50*(9*a^2 - 2*b^2)*c^5*e*g*h + (25*(9*a^2 - 2*b^2)*c^5*d - 24*b^2*c^3*f)*h 
^2)*x^3 + 3375*(8*(2*a^2 - b^2)*c^5*e*g^2 - 3*b^2*c^3*e*h^2 + 2*(8*(2*a^2 
- b^2)*c^5*d - 3*b^2*c^3*f)*g*h)*x^2 + 225*(96*b^2*c^5*f*h^2*x^5 + 480*b^2 
*c^5*d*g^2*x - 120*b^2*c^3*e*g^2 - 45*b^2*c*e*h^2 + 120*(2*b^2*c^5*f*g*h + 
 b^2*c^5*e*h^2)*x^4 + 160*(b^2*c^5*f*g^2 + 2*b^2*c^5*e*g*h + b^2*c^5*d*h^2 
)*x^3 - 30*(8*b^2*c^3*d + 3*b^2*c*f)*g*h + 240*(b^2*c^5*e*g^2 + 2*b^2*c^5* 
d*g*h)*x^2)*arcsin(c*x)^2 - 480*(200*b^2*c^3*e*g*h - 25*(9*(a^2 - 2*b^2)*c 
^5*d - 4*b^2*c^3*f)*g^2 + 4*(25*b^2*c^3*d + 12*b^2*c*f)*h^2)*x + 450*(96*a 
*b*c^5*f*h^2*x^5 + 480*a*b*c^5*d*g^2*x - 120*a*b*c^3*e*g^2 - 45*a*b*c*e*h^ 
2 + 120*(2*a*b*c^5*f*g*h + a*b*c^5*e*h^2)*x^4 + 160*(a*b*c^5*f*g^2 + 2*a*b 
*c^5*e*g*h + a*b*c^5*d*h^2)*x^3 - 30*(8*a*b*c^3*d + 3*a*b*c*f)*g*h + 240*( 
a*b*c^5*e*g^2 + 2*a*b*c^5*d*g*h)*x^2)*arcsin(c*x) + 30*(288*a*b*c^4*f*h^2* 
x^4 + 3200*a*b*c^2*e*g*h + 450*(2*a*b*c^4*f*g*h + a*b*c^4*e*h^2)*x^3 + 800 
*(9*a*b*c^4*d + 2*a*b*c^2*f)*g^2 + 64*(25*a*b*c^2*d + 12*a*b*f)*h^2 + 32*( 
25*a*b*c^4*f*g^2 + 50*a*b*c^4*e*g*h + (25*a*b*c^4*d + 12*a*b*c^2*f)*h^2)*x 
^2 + 225*(8*a*b*c^4*e*g^2 + 3*a*b*c^2*e*h^2 + 2*(8*a*b*c^4*d + 3*a*b*c^2*f 
)*g*h)*x + (288*b^2*c^4*f*h^2*x^4 + 3200*b^2*c^2*e*g*h + 450*(2*b^2*c^4*f* 
g*h + b^2*c^4*e*h^2)*x^3 + 800*(9*b^2*c^4*d + 2*b^2*c^2*f)*g^2 + 64*(25...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1935 vs. \(2 (694) = 1388\).

Time = 0.74 (sec) , antiderivative size = 1935, normalized size of antiderivative = 2.76 \[ \int (g+h x)^2 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**2*d*g**2*x + a**2*d*g*h*x**2 + a**2*d*h**2*x**3/3 + a**2*e*g 
**2*x**2/2 + 2*a**2*e*g*h*x**3/3 + a**2*e*h**2*x**4/4 + a**2*f*g**2*x**3/3 
 + a**2*f*g*h*x**4/2 + a**2*f*h**2*x**5/5 + 2*a*b*d*g**2*x*asin(c*x) + 2*a 
*b*d*g*h*x**2*asin(c*x) + 2*a*b*d*h**2*x**3*asin(c*x)/3 + a*b*e*g**2*x**2* 
asin(c*x) + 4*a*b*e*g*h*x**3*asin(c*x)/3 + a*b*e*h**2*x**4*asin(c*x)/2 + 2 
*a*b*f*g**2*x**3*asin(c*x)/3 + a*b*f*g*h*x**4*asin(c*x) + 2*a*b*f*h**2*x** 
5*asin(c*x)/5 + 2*a*b*d*g**2*sqrt(-c**2*x**2 + 1)/c + a*b*d*g*h*x*sqrt(-c* 
*2*x**2 + 1)/c + 2*a*b*d*h**2*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + a*b*e*g**2 
*x*sqrt(-c**2*x**2 + 1)/(2*c) + 4*a*b*e*g*h*x**2*sqrt(-c**2*x**2 + 1)/(9*c 
) + a*b*e*h**2*x**3*sqrt(-c**2*x**2 + 1)/(8*c) + 2*a*b*f*g**2*x**2*sqrt(-c 
**2*x**2 + 1)/(9*c) + a*b*f*g*h*x**3*sqrt(-c**2*x**2 + 1)/(4*c) + 2*a*b*f* 
h**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - a*b*d*g*h*asin(c*x)/c**2 - a*b*e*g 
**2*asin(c*x)/(2*c**2) + 4*a*b*d*h**2*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*a* 
b*e*g*h*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*a*b*e*h**2*x*sqrt(-c**2*x**2 + 1 
)/(16*c**3) + 4*a*b*f*g**2*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*a*b*f*g*h*x*s 
qrt(-c**2*x**2 + 1)/(8*c**3) + 8*a*b*f*h**2*x**2*sqrt(-c**2*x**2 + 1)/(75* 
c**3) - 3*a*b*e*h**2*asin(c*x)/(16*c**4) - 3*a*b*f*g*h*asin(c*x)/(8*c**4) 
+ 16*a*b*f*h**2*sqrt(-c**2*x**2 + 1)/(75*c**5) + b**2*d*g**2*x*asin(c*x)** 
2 - 2*b**2*d*g**2*x + b**2*d*g*h*x**2*asin(c*x)**2 - b**2*d*g*h*x**2/2 + b 
**2*d*h**2*x**3*asin(c*x)**2/3 - 2*b**2*d*h**2*x**3/27 + b**2*e*g**2*x*...
 

Maxima [F]

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

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

Output:

1/5*a^2*f*h^2*x^5 + 1/2*a^2*f*g*h*x^4 + 1/4*a^2*e*h^2*x^4 + 1/3*a^2*f*g^2* 
x^3 + 2/3*a^2*e*g*h*x^3 + 1/3*a^2*d*h^2*x^3 + b^2*d*g^2*x*arcsin(c*x)^2 + 
1/2*a^2*e*g^2*x^2 + a^2*d*g*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^2 + 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^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*h 
+ 4/9*(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*e*g*h + 1/8*(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*h + 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*d*h^2 + 1/16*(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*h^2 + 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*f*h^2 - 2*b^2*d*g^2*(x - sqrt(-c^2 
*x^2 + 1)*arcsin(c*x)/c) + a^2*d*g^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^ 
2 + 1))*a*b*d*g^2/c + 1/60*(12*b^2*f*h^2*x^5 + 15*(2*b^2*f*g*h + b^2*e*h^2 
)*x^4 + 20*(b^2*f*g^2 + 2*b^2*e*g*h + b^2*d*h^2)*x^3 + 30*(b^2*e*g^2 + 2*b 
^2*d*g*h)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/ 
30*(12*b^2*c*f*h^2*x^5 + 15*(2*b^2*c*f*g*h + b^2*c*e*h^2)*x^4 + 20*(b^2*c* 
f*g^2 + 2*b^2*c*e*g*h + b^2*c*d*h^2)*x^3 + 30*(b^2*c*e*g^2 + 2*b^2*c*d*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2166 vs. \(2 (639) = 1278\).

Time = 0.20 (sec) , antiderivative size = 2166, normalized size of antiderivative = 3.09 \[ \int (g+h x)^2 \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3600*asin(c*x)**2*b**2*c**5*d*g**2*x + 3600*asin(c*x)**2*b**2*c**5*d*g*h* 
x**2 + 1800*asin(c*x)**2*b**2*c**5*e*g**2*x**2 - 1800*asin(c*x)**2*b**2*c* 
*3*d*g*h - 900*asin(c*x)**2*b**2*c**3*e*g**2 + 7200*sqrt( - c**2*x**2 + 1) 
*asin(c*x)*b**2*c**4*d*g**2 + 3600*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c 
**4*d*g*h*x + 1800*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c**4*e*g**2*x + 7 
200*asin(c*x)*a*b*c**5*d*g**2*x + 7200*asin(c*x)*a*b*c**5*d*g*h*x**2 + 240 
0*asin(c*x)*a*b*c**5*d*h**2*x**3 + 3600*asin(c*x)*a*b*c**5*e*g**2*x**2 + 4 
800*asin(c*x)*a*b*c**5*e*g*h*x**3 + 1800*asin(c*x)*a*b*c**5*e*h**2*x**4 + 
2400*asin(c*x)*a*b*c**5*f*g**2*x**3 + 3600*asin(c*x)*a*b*c**5*f*g*h*x**4 + 
 1440*asin(c*x)*a*b*c**5*f*h**2*x**5 - 3600*asin(c*x)*a*b*c**3*d*g*h - 180 
0*asin(c*x)*a*b*c**3*e*g**2 - 675*asin(c*x)*a*b*c*e*h**2 - 1350*asin(c*x)* 
a*b*c*f*g*h + 7200*sqrt( - c**2*x**2 + 1)*a*b*c**4*d*g**2 + 3600*sqrt( - c 
**2*x**2 + 1)*a*b*c**4*d*g*h*x + 800*sqrt( - c**2*x**2 + 1)*a*b*c**4*d*h** 
2*x**2 + 1800*sqrt( - c**2*x**2 + 1)*a*b*c**4*e*g**2*x + 1600*sqrt( - c**2 
*x**2 + 1)*a*b*c**4*e*g*h*x**2 + 450*sqrt( - c**2*x**2 + 1)*a*b*c**4*e*h** 
2*x**3 + 800*sqrt( - c**2*x**2 + 1)*a*b*c**4*f*g**2*x**2 + 900*sqrt( - c** 
2*x**2 + 1)*a*b*c**4*f*g*h*x**3 + 288*sqrt( - c**2*x**2 + 1)*a*b*c**4*f*h* 
*2*x**4 + 1600*sqrt( - c**2*x**2 + 1)*a*b*c**2*d*h**2 + 3200*sqrt( - c**2* 
x**2 + 1)*a*b*c**2*e*g*h + 675*sqrt( - c**2*x**2 + 1)*a*b*c**2*e*h**2*x + 
1600*sqrt( - c**2*x**2 + 1)*a*b*c**2*f*g**2 + 1350*sqrt( - c**2*x**2 + ...