3.116 \(\int (g+h x)^2 (d+e x+f x^2) (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=701 \[ \frac{2 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c}+\frac{4 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c^3}+\frac{b g x \sqrt{1-c^2 x^2} (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac{g (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{2 b d g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b h x^3 \sqrt{1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b h x \sqrt{1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac{3 h (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{16 b f h^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac{1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 x \left (h (d h+2 e g)+f g^2\right )}{9 c^2}-\frac{3 b^2 h x^2 (e h+2 f g)}{32 c^2}-\frac{8 b^2 f h^2 x^3}{225 c^2}-\frac{16 b^2 f h^2 x}{75 c^4}-\frac{2}{27} b^2 x^3 \left (h (d h+2 e g)+f g^2\right )-\frac{1}{4} b^2 g x^2 (2 d h+e g)-2 b^2 d g^2 x-\frac{1}{32} b^2 h x^4 (e h+2 f g)-\frac{2}{125} b^2 f h^2 x^5 \]

[Out]

-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*ArcSin[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

________________________________________________________________________________________

Rubi [A]  time = 1.12291, antiderivative size = 701, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4751, 4619, 4677, 8, 4627, 4707, 4641, 30} \[ \frac{2 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c}+\frac{4 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c^3}+\frac{b g x \sqrt{1-c^2 x^2} (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac{g (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{2 b d g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b h x^3 \sqrt{1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b h x \sqrt{1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac{3 h (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{16 b f h^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac{1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 x \left (h (d h+2 e g)+f g^2\right )}{9 c^2}-\frac{3 b^2 h x^2 (e h+2 f g)}{32 c^2}-\frac{8 b^2 f h^2 x^3}{225 c^2}-\frac{16 b^2 f h^2 x}{75 c^4}-\frac{2}{27} b^2 x^3 \left (h (d h+2 e g)+f g^2\right )-\frac{1}{4} b^2 g x^2 (2 d h+e g)-2 b^2 d g^2 x-\frac{1}{32} b^2 h x^4 (e h+2 f g)-\frac{2}{125} b^2 f h^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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*ArcSin[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

Rule 4751

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]

Rule 4619

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 8

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

Rule 4627

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 4707

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

Rule 4641

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

Rule 30

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

Rubi steps

\begin{align*} \int (g+h x)^2 \left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d g^2 \left (a+b \sin ^{-1}(c x)\right )^2+g (e g+2 d h) x \left (a+b \sin ^{-1}(c x)\right )^2+\left (f g^2+h (2 e g+d h)\right ) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+h (2 f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+f h^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=\left (d g^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (f h^2\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(g (e g+2 d h)) \int x \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(h (2 f g+e h)) \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (f g^2+h (2 e g+d h)\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d g^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{5} \left (2 b c f h^2\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-(b c g (e g+2 d h)) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} (b c h (2 f g+e h)) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} \left (2 b c \left (f g^2+h (2 e g+d h)\right )\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b g (e g+2 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{b h (2 f g+e h) x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d g^2\right ) \int 1 \, dx-\frac{1}{25} \left (2 b^2 f h^2\right ) \int x^4 \, dx-\frac{\left (8 b f h^2\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 c}-\frac{1}{2} \left (b^2 g (e g+2 d h)\right ) \int x \, dx-\frac{(b g (e g+2 d h)) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c}-\frac{1}{8} \left (b^2 h (2 f g+e h)\right ) \int x^3 \, dx-\frac{(3 b h (2 f g+e h)) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{8 c}-\frac{1}{9} \left (2 b^2 \left (f g^2+h (2 e g+d h)\right )\right ) \int x^2 \, dx-\frac{\left (4 b \left (f g^2+h (2 e g+d h)\right )\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{9 c}\\ &=-2 b^2 d g^2 x-\frac{1}{4} b^2 g (e g+2 d h) x^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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{b g (e g+2 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{3 b h (2 f g+e h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{b h (2 f g+e h) x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac{g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (16 b f h^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 c^3}-\frac{\left (8 b^2 f h^2\right ) \int x^2 \, dx}{75 c^2}-\frac{(3 b h (2 f g+e h)) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c^3}-\frac{\left (3 b^2 h (2 f g+e h)\right ) \int x \, dx}{16 c^2}-\frac{\left (4 b^2 \left (f g^2+h (2 e g+d h)\right )\right ) \int 1 \, dx}{9 c^2}\\ &=-2 b^2 d g^2 x-\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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{16 b f h^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{b g (e g+2 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{3 b h (2 f g+e h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{b h (2 f g+e h) x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac{g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 h (2 f g+e h) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (16 b^2 f h^2\right ) \int 1 \, dx}{75 c^4}\\ &=-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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{16 b f h^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{b g (e g+2 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{3 b h (2 f g+e h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{b h (2 f g+e h) x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac{g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 h (2 f g+e h) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.579099, size = 534, normalized size = 0.76 \[ -\frac{2 b \left (-3 a \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{27 c^3}-\frac{1}{4} b g (2 d h+e g) \left (-\frac{2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-2 b d g^2 \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{1}{32} b h (e h+2 f g) \left (-\frac{4 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )^2}{b c^4}+\frac{3 b x^2}{c^2}+b x^4\right )-\frac{2 b f h^2 \left (-15 a \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )+b c x \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 b \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right ) \sin ^{-1}(c x)\right )}{1125 c^5}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac{1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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]))/(1
125*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

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Maple [B]  time = 0.177, size = 1633, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^4*(1/5*h^2*f*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*(1/32*h^2*c*e*(8*arcsin(c*x)^2*c^4*x^4+4*arcsin(c*x)*
(-c^2*x^2+1)^(1/2)*c^3*x^3-16*arcsin(c*x)^2*c^2*x^2-c^4*x^4-10*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+5*arcsin(c*x
)^2+5*c^2*x^2-4)+1/16*c*g*h*f*(8*arcsin(c*x)^2*c^4*x^4+4*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-16*arcsin(c*x)
^2*c^2*x^2-c^4*x^4-10*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+5*arcsin(c*x)^2+5*c^2*x^2-4)+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)+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)+1/3375*h^2*f*(675*arcsin(c*x)^2*c^5*x^5+270*ar
csin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-2250*c^3*x^3*arcsin(c*x)^2-54*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^2*x^2+3375*arcsin(c*x)^2*c*x+380*c^3*x^3+4470*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4470*c*x)+1/27*h^2*c^2*d*(9*c^
3*x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c
^2*x^2+1)^(1/2)+42*c*x)+2/27*c^2*g*h*e*(9*c^3*x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*ar
csin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+1/27*c^2*g^2*f*(9*c^3*x^3*arcsin(c*x)^2+6*
arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x
)+c^4*g^2*d*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*h^2*c*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)+1/2*c*g*h*f*(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*h^2*f*(9*c^3*x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1
/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+h^2*c^2*d*(arcsin(c*x)^2*
c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+2*c^2*g*h*e*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1
/2))+c^2*g^2*f*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+h^2*f*(arcsin(c*x)^2*c*x-2*c*x+2*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c^4*(1/5*arcsin(c*x)*h^2*f*c^5*x^5+1/4*arcsin(c*x)*c^5*x^4*e*h^2+1/2*arcsi
n(c*x)*c^5*x^4*f*g*h+1/3*arcsin(c*x)*c^5*x^3*d*h^2+2/3*arcsin(c*x)*c^5*x^3*e*g*h+1/3*arcsin(c*x)*c^5*x^3*f*g^2
+arcsin(c*x)*c^5*x^2*d*g*h+1/2*arcsin(c*x)*c^5*x^2*e*g^2+arcsin(c*x)*c^5*g^2*d*x-1/5*h^2*f*(-1/5*c^4*x^4*(-c^2
*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/60*(15*c*e*h^2+30*c*f*g*h)*(-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))-1/60*(20*c^2*d*h^2+40*c^2*e*g*h+20*c^2*f*g^
2)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-1/60*(60*c^3*d*g*h+30*c^3*e*g^2)*(-1/2*c*x*(-c^2*x
^2+1)^(1/2)+1/2*arcsin(c*x))+c^4*g^2*d*(-c^2*x^2+1)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*e*h^2 + 2/75*(15*x^5*ar
csin(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*g*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)

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Fricas [A]  time = 3.70757, size = 2303, normalized size = 3.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*b^2*c^2*d + 12*b^2*f)*h^2 + 32*(25*b^2*c^4*f*g^2 + 50*b^2*c^4*
e*g*h + (25*b^2*c^4*d + 12*b^2*c^2*f)*h^2)*x^2 + 225*(8*b^2*c^4*e*g^2 + 3*b^2*c^2*e*h^2 + 2*(8*b^2*c^4*d + 3*b
^2*c^2*f)*g*h)*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 12.9761, size = 1935, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*sqrt(-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**2*asin(c*x)**2/2 - b**2*e*g**2*x**2/4 + 2*b**2*e*g*h*x**3*asin
(c*x)**2/3 - 4*b**2*e*g*h*x**3/27 + b**2*e*h**2*x**4*asin(c*x)**2/4 - b**2*e*h**2*x**4/32 + b**2*f*g**2*x**3*a
sin(c*x)**2/3 - 2*b**2*f*g**2*x**3/27 + b**2*f*g*h*x**4*asin(c*x)**2/2 - b**2*f*g*h*x**4/16 + b**2*f*h**2*x**5
*asin(c*x)**2/5 - 2*b**2*f*h**2*x**5/125 + 2*b**2*d*g**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + b**2*d*g*h*x*sqrt(
-c**2*x**2 + 1)*asin(c*x)/c + 2*b**2*d*h**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) + b**2*e*g**2*x*sqrt(-c*
*2*x**2 + 1)*asin(c*x)/(2*c) + 4*b**2*e*g*h*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) + b**2*e*h**2*x**3*sqrt(
-c**2*x**2 + 1)*asin(c*x)/(8*c) + 2*b**2*f*g**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) + b**2*f*g*h*x**3*sq
rt(-c**2*x**2 + 1)*asin(c*x)/(4*c) + 2*b**2*f*h**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) - b**2*d*g*h*asi
n(c*x)**2/(2*c**2) - 4*b**2*d*h**2*x/(9*c**2) - b**2*e*g**2*asin(c*x)**2/(4*c**2) - 8*b**2*e*g*h*x/(9*c**2) -
3*b**2*e*h**2*x**2/(32*c**2) - 4*b**2*f*g**2*x/(9*c**2) - 3*b**2*f*g*h*x**2/(16*c**2) - 8*b**2*f*h**2*x**3/(22
5*c**2) + 4*b**2*d*h**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 8*b**2*e*g*h*sqrt(-c**2*x**2 + 1)*asin(c*x)/
(9*c**3) + 3*b**2*e*h**2*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) + 4*b**2*f*g**2*sqrt(-c**2*x**2 + 1)*asin(
c*x)/(9*c**3) + 3*b**2*f*g*h*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c**3) + 8*b**2*f*h**2*x**2*sqrt(-c**2*x**2 +
1)*asin(c*x)/(75*c**3) - 3*b**2*e*h**2*asin(c*x)**2/(32*c**4) - 3*b**2*f*g*h*asin(c*x)**2/(16*c**4) - 16*b**2*
f*h**2*x/(75*c**4) + 16*b**2*f*h**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**5), Ne(c, 0)), (a**2*(d*g**2*x + d*g
*h*x**2 + d*h**2*x**3/3 + e*g**2*x**2/2 + 2*e*g*h*x**3/3 + e*h**2*x**4/4 + f*g**2*x**3/3 + f*g*h*x**4/2 + f*h*
*2*x**5/5), True))

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Giac [B]  time = 1.39061, size = 3055, normalized size = 4.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*f*h^2*x^5 + 1/3*a^2*f*g^2*x^3 + 1/3*a^2*d*h^2*x^3 + b^2*d*g^2*x*arcsin(c*x)^2 + 2/3*a^2*g*h*x^3*e + 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 + 1/3*(c^2*x^2 - 1)*b^2*d*h^2*x*arcs
in(c*x)^2/c^2 + 2/3*(c^2*x^2 - 1)*b^2*g*h*x*arcsin(c*x)^2*e/c^2 + sqrt(-c^2*x^2 + 1)*b^2*d*g*h*x*arcsin(c*x)/c
+ 1/2*sqrt(-c^2*x^2 + 1)*b^2*g^2*x*arcsin(c*x)*e/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 + 2/3*(c^2*x^2 - 1)*a*b*d*h^2*x*arcsin(c*x)/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 + 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*ar
csin(c*x)^2/c^4 + 4/3*(c^2*x^2 - 1)*a*b*g*h*x*arcsin(c*x)*e/c^2 + 1/2*(c^2*x^2 - 1)*b^2*g^2*arcsin(c*x)^2*e/c^
2 + 2/3*b^2*g*h*x*arcsin(c*x)^2*e/c^2 + 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*arcs
in(c*x)/c - 1/4*(-c^2*x^2 + 1)^(3/2)*b^2*f*g*h*x*arcsin(c*x)/c^3 + 1/2*sqrt(-c^2*x^2 + 1)*a*b*g^2*x*e/c - 1/8*
(-c^2*x^2 + 1)^(3/2)*b^2*h^2*x*arcsin(c*x)*e/c^3 - 2/27*(c^2*x^2 - 1)*b^2*f*g^2*x/c^2 - 2/27*(c^2*x^2 - 1)*b^2
*d*h^2*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 + 2/3*a*b*d*h^2*x*a
rcsin(c*x)/c^2 + 2/5*(c^2*x^2 - 1)^2*a*b*f*h^2*x*arcsin(c*x)/c^4 + 1/2*b^2*d*g*h*arcsin(c*x)^2/c^2 + 1/2*(c^2*
x^2 - 1)^2*b^2*f*g*h*arcsin(c*x)^2/c^4 + 2/5*(c^2*x^2 - 1)*b^2*f*h^2*x*arcsin(c*x)^2/c^4 - 4/27*(c^2*x^2 - 1)*
b^2*g*h*x*e/c^2 + (c^2*x^2 - 1)*a*b*g^2*arcsin(c*x)*e/c^2 + 4/3*a*b*g*h*x*arcsin(c*x)*e/c^2 + 1/4*b^2*g^2*arcs
in(c*x)^2*e/c^2 + 1/4*(c^2*x^2 - 1)^2*b^2*h^2*arcsin(c*x)^2*e/c^4 + 2*sqrt(-c^2*x^2 + 1)*a*b*d*g^2/c - 1/4*(-c
^2*x^2 + 1)^(3/2)*a*b*f*g*h*x/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*f*g^2*arcsin(c*x)/c^3 - 2/9*(-c^2*x^2 + 1)^(3
/2)*b^2*d*h^2*arcsin(c*x)/c^3 + 5/8*sqrt(-c^2*x^2 + 1)*b^2*f*g*h*x*arcsin(c*x)/c^3 - 1/8*(-c^2*x^2 + 1)^(3/2)*
a*b*h^2*x*e/c^3 - 4/9*(-c^2*x^2 + 1)^(3/2)*b^2*g*h*arcsin(c*x)*e/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*b^2*h^2*x*arcsi
n(c*x)*e/c^3 + (c^2*x^2 - 1)*a^2*d*g*h/c^2 - 1/2*(c^2*x^2 - 1)*b^2*d*g*h/c^2 - 14/27*b^2*f*g^2*x/c^2 - 14/27*b
^2*d*h^2*x/c^2 - 2/125*(c^2*x^2 - 1)^2*b^2*f*h^2*x/c^4 + a*b*d*g*h*arcsin(c*x)/c^2 + (c^2*x^2 - 1)^2*a*b*f*g*h
*arcsin(c*x)/c^4 + 4/5*(c^2*x^2 - 1)*a*b*f*h^2*x*arcsin(c*x)/c^4 + (c^2*x^2 - 1)*b^2*f*g*h*arcsin(c*x)^2/c^4 +
1/5*b^2*f*h^2*x*arcsin(c*x)^2/c^4 + 1/2*(c^2*x^2 - 1)*a^2*g^2*e/c^2 - 1/4*(c^2*x^2 - 1)*b^2*g^2*e/c^2 - 28/27
*b^2*g*h*x*e/c^2 + 1/2*a*b*g^2*arcsin(c*x)*e/c^2 + 1/2*(c^2*x^2 - 1)^2*a*b*h^2*arcsin(c*x)*e/c^4 + 1/2*(c^2*x^
2 - 1)*b^2*h^2*arcsin(c*x)^2*e/c^4 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*f*g^2/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*d*h
^2/c^3 + 5/8*sqrt(-c^2*x^2 + 1)*a*b*f*g*h*x/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*f*g^2*arcsin(c*x)/c^3 + 2/3*sqrt(
-c^2*x^2 + 1)*b^2*d*h^2*arcsin(c*x)/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*f*h^2*arcsin(c*x)/c^5 -
4/9*(-c^2*x^2 + 1)^(3/2)*a*b*g*h*e/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*a*b*h^2*x*e/c^3 + 4/3*sqrt(-c^2*x^2 + 1)*b^2*
g*h*arcsin(c*x)*e/c^3 - 1/4*b^2*d*g*h/c^2 + 1/2*(c^2*x^2 - 1)^2*a^2*f*g*h/c^4 - 1/16*(c^2*x^2 - 1)^2*b^2*f*g*h
/c^4 - 76/1125*(c^2*x^2 - 1)*b^2*f*h^2*x/c^4 + 2*(c^2*x^2 - 1)*a*b*f*g*h*arcsin(c*x)/c^4 + 2/5*a*b*f*h^2*x*arc
sin(c*x)/c^4 + 5/16*b^2*f*g*h*arcsin(c*x)^2/c^4 - 1/8*b^2*g^2*e/c^2 + 1/4*(c^2*x^2 - 1)^2*a^2*h^2*e/c^4 - 1/32
*(c^2*x^2 - 1)^2*b^2*h^2*e/c^4 + (c^2*x^2 - 1)*a*b*h^2*arcsin(c*x)*e/c^4 + 5/32*b^2*h^2*arcsin(c*x)^2*e/c^4 +
2/3*sqrt(-c^2*x^2 + 1)*a*b*f*g^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*d*h^2/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x
^2 + 1)*a*b*f*h^2/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b^2*f*h^2*arcsin(c*x)/c^5 + 4/3*sqrt(-c^2*x^2 + 1)*a*b*g*h*e
/c^3 + (c^2*x^2 - 1)*a^2*f*g*h/c^4 - 5/16*(c^2*x^2 - 1)*b^2*f*g*h/c^4 - 298/1125*b^2*f*h^2*x/c^4 + 5/8*a*b*f*g
*h*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)*a^2*h^2*e/c^4 - 5/32*(c^2*x^2 - 1)*b^2*h^2*e/c^4 + 5/16*a*b*h^2*arcsin(
c*x)*e/c^4 - 4/15*(-c^2*x^2 + 1)^(3/2)*a*b*f*h^2/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*b^2*f*h^2*arcsin(c*x)/c^5 - 17/1
28*b^2*f*g*h/c^4 - 17/256*b^2*h^2*e/c^4 + 2/5*sqrt(-c^2*x^2 + 1)*a*b*f*h^2/c^5