\(\int (g+h x) (d+e x+f x^2) (a+b \arcsin (c x))^2 \, dx\) [117]

Optimal result

Integrand size = 26, antiderivative size = 425 \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d g x-\frac {4 b^2 (f g+e h) x}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}-\frac {3 f h (a+b \arcsin (c x))^2}{32 c^4}-\frac {(e g+d h) (a+b \arcsin (c x))^2}{4 c^2}+d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2 \]

[Out]

-2*b^2*d*g*x-4/9*b^2*(e*h+f*g)*x/c^2-3/32*b^2*f*h*x^2/c^2-1/4*b^2*(d*h+e*g)*x^2-2/27*b^2*(e*h+f*g)*x^3-1/32*b^
2*f*h*x^4-3/32*f*h*(a+b*arcsin(c*x))^2/c^4-1/4*(d*h+e*g)*(a+b*arcsin(c*x))^2/c^2+d*g*x*(a+b*arcsin(c*x))^2+1/2
*(d*h+e*g)*x^2*(a+b*arcsin(c*x))^2+1/3*(e*h+f*g)*x^3*(a+b*arcsin(c*x))^2+1/4*f*h*x^4*(a+b*arcsin(c*x))^2+2*b*d
*g*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+4/9*b*(e*h+f*g)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+3/16*b*f*h*
x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+1/2*b*(d*h+e*g)*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+2/9*b*(e*h
+f*g)*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+1/8*b*f*h*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4835, 4715, 4767, 8, 4723, 4795, 4737, 30} \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {3 f h (a+b \arcsin (c x))^2}{32 c^4}+\frac {b x \sqrt {1-c^2 x^2} (d h+e g) (a+b \arcsin (c x))}{2 c}-\frac {(d h+e g) (a+b \arcsin (c x))^2}{4 c^2}+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (e h+f g) (a+b \arcsin (c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {4 b \sqrt {1-c^2 x^2} (e h+f g) (a+b \arcsin (c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {1}{2} x^2 (d h+e g) (a+b \arcsin (c x))^2+d g x (a+b \arcsin (c x))^2+\frac {1}{3} x^3 (e h+f g) (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2-\frac {4 b^2 x (e h+f g)}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac {2}{27} b^2 x^3 (e h+f g)-\frac {1}{32} b^2 f h x^4 \]

[In]

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

[Out]

-2*b^2*d*g*x - (4*b^2*(f*g + e*h)*x)/(9*c^2) - (3*b^2*f*h*x^2)/(32*c^2) - (b^2*(e*g + d*h)*x^2)/4 - (2*b^2*(f*
g + e*h)*x^3)/27 - (b^2*f*h*x^4)/32 + (2*b*d*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*(f*g + e*h)*Sqr
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (3*b*f*h*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (b*
(e*g + d*h)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (2*b*(f*g + e*h)*x^2*Sqrt[1 - c^2*x^2]*(a + b*Arc
Sin[c*x]))/(9*c) + (b*f*h*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (3*f*h*(a + b*ArcSin[c*x])^2)/(32
*c^4) - ((e*g + d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2) + d*g*x*(a + b*ArcSin[c*x])^2 + ((e*g + d*h)*x^2*(a + b*Ar
cSin[c*x])^2)/2 + ((f*g + e*h)*x^3*(a + b*ArcSin[c*x])^2)/3 + (f*h*x^4*(a + b*ArcSin[c*x])^2)/4

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4835

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

Rubi steps \begin{align*} \text {integral}& = \int \left (d g (a+b \arcsin (c x))^2+(e g+d h) x (a+b \arcsin (c x))^2+(f g+e h) x^2 (a+b \arcsin (c x))^2+f h x^3 (a+b \arcsin (c x))^2\right ) \, dx \\ & = (d g) \int (a+b \arcsin (c x))^2 \, dx+(f h) \int x^3 (a+b \arcsin (c x))^2 \, dx+(e g+d h) \int x (a+b \arcsin (c x))^2 \, dx+(f g+e h) \int x^2 (a+b \arcsin (c x))^2 \, dx \\ & = d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2-(2 b c d g) \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} (b c f h) \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx-(b c (e g+d h)) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} (2 b c (f g+e h)) \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {2 b d g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2-\left (2 b^2 d g\right ) \int 1 \, dx-\frac {1}{8} \left (b^2 f h\right ) \int x^3 \, dx-\frac {(3 b f h) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{8 c}-\frac {1}{2} \left (b^2 (e g+d h)\right ) \int x \, dx-\frac {(b (e g+d h)) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}-\frac {1}{9} \left (2 b^2 (f g+e h)\right ) \int x^2 \, dx-\frac {(4 b (f g+e h)) \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{9 c} \\ & = -2 b^2 d g x-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}-\frac {(e g+d h) (a+b \arcsin (c x))^2}{4 c^2}+d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2-\frac {(3 b f h) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 f h\right ) \int x \, dx}{16 c^2}-\frac {\left (4 b^2 (f g+e h)\right ) \int 1 \, dx}{9 c^2} \\ & = -2 b^2 d g x-\frac {4 b^2 (f g+e h) x}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}-\frac {3 f h (a+b \arcsin (c x))^2}{32 c^4}-\frac {(e g+d h) (a+b \arcsin (c x))^2}{4 c^2}+d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.86 \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=d g x (a+b \arcsin (c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \arcsin (c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \arcsin (c x))^2+\frac {1}{4} f h x^4 (a+b \arcsin (c x))^2-\frac {2 b (f g+e h) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^3}-2 b d g \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )-\frac {1}{32} b f h \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3}-\frac {4 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {3 (a+b \arcsin (c x))^2}{b c^4}\right )-\frac {1}{4} b (e g+d h) \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right ) \]

[In]

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

[Out]

d*g*x*(a + b*ArcSin[c*x])^2 + ((e*g + d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + ((f*g + e*h)*x^3*(a + b*ArcSin[c*x])
^2)/3 + (f*h*x^4*(a + b*ArcSin[c*x])^2)/4 - (2*b*(f*g + e*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*d*g*(b*x - (Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x]))/c) - (b*f*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*(e*g + 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

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.60

method	result	size
parts	\(a^{2} \left (\frac {h f \,x^{4}}{4}+\frac {\left (e h +f g \right ) x^{3}}{3}+\frac {\left (d h +e g \right ) x^{2}}{2}+d g x \right )+\frac {b^{2} \left (\frac {h f \left (32 \arcsin \left (c x \right )^{2} x^{4} c^{4}+16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-4 c^{4} x^{4}+24 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -12 \arcsin \left (c x \right )^{2}-12 c^{2} x^{2}-9\right )}{128 c^{3}}+\frac {h e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+\frac {h d \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4 c}+\frac {g f \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+\frac {g e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4 c}+d g \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) h f \,x^{4}}{4}+\frac {c \arcsin \left (c x \right ) e h \,x^{3}}{3}+\frac {c \arcsin \left (c x \right ) x^{3} f g}{3}+\frac {c \arcsin \left (c x \right ) x^{2} d h}{2}+\frac {c \arcsin \left (c x \right ) e g \,x^{2}}{2}+c \arcsin \left (c x \right ) x d g -\frac {3 h f \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )-12 g \,c^{3} d \sqrt {-c^{2} x^{2}+1}+\left (4 e c h +4 c f g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+\left (6 d \,c^{2} h +6 e \,c^{2} g \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12 c^{3}}\right )}{c}\)	\(679\)
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {h f \,c^{4} x^{4}}{4}+\frac {\left (e c h +c f g \right ) c^{3} x^{3}}{3}+\frac {\left (d \,c^{2} h +e \,c^{2} g \right ) c^{2} x^{2}}{2}+g \,c^{4} d x \right )}{c^{3}}+\frac {b^{2} \left (c^{3} d g \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{2} e g \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c f g \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {c^{2} d h \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c e h \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {h f \left (32 \arcsin \left (c x \right )^{2} x^{4} c^{4}+16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-4 c^{4} x^{4}+24 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -12 \arcsin \left (c x \right )^{2}-12 c^{2} x^{2}-9\right )}{128}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) h f \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} e h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} f g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{2}}{2}+\arcsin \left (c x \right ) g \,c^{4} d x -\frac {h f \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+g \,c^{3} d \sqrt {-c^{2} x^{2}+1}-\frac {\left (4 e c h +4 c f g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{12}-\frac {\left (6 d \,c^{2} h +6 e \,c^{2} g \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12}\right )}{c^{3}}}{c}\)	\(709\)
default	\(\frac {\frac {a^{2} \left (\frac {h f \,c^{4} x^{4}}{4}+\frac {\left (e c h +c f g \right ) c^{3} x^{3}}{3}+\frac {\left (d \,c^{2} h +e \,c^{2} g \right ) c^{2} x^{2}}{2}+g \,c^{4} d x \right )}{c^{3}}+\frac {b^{2} \left (c^{3} d g \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{2} e g \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c f g \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {c^{2} d h \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c e h \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {h f \left (32 \arcsin \left (c x \right )^{2} x^{4} c^{4}+16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-4 c^{4} x^{4}+24 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -12 \arcsin \left (c x \right )^{2}-12 c^{2} x^{2}-9\right )}{128}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) h f \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} e h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} f g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{2}}{2}+\arcsin \left (c x \right ) g \,c^{4} d x -\frac {h f \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+g \,c^{3} d \sqrt {-c^{2} x^{2}+1}-\frac {\left (4 e c h +4 c f g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{12}-\frac {\left (6 d \,c^{2} h +6 e \,c^{2} g \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12}\right )}{c^{3}}}{c}\)	\(709\)

[In]

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

[Out]

a^2*(1/4*h*f*x^4+1/3*(e*h+f*g)*x^3+1/2*(d*h+e*g)*x^2+d*g*x)+b^2/c*(1/128*h*f*(32*arcsin(c*x)^2*x^4*c^4+16*(-c^
2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3-4*c^4*x^4+24*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-12*arcsin(c*x)^2-12*c^2*x^2
-9)/c^3+1/27*h*e*(9*c^3*x^3*arcsin(c*x)^2+6*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2-2*c^3*x^3+12*arcsin(c*x)*(-
c^2*x^2+1)^(1/2)-12*c*x)/c^2+1/4*h*d*(2*arcsin(c*x)^2*x^2*c^2+2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-arcsin(c*x)
^2-c^2*x^2)/c+1/27*g*f*(9*c^3*x^3*arcsin(c*x)^2+6*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2-2*c^3*x^3+12*arcsin(c
*x)*(-c^2*x^2+1)^(1/2)-12*c*x)/c^2+1/4*g*e*(2*arcsin(c*x)^2*x^2*c^2+2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-arcsi
n(c*x)^2-c^2*x^2)/c+d*g*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c*(1/4*c*arcsin(c*x)
*h*f*x^4+1/3*c*arcsin(c*x)*e*h*x^3+1/3*c*arcsin(c*x)*x^3*f*g+1/2*c*arcsin(c*x)*x^2*d*h+1/2*c*arcsin(c*x)*e*g*x
^2+c*arcsin(c*x)*x*d*g-1/12/c^3*(3*h*f*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*arcsin(
c*x))-12*g*c^3*d*(-c^2*x^2+1)^(1/2)+(4*c*e*h+4*c*f*g)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))
+(6*c^2*d*h+6*c^2*e*g)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.36 \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {27 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} f h x^{4} + 32 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} f g + {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} e h\right )} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} e g + {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d - 3 \, b^{2} c^{2} f\right )} h\right )} x^{2} + 9 \, {\left (24 \, b^{2} c^{4} f h x^{4} + 96 \, b^{2} c^{4} d g x - 24 \, b^{2} c^{2} e g + 32 \, {\left (b^{2} c^{4} f g + b^{2} c^{4} e h\right )} x^{3} + 48 \, {\left (b^{2} c^{4} e g + b^{2} c^{4} d h\right )} x^{2} - 3 \, {\left (8 \, b^{2} c^{2} d + 3 \, b^{2} f\right )} h\right )} \arcsin \left (c x\right )^{2} - 96 \, {\left (4 \, b^{2} c^{2} e h - {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d - 4 \, b^{2} c^{2} f\right )} g\right )} x + 18 \, {\left (24 \, a b c^{4} f h x^{4} + 96 \, a b c^{4} d g x - 24 \, a b c^{2} e g + 32 \, {\left (a b c^{4} f g + a b c^{4} e h\right )} x^{3} + 48 \, {\left (a b c^{4} e g + a b c^{4} d h\right )} x^{2} - 3 \, {\left (8 \, a b c^{2} d + 3 \, a b f\right )} h\right )} \arcsin \left (c x\right ) + 6 \, {\left (18 \, a b c^{3} f h x^{3} + 64 \, a b c e h + 32 \, {\left (a b c^{3} f g + a b c^{3} e h\right )} x^{2} + 32 \, {\left (9 \, a b c^{3} d + 2 \, a b c f\right )} g + 9 \, {\left (8 \, a b c^{3} e g + {\left (8 \, a b c^{3} d + 3 \, a b c f\right )} h\right )} x + {\left (18 \, b^{2} c^{3} f h x^{3} + 64 \, b^{2} c e h + 32 \, {\left (b^{2} c^{3} f g + b^{2} c^{3} e h\right )} x^{2} + 32 \, {\left (9 \, b^{2} c^{3} d + 2 \, b^{2} c f\right )} g + 9 \, {\left (8 \, b^{2} c^{3} e g + {\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} h\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{4}} \]

[In]

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

[Out]

1/864*(27*(8*a^2 - b^2)*c^4*f*h*x^4 + 32*((9*a^2 - 2*b^2)*c^4*f*g + (9*a^2 - 2*b^2)*c^4*e*h)*x^3 + 27*(8*(2*a^
2 - b^2)*c^4*e*g + (8*(2*a^2 - b^2)*c^4*d - 3*b^2*c^2*f)*h)*x^2 + 9*(24*b^2*c^4*f*h*x^4 + 96*b^2*c^4*d*g*x - 2
4*b^2*c^2*e*g + 32*(b^2*c^4*f*g + b^2*c^4*e*h)*x^3 + 48*(b^2*c^4*e*g + b^2*c^4*d*h)*x^2 - 3*(8*b^2*c^2*d + 3*b
^2*f)*h)*arcsin(c*x)^2 - 96*(4*b^2*c^2*e*h - (9*(a^2 - 2*b^2)*c^4*d - 4*b^2*c^2*f)*g)*x + 18*(24*a*b*c^4*f*h*x
^4 + 96*a*b*c^4*d*g*x - 24*a*b*c^2*e*g + 32*(a*b*c^4*f*g + a*b*c^4*e*h)*x^3 + 48*(a*b*c^4*e*g + a*b*c^4*d*h)*x
^2 - 3*(8*a*b*c^2*d + 3*a*b*f)*h)*arcsin(c*x) + 6*(18*a*b*c^3*f*h*x^3 + 64*a*b*c*e*h + 32*(a*b*c^3*f*g + a*b*c
^3*e*h)*x^2 + 32*(9*a*b*c^3*d + 2*a*b*c*f)*g + 9*(8*a*b*c^3*e*g + (8*a*b*c^3*d + 3*a*b*c*f)*h)*x + (18*b^2*c^3
*f*h*x^3 + 64*b^2*c*e*h + 32*(b^2*c^3*f*g + b^2*c^3*e*h)*x^2 + 32*(9*b^2*c^3*d + 2*b^2*c*f)*g + 9*(8*b^2*c^3*e
*g + (8*b^2*c^3*d + 3*b^2*c*f)*h)*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (416) = 832\).

Time = 0.51 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.49 \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**2*d*g*x + a**2*d*h*x**2/2 + a**2*e*g*x**2/2 + a**2*e*h*x**3/3 + a**2*f*g*x**3/3 + a**2*f*h*x**4/
4 + 2*a*b*d*g*x*asin(c*x) + a*b*d*h*x**2*asin(c*x) + a*b*e*g*x**2*asin(c*x) + 2*a*b*e*h*x**3*asin(c*x)/3 + 2*a
*b*f*g*x**3*asin(c*x)/3 + a*b*f*h*x**4*asin(c*x)/2 + 2*a*b*d*g*sqrt(-c**2*x**2 + 1)/c + a*b*d*h*x*sqrt(-c**2*x
**2 + 1)/(2*c) + a*b*e*g*x*sqrt(-c**2*x**2 + 1)/(2*c) + 2*a*b*e*h*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 2*a*b*f*g*
x**2*sqrt(-c**2*x**2 + 1)/(9*c) + a*b*f*h*x**3*sqrt(-c**2*x**2 + 1)/(8*c) - a*b*d*h*asin(c*x)/(2*c**2) - a*b*e
*g*asin(c*x)/(2*c**2) + 4*a*b*e*h*sqrt(-c**2*x**2 + 1)/(9*c**3) + 4*a*b*f*g*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*
a*b*f*h*x*sqrt(-c**2*x**2 + 1)/(16*c**3) - 3*a*b*f*h*asin(c*x)/(16*c**4) + b**2*d*g*x*asin(c*x)**2 - 2*b**2*d*
g*x + b**2*d*h*x**2*asin(c*x)**2/2 - b**2*d*h*x**2/4 + b**2*e*g*x**2*asin(c*x)**2/2 - b**2*e*g*x**2/4 + b**2*e
*h*x**3*asin(c*x)**2/3 - 2*b**2*e*h*x**3/27 + b**2*f*g*x**3*asin(c*x)**2/3 - 2*b**2*f*g*x**3/27 + b**2*f*h*x**
4*asin(c*x)**2/4 - b**2*f*h*x**4/32 + 2*b**2*d*g*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + b**2*d*h*x*sqrt(-c**2*x**2
+ 1)*asin(c*x)/(2*c) + b**2*e*g*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2*c) + 2*b**2*e*h*x**2*sqrt(-c**2*x**2 + 1)
*asin(c*x)/(9*c) + 2*b**2*f*g*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) + b**2*f*h*x**3*sqrt(-c**2*x**2 + 1)*a
sin(c*x)/(8*c) - b**2*d*h*asin(c*x)**2/(4*c**2) - b**2*e*g*asin(c*x)**2/(4*c**2) - 4*b**2*e*h*x/(9*c**2) - 4*b
**2*f*g*x/(9*c**2) - 3*b**2*f*h*x**2/(32*c**2) + 4*b**2*e*h*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 4*b**2*f
*g*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 3*b**2*f*h*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) - 3*b**2*f*
h*asin(c*x)**2/(32*c**4), Ne(c, 0)), (a**2*(d*g*x + d*h*x**2/2 + e*g*x**2/2 + e*h*x**3/3 + f*g*x**3/3 + f*h*x*
*4/4), True))

Maxima [F]

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

[In]

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

[Out]

1/4*a^2*f*h*x^4 + 1/3*a^2*f*g*x^3 + 1/3*a^2*e*h*x^3 + b^2*d*g*x*arcsin(c*x)^2 + 1/2*a^2*e*g*x^2 + 1/2*a^2*d*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 + 2/9*(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 + 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*d*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*e*h + 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*f*h - 2*b^2*d*g*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*g*x + 2*(c*x*a
rcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d*g/c + 1/12*(3*b^2*f*h*x^4 + 4*(b^2*f*g + b^2*e*h)*x^3 + 6*(b^2*e*g + b^
2*d*h)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/6*(3*b^2*c*f*h*x^4 + 4*(b^2*c*f*g + b^2
*c*e*h)*x^3 + 6*(b^2*c*e*g + b^2*c*d*h)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1))/(c^2*x^2 - 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (383) = 766\).

Time = 0.33 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.69 \[ \int (g+h x) \left (d+e x+f x^2\right ) (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*a^2*f*h*x^4 + 1/3*a^2*f*g*x^3 + 1/3*a^2*e*h*x^3 + b^2*d*g*x*arcsin(c*x)^2 + 2*a*b*d*g*x*arcsin(c*x) + 1/3*
(c^2*x^2 - 1)*b^2*f*g*x*arcsin(c*x)^2/c^2 + 1/3*(c^2*x^2 - 1)*b^2*e*h*x*arcsin(c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2
+ 1)*b^2*e*g*x*arcsin(c*x)/c + 1/2*sqrt(-c^2*x^2 + 1)*b^2*d*h*x*arcsin(c*x)/c + a^2*d*g*x - 2*b^2*d*g*x + 2/3*
(c^2*x^2 - 1)*a*b*f*g*x*arcsin(c*x)/c^2 + 2/3*(c^2*x^2 - 1)*a*b*e*h*x*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1)*b^2*
e*g*arcsin(c*x)^2/c^2 + 1/2*(c^2*x^2 - 1)*b^2*d*h*arcsin(c*x)^2/c^2 + 1/3*b^2*f*g*x*arcsin(c*x)^2/c^2 + 1/3*b^
2*e*h*x*arcsin(c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*a*b*e*g*x/c + 1/2*sqrt(-c^2*x^2 + 1)*a*b*d*h*x/c + 2*sqrt(-
c^2*x^2 + 1)*b^2*d*g*arcsin(c*x)/c - 1/8*(-c^2*x^2 + 1)^(3/2)*b^2*f*h*x*arcsin(c*x)/c^3 - 2/27*(c^2*x^2 - 1)*b
^2*f*g*x/c^2 - 2/27*(c^2*x^2 - 1)*b^2*e*h*x/c^2 + (c^2*x^2 - 1)*a*b*e*g*arcsin(c*x)/c^2 + (c^2*x^2 - 1)*a*b*d*
h*arcsin(c*x)/c^2 + 2/3*a*b*f*g*x*arcsin(c*x)/c^2 + 2/3*a*b*e*h*x*arcsin(c*x)/c^2 + 1/4*b^2*e*g*arcsin(c*x)^2/
c^2 + 1/4*b^2*d*h*arcsin(c*x)^2/c^2 + 1/4*(c^2*x^2 - 1)^2*b^2*f*h*arcsin(c*x)^2/c^4 + 2*sqrt(-c^2*x^2 + 1)*a*b
*d*g/c - 1/8*(-c^2*x^2 + 1)^(3/2)*a*b*f*h*x/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*f*g*arcsin(c*x)/c^3 - 2/9*(-c^2
*x^2 + 1)^(3/2)*b^2*e*h*arcsin(c*x)/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*b^2*f*h*x*arcsin(c*x)/c^3 + 1/2*(c^2*x^2 - 1
)*a^2*e*g/c^2 - 1/4*(c^2*x^2 - 1)*b^2*e*g/c^2 + 1/2*(c^2*x^2 - 1)*a^2*d*h/c^2 - 1/4*(c^2*x^2 - 1)*b^2*d*h/c^2
- 14/27*b^2*f*g*x/c^2 - 14/27*b^2*e*h*x/c^2 + 1/2*a*b*e*g*arcsin(c*x)/c^2 + 1/2*a*b*d*h*arcsin(c*x)/c^2 + 1/2*
(c^2*x^2 - 1)^2*a*b*f*h*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)*b^2*f*h*arcsin(c*x)^2/c^4 - 2/9*(-c^2*x^2 + 1)^(3/
2)*a*b*f*g/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*e*h/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*a*b*f*h*x/c^3 + 2/3*sqrt(-c^2*
x^2 + 1)*b^2*f*g*arcsin(c*x)/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*e*h*arcsin(c*x)/c^3 - 1/8*b^2*e*g/c^2 - 1/8*b^2*
d*h/c^2 - 1/32*(c^2*x^2 - 1)^2*b^2*f*h/c^4 + (c^2*x^2 - 1)*a*b*f*h*arcsin(c*x)/c^4 + 5/32*b^2*f*h*arcsin(c*x)^
2/c^4 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*f*g/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e*h/c^3 - 5/32*(c^2*x^2 - 1)*b^2*f*h/c
^4 + 5/16*a*b*f*h*arcsin(c*x)/c^4 - 17/256*b^2*f*h/c^4

Mupad [F(-1)]

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

[In]

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

[Out]

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