3.115 \(\int (g+h x)^3 (d+e x+f x^2) (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=1016 \[ \text{result too large to display} \]

[Out]

-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)*Sqrt[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*ArcSin[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) + d*g^3*x*(a + b*ArcSin[c*x])^2 + (g^2*(e*g + 3*d*h)*x^2*(a + b*A
rcSin[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

________________________________________________________________________________________

Rubi [A]  time = 1.58017, antiderivative size = 1016, normalized size of antiderivative = 1., number of steps used = 35, 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{1}{108} b^2 f h^3 x^6+\frac{1}{6} f h^3 \left (a+b \sin ^{-1}(c x)\right )^2 x^6+\frac{1}{5} h^2 (3 f g+e h) \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right ) 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 ) \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right ) x^4}{25 c}+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right ) 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} \left (a+b \sin ^{-1}(c x)\right ) 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) \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right ) 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} \left (a+b \sin ^{-1}(c x)\right ) x^2}{9 c}-2 b^2 d g^3 x+d g^3 \left (a+b \sin ^{-1}(c x)\right )^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} \left (a+b \sin ^{-1}(c x)\right ) x}{48 c^5}+\frac{b g^2 (e g+3 d h) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) 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} \left (a+b \sin ^{-1}(c x)\right ) x}{16 c^3}-\frac{5 f h^3 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^6}-\frac{g^2 (e g+3 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 h \left (3 f g^2+h (3 e g+d h)\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac{2 b d g^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{16 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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)*Sqrt[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*ArcSin[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) + d*g^3*x*(a + b*ArcSin[c*x])^2 + (g^2*(e*g + 3*d*h)*x^2*(a + b*A
rcSin[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

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)^3 \left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d g^3 \left (a+b \sin ^{-1}(c x)\right )^2+g^2 (e g+3 d h) x \left (a+b \sin ^{-1}(c x)\right )^2+g \left (f g^2+3 h (e g+d h)\right ) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+h \left (3 f g^2+h (3 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+h^2 (3 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+f h^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=\left (d g^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (f h^3\right ) \int x^5 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (g^2 (e g+3 d h)\right ) \int x \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (h^2 (3 f g+e h)\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (g \left (f g^2+3 h (e g+d h)\right )\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g^2 (e g+3 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 (3 f g+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d g^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} \left (b c f h^3\right ) \int \frac{x^6 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\left (b c g^2 (e g+3 d h)\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{5} \left (2 b c h^2 (3 f g+e h)\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} \left (2 b c g \left (f g^2+3 h (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{1}{2} \left (b c h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d g^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b g^2 (e g+3 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{2 b g \left (f g^2+3 h (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 \left (3 f g^2+h (3 e g+d h)\right ) x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b h^2 (3 f g+e h) x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{b f h^3 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g^2 (e g+3 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 (3 f g+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d g^3\right ) \int 1 \, dx-\frac{1}{18} \left (b^2 f h^3\right ) \int x^5 \, dx-\frac{\left (5 b f h^3\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{18 c}-\frac{1}{2} \left (b^2 g^2 (e g+3 d h)\right ) \int x \, dx-\frac{\left (b g^2 (e g+3 d h)\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c}-\frac{1}{25} \left (2 b^2 h^2 (3 f g+e h)\right ) \int x^4 \, dx-\frac{\left (8 b h^2 (3 f g+e h)\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 c}-\frac{1}{9} \left (2 b^2 g \left (f g^2+3 h (e g+d h)\right )\right ) \int x^2 \, dx-\frac{\left (4 b g \left (f g^2+3 h (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}-\frac{1}{8} \left (b^2 h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int x^3 \, dx-\frac{\left (3 b h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{8 c}\\ &=-2 b^2 d g^3 x-\frac{1}{4} b^2 g^2 (e g+3 d h) x^2-\frac{2}{27} b^2 g \left (f g^2+3 h (e g+d h)\right ) x^3-\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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b g \left (f g^2+3 h (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^2 (e g+3 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b h^2 (3 f g+e h) x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{5 b f h^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b h^2 (3 f g+e h) x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{b f h^3 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{g^2 (e g+3 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g^2 (e g+3 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 (3 f g+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (5 b f h^3\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{24 c^3}-\frac{\left (5 b^2 f h^3\right ) \int x^3 \, dx}{72 c^2}-\frac{\left (16 b h^2 (3 f g+e h)\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 h^2 (3 f g+e h)\right ) \int x^2 \, dx}{75 c^2}-\frac{\left (4 b^2 g \left (f g^2+3 h (e g+d h)\right )\right ) \int 1 \, dx}{9 c^2}-\frac{\left (3 b h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c^3}-\frac{\left (3 b^2 h \left (3 f g^2+h (3 e g+d h)\right )\right ) \int x \, dx}{16 c^2}\\ &=-2 b^2 d g^3 x-\frac{4 b^2 g \left (f g^2+3 h (e g+d h)\right ) x}{9 c^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^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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{16 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{5 b f h^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c^5}+\frac{b g^2 (e g+3 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b h^2 (3 f g+e h) x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{5 b f h^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b h^2 (3 f g+e h) x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{b f h^3 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{g^2 (e g+3 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 h \left (3 f g^2+h (3 e g+d h)\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g^2 (e g+3 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 (3 f g+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (5 b f h^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{48 c^5}-\frac{\left (5 b^2 f h^3\right ) \int x \, dx}{48 c^4}-\frac{\left (16 b^2 h^2 (3 f g+e h)\right ) \int 1 \, dx}{75 c^4}\\ &=-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} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{16 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{5 b f h^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c^5}+\frac{b g^2 (e g+3 d h) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{8 b h^2 (3 f g+e h) x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{5 b f h^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{2 b h^2 (3 f g+e h) x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{b f h^3 x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{5 f h^3 \left (a+b \sin ^{-1}(c x)\right )^2}{96 c^6}-\frac{g^2 (e g+3 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 h \left (3 f g^2+h (3 e g+d h)\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} g^2 (e g+3 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 (3 f g+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 1.0357, size = 734, normalized size = 0.72 \[ -\frac{f h^3 \left (45 a^2-6 a b c x \sqrt{1-c^2 x^2} \left (8 c^4 x^4+10 c^2 x^2+15\right )-6 b \sin ^{-1}(c x) \left (b c x \sqrt{1-c^2 x^2} \left (8 c^4 x^4+10 c^2 x^2+15\right )-15 a\right )+b^2 c^2 x^2 \left (8 c^4 x^4+15 c^2 x^2+45\right )+45 b^2 \sin ^{-1}(c x)^2\right )}{864 c^6}-\frac{2 b g \left (-3 a \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right ) \left (3 h (d h+e g)+f g^2\right )}{27 c^3}-\frac{1}{32} b h \left (-\frac{4 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )^2}{b c^4}+\frac{3 b x^2}{c^2}+b x^4\right ) \left (h (d h+3 e g)+3 f g^2\right )-\frac{1}{4} b g^2 (3 d h+e g) \left (-\frac{2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-2 b d g^3 \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{2 b h^2 (e h+3 f g) \left (-15 a \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )+b c x \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 b \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right ) \sin ^{-1}(c x)\right )}{1125 c^5}+\frac{1}{4} h x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+3 e g)+3 f g^2\right )+\frac{1}{3} g x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (3 h (d h+e g)+f g^2\right )+\frac{1}{2} g^2 x^2 (3 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} h^2 x^5 (e h+3 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} f h^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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*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^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

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Maple [B]  time = 0.268, size = 2600, normalized size = 2.6 \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)^3*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x)

[Out]

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*g^3*d*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+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)+1/4*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)+1/9*c^3*g*h^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)+1/9*c^
3*g^2*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*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)+1/27*c^3*g^3*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)+1/32*h^3*c^2*d*(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)+3/32*c^2*g*h^2*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)+3/32*c^2*g^2*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)+1/3375*h^3*c*e*(675*arcsin(c*x
)^2*c^5*x^5+270*arcsin(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/
1125*c*g*h^2*f*(675*arcsin(c*x)^2*c^5*x^5+270*arcsin(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)+1/864*h^3*f*(144*arcsin(c*x)^2*c^6*x^6+48*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5*x^5-4
32*arcsin(c*x)^2*c^4*x^4-8*c^6*x^6-156*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3+432*arcsin(c*x)^2*c^2*x^2+39*c^4
*x^4+198*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-99*arcsin(c*x)^2-99*c^2*x^2+68)+3*c^3*g*h^2*d*(arcsin(c*x)^2*c*x-2
*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+3*c^3*g^2*h*e*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
)+c^3*g^3*f*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*h^3*c^2*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)+3/4*c^2*g*h^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)+3/4*c^2*g^2*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^3*c*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*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)+2/9*c*g*h^2*f*(9*c^3*x^3*a
rcsin(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)+1/16*h^3*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)+h^3*c*e*(arcsin(c*x)^2*c*x
-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+3*c*g*h^2*f*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
)+1/4*h^3*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*a*b/c^5*(1
/6*arcsin(c*x)*h^3*f*c^6*x^6+1/5*arcsin(c*x)*c^6*x^5*e*h^3+3/5*arcsin(c*x)*c^6*x^5*f*g*h^2+1/4*arcsin(c*x)*c^6
*x^4*d*h^3+3/4*arcsin(c*x)*c^6*x^4*e*g*h^2+3/4*arcsin(c*x)*c^6*x^4*f*g^2*h+arcsin(c*x)*c^6*x^3*d*g*h^2+arcsin(
c*x)*c^6*x^3*e*g^2*h+1/3*arcsin(c*x)*c^6*x^3*f*g^3+3/2*arcsin(c*x)*c^6*x^2*d*g^2*h+1/2*arcsin(c*x)*c^6*x^2*e*g
^3+arcsin(c*x)*c^6*g^3*d*x-1/6*h^3*f*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^2+1)^(1/2)-5/16*c*x
*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))-1/60*(12*c*e*h^3+36*c*f*g*h^2)*(-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^2*d*h^3+45*c^2*e*g*h^2+45*c^2*f*g^2*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*(60*c^3*d*g*h^2+60*c^3*e*g^2*h+20*c^3*f
*g^3)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-1/60*(90*c^4*d*g^2*h+30*c^4*e*g^3)*(-1/2*c*x*(-
c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^5*g^3*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)^3*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

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^2*x/sqrt(c^2))
/(sqrt(c^2)*c^2)))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^2
)))*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^2*x/sq
rt(c^2))/(sqrt(c^2)*c^4))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*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*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*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*sq
rt(-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*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c)*a*b*f*h^3 - 2*
b^2*d*g^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*g^3*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*
d*g^3/c + 1/60*(10*b^2*f*h^3*x^6 + 12*(3*b^2*f*g*h^2 + b^2*e*h^3)*x^5 + 15*(3*b^2*f*g^2*h + 3*b^2*e*g*h^2 + b^
2*d*h^3)*x^4 + 20*(b^2*f*g^3 + 3*b^2*e*g^2*h + 3*b^2*d*g*h^2)*x^3 + 30*(b^2*e*g^3 + 3*b^2*d*g^2*h)*x^2)*arctan
2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/30*(10*b^2*c*f*h^3*x^6 + 12*(3*b^2*c*f*g*h^2 + b^2*c*e*h^
3)*x^5 + 15*(3*b^2*c*f*g^2*h + 3*b^2*c*e*g*h^2 + b^2*c*d*h^3)*x^4 + 20*(b^2*c*f*g^3 + 3*b^2*c*e*g^2*h + 3*b^2*
c*d*g*h^2)*x^3 + 30*(b^2*c*e*g^3 + 3*b^2*c*d*g^2*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 = 4.06177, size = 3374, normalized size = 3.32 \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)^3*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

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 + 1
20*(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 + 12*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*h^3*x^5 + 4800*a*b*c^3*e*g^2*h + 768*a*b*c*e*h^3 + 288*(3*a*b*c
^5*f*g*h^2 + a*b*c^5*e*h^3)*x^4 + 800*(9*a*b*c^5*d + 2*a*b*c^3*f)*g^3 + 192*(25*a*b*c^3*d + 12*a*b*c*f)*g*h^2
+ 50*(27*a*b*c^5*f*g^2*h + 27*a*b*c^5*e*g*h^2 + (9*a*b*c^5*d + 5*a*b*c^3*f)*h^3)*x^3 + 32*(25*a*b*c^5*f*g^3 +
75*a*b*c^5*e*g^2*h + 12*a*b*c^3*e*h^3 + 3*(25*a*b*c^5*d + 12*a*b*c^3*f)*g*h^2)*x^2 + 75*(24*a*b*c^5*e*g^3 + 27
*a*b*c^3*e*g*h^2 + 9*(8*a*b*c^5*d + 3*a*b*c^3*f)*g^2*h + (9*a*b*c^3*d + 5*a*b*c*f)*h^3)*x + (200*b^2*c^5*f*h^3
*x^5 + 4800*b^2*c^3*e*g^2*h + 768*b^2*c*e*h^3 + 288*(3*b^2*c^5*f*g*h^2 + b^2*c^5*e*h^3)*x^4 + 800*(9*b^2*c^5*d
+ 2*b^2*c^3*f)*g^3 + 192*(25*b^2*c^3*d + 12*b^2*c*f)*g*h^2 + 50*(27*b^2*c^5*f*g^2*h + 27*b^2*c^5*e*g*h^2 + (9
*b^2*c^5*d + 5*b^2*c^3*f)*h^3)*x^3 + 32*(25*b^2*c^5*f*g^3 + 75*b^2*c^5*e*g^2*h + 12*b^2*c^3*e*h^3 + 3*(25*b^2*
c^5*d + 12*b^2*c^3*f)*g*h^2)*x^2 + 75*(24*b^2*c^5*e*g^3 + 27*b^2*c^3*e*g*h^2 + 9*(8*b^2*c^5*d + 3*b^2*c^3*f)*g
^2*h + (9*b^2*c^3*d + 5*b^2*c*f)*h^3)*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^6

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Sympy [A]  time = 25.7425, size = 2992, normalized size = 2.94 \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)**3*(f*x**2+e*x+d)*(a+b*asin(c*x))**2,x)

[Out]

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*asin(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*sqrt(-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*sqr
t(-c**2*x**2 + 1)/(25*c) + 2*a*b*f*g**3*x**2*sqrt(-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*asin(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*h**3*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 4*a*b*f*g**3*sqrt(-c
**2*x**2 + 1)/(9*c**3) + 9*a*b*f*g**2*h*x*sqrt(-c**2*x**2 + 1)/(16*c**3) + 8*a*b*f*g*h**2*x**2*sqrt(-c**2*x**2
+ 1)/(25*c**3) + 5*a*b*f*h**3*x**3*sqrt(-c**2*x**2 + 1)/(72*c**3) - 3*a*b*d*h**3*asin(c*x)/(16*c**4) - 9*a*b*
e*g*h**2*asin(c*x)/(16*c**4) - 9*a*b*f*g**2*h*asin(c*x)/(16*c**4) + 16*a*b*e*h**3*sqrt(-c**2*x**2 + 1)/(75*c**
5) + 16*a*b*f*g*h**2*sqrt(-c**2*x**2 + 1)/(25*c**5) + 5*a*b*f*h**3*x*sqrt(-c**2*x**2 + 1)/(48*c**5) - 5*a*b*f*
h**3*asin(c*x)/(48*c**6) + b**2*d*g**3*x*asin(c*x)**2 - 2*b**2*d*g**3*x + 3*b**2*d*g**2*h*x**2*asin(c*x)**2/2
- 3*b**2*d*g**2*h*x**2/4 + b**2*d*g*h**2*x**3*asin(c*x)**2 - 2*b**2*d*g*h**2*x**3/9 + b**2*d*h**3*x**4*asin(c*
x)**2/4 - b**2*d*h**3*x**4/32 + b**2*e*g**3*x**2*asin(c*x)**2/2 - b**2*e*g**3*x**2/4 + b**2*e*g**2*h*x**3*asin
(c*x)**2 - 2*b**2*e*g**2*h*x**3/9 + 3*b**2*e*g*h**2*x**4*asin(c*x)**2/4 - 3*b**2*e*g*h**2*x**4/32 + b**2*e*h**
3*x**5*asin(c*x)**2/5 - 2*b**2*e*h**3*x**5/125 + b**2*f*g**3*x**3*asin(c*x)**2/3 - 2*b**2*f*g**3*x**3/27 + 3*b
**2*f*g**2*h*x**4*asin(c*x)**2/4 - 3*b**2*f*g**2*h*x**4/32 + 3*b**2*f*g*h**2*x**5*asin(c*x)**2/5 - 6*b**2*f*g*
h**2*x**5/125 + b**2*f*h**3*x**6*asin(c*x)**2/6 - b**2*f*h**3*x**6/108 + 2*b**2*d*g**3*sqrt(-c**2*x**2 + 1)*as
in(c*x)/c + 3*b**2*d*g**2*h*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2*c) + 2*b**2*d*g*h**2*x**2*sqrt(-c**2*x**2 + 1)
*asin(c*x)/(3*c) + b**2*d*h**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c) + b**2*e*g**3*x*sqrt(-c**2*x**2 + 1)*
asin(c*x)/(2*c) + 2*b**2*e*g**2*h*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c) + 3*b**2*e*g*h**2*x**3*sqrt(-c**2*
x**2 + 1)*asin(c*x)/(8*c) + 2*b**2*e*h**3*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) + 2*b**2*f*g**3*x**2*sqrt
(-c**2*x**2 + 1)*asin(c*x)/(9*c) + 3*b**2*f*g**2*h*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c) + 6*b**2*f*g*h**2
*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) + b**2*f*h**3*x**5*sqrt(-c**2*x**2 + 1)*asin(c*x)/(18*c) - 3*b**2*
d*g**2*h*asin(c*x)**2/(4*c**2) - 4*b**2*d*g*h**2*x/(3*c**2) - 3*b**2*d*h**3*x**2/(32*c**2) - b**2*e*g**3*asin(
c*x)**2/(4*c**2) - 4*b**2*e*g**2*h*x/(3*c**2) - 9*b**2*e*g*h**2*x**2/(32*c**2) - 8*b**2*e*h**3*x**3/(225*c**2)
- 4*b**2*f*g**3*x/(9*c**2) - 9*b**2*f*g**2*h*x**2/(32*c**2) - 8*b**2*f*g*h**2*x**3/(75*c**2) - 5*b**2*f*h**3*
x**4/(288*c**2) + 4*b**2*d*g*h**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + 3*b**2*d*h**3*x*sqrt(-c**2*x**2 +
1)*asin(c*x)/(16*c**3) + 4*b**2*e*g**2*h*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + 9*b**2*e*g*h**2*x*sqrt(-c**
2*x**2 + 1)*asin(c*x)/(16*c**3) + 8*b**2*e*h**3*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**3) + 4*b**2*f*g**3*
sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 9*b**2*f*g**2*h*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) + 8*b**2*
f*g*h**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c**3) + 5*b**2*f*h**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(72
*c**3) - 3*b**2*d*h**3*asin(c*x)**2/(32*c**4) - 9*b**2*e*g*h**2*asin(c*x)**2/(32*c**4) - 16*b**2*e*h**3*x/(75*
c**4) - 9*b**2*f*g**2*h*asin(c*x)**2/(32*c**4) - 16*b**2*f*g*h**2*x/(25*c**4) - 5*b**2*f*h**3*x**2/(96*c**4) +
16*b**2*e*h**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**5) + 16*b**2*f*g*h**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25
*c**5) + 5*b**2*f*h**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(48*c**5) - 5*b**2*f*h**3*asin(c*x)**2/(96*c**6), Ne(c
, 0)), (a**2*(d*g**3*x + 3*d*g**2*h*x**2/2 + d*g*h**2*x**3 + d*h**3*x**4/4 + e*g**3*x**2/2 + e*g**2*h*x**3 + 3
*e*g*h**2*x**4/4 + e*h**3*x**5/5 + f*g**3*x**3/3 + 3*f*g**2*h*x**4/4 + 3*f*g*h**2*x**5/5 + f*h**3*x**6/6), Tru
e))

________________________________________________________________________________________

Giac [B]  time = 1.47464, size = 4925, normalized size = 4.85 \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)^3*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

3/5*a^2*f*g*h^2*x^5 + 1/5*a^2*h^3*x^5*e + 1/3*a^2*f*g^3*x^3 + a^2*d*g*h^2*x^3 + b^2*d*g^3*x*arcsin(c*x)^2 + a^
2*g^2*h*x^3*e + 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*d*g*h^2*x*arcsin(c*x)^2/c^2 + (c^2*x^2 - 1)*b^2*g^2*h*x*arcsin(c*x)^2*e/c^2 + 3/2*sqrt(-c^2*x^2 + 1)*b^2*d*g
^2*h*x*arcsin(c*x)/c + 1/2*sqrt(-c^2*x^2 + 1)*b^2*g^3*x*arcsin(c*x)*e/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*d*g*h^2*x*arcsin(c*x)/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*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 + 2*(c^2*x^2 - 1)*a*b*g^2*h*x*arcsin(c*x)*e/c^2 + 1/2*(c^2*x^2 - 1)*
b^2*g^3*arcsin(c*x)^2*e/c^2 + b^2*g^2*h*x*arcsin(c*x)^2*e/c^2 + 1/5*(c^2*x^2 - 1)^2*b^2*h^3*x*arcsin(c*x)^2*e/
c^4 + 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 - 1/8*(-c^2*x^2 + 1)^(3/2)*b^2*d*h^3*x*arcsin(c*x)/c^3 + 1/2*sqrt(-c^2*
x^2 + 1)*a*b*g^3*x*e/c - 3/8*(-c^2*x^2 + 1)^(3/2)*b^2*g*h^2*x*arcsin(c*x)*e/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*d*g*h^2*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*g^3*x*a
rcsin(c*x)/c^2 + 2*a*b*d*g*h^2*x*arcsin(c*x)/c^2 + 6/5*(c^2*x^2 - 1)^2*a*b*f*g*h^2*x*arcsin(c*x)/c^4 + 3/4*b^2
*d*g^2*h*arcsin(c*x)^2/c^2 + 3/4*(c^2*x^2 - 1)^2*b^2*f*g^2*h*arcsin(c*x)^2/c^4 + 1/4*(c^2*x^2 - 1)^2*b^2*d*h^3
*arcsin(c*x)^2/c^4 + 6/5*(c^2*x^2 - 1)*b^2*f*g*h^2*x*arcsin(c*x)^2/c^4 - 2/9*(c^2*x^2 - 1)*b^2*g^2*h*x*e/c^2 +
(c^2*x^2 - 1)*a*b*g^3*arcsin(c*x)*e/c^2 + 2*a*b*g^2*h*x*arcsin(c*x)*e/c^2 + 2/5*(c^2*x^2 - 1)^2*a*b*h^3*x*arc
sin(c*x)*e/c^4 + 1/4*b^2*g^3*arcsin(c*x)^2*e/c^2 + 3/4*(c^2*x^2 - 1)^2*b^2*g*h^2*arcsin(c*x)^2*e/c^4 + 2/5*(c^
2*x^2 - 1)*b^2*h^3*x*arcsin(c*x)^2*e/c^4 + 2*sqrt(-c^2*x^2 + 1)*a*b*d*g^3/c - 3/8*(-c^2*x^2 + 1)^(3/2)*a*b*f*g
^2*h*x/c^3 - 1/8*(-c^2*x^2 + 1)^(3/2)*a*b*d*h^3*x/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*f*g^3*arcsin(c*x)/c^3 - 2
/3*(-c^2*x^2 + 1)^(3/2)*b^2*d*g*h^2*arcsin(c*x)/c^3 + 15/16*sqrt(-c^2*x^2 + 1)*b^2*f*g^2*h*x*arcsin(c*x)/c^3 +
5/16*sqrt(-c^2*x^2 + 1)*b^2*d*h^3*x*arcsin(c*x)/c^3 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*f*h^3*x*arc
sin(c*x)/c^5 - 3/8*(-c^2*x^2 + 1)^(3/2)*a*b*g*h^2*x*e/c^3 - 2/3*(-c^2*x^2 + 1)^(3/2)*b^2*g^2*h*arcsin(c*x)*e/c
^3 + 15/16*sqrt(-c^2*x^2 + 1)*b^2*g*h^2*x*arcsin(c*x)*e/c^3 + 3/2*(c^2*x^2 - 1)*a^2*d*g^2*h/c^2 - 3/4*(c^2*x^2
- 1)*b^2*d*g^2*h/c^2 - 14/27*b^2*f*g^3*x/c^2 - 14/9*b^2*d*g*h^2*x/c^2 - 6/125*(c^2*x^2 - 1)^2*b^2*f*g*h^2*x/c
^4 + 3/2*a*b*d*g^2*h*arcsin(c*x)/c^2 + 3/2*(c^2*x^2 - 1)^2*a*b*f*g^2*h*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^2*a
*b*d*h^3*arcsin(c*x)/c^4 + 12/5*(c^2*x^2 - 1)*a*b*f*g*h^2*x*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)*b^2*f*g^2*h*ar
csin(c*x)^2/c^4 + 1/2*(c^2*x^2 - 1)*b^2*d*h^3*arcsin(c*x)^2/c^4 + 1/6*(c^2*x^2 - 1)^3*b^2*f*h^3*arcsin(c*x)^2/
c^6 + 3/5*b^2*f*g*h^2*x*arcsin(c*x)^2/c^4 + 1/2*(c^2*x^2 - 1)*a^2*g^3*e/c^2 - 1/4*(c^2*x^2 - 1)*b^2*g^3*e/c^2
- 14/9*b^2*g^2*h*x*e/c^2 - 2/125*(c^2*x^2 - 1)^2*b^2*h^3*x*e/c^4 + 1/2*a*b*g^3*arcsin(c*x)*e/c^2 + 3/2*(c^2*x^
2 - 1)^2*a*b*g*h^2*arcsin(c*x)*e/c^4 + 4/5*(c^2*x^2 - 1)*a*b*h^3*x*arcsin(c*x)*e/c^4 + 3/2*(c^2*x^2 - 1)*b^2*g
*h^2*arcsin(c*x)^2*e/c^4 + 1/5*b^2*h^3*x*arcsin(c*x)^2*e/c^4 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*f*g^3/c^3 - 2/3*(-
c^2*x^2 + 1)^(3/2)*a*b*d*g*h^2/c^3 + 15/16*sqrt(-c^2*x^2 + 1)*a*b*f*g^2*h*x/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*a*b*
d*h^3*x/c^3 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*f*h^3*x/c^5 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*f*g^3*arcsi
n(c*x)/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*d*g*h^2*arcsin(c*x)/c^3 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*f*
g*h^2*arcsin(c*x)/c^5 - 13/72*(-c^2*x^2 + 1)^(3/2)*b^2*f*h^3*x*arcsin(c*x)/c^5 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b*
g^2*h*e/c^3 + 15/16*sqrt(-c^2*x^2 + 1)*a*b*g*h^2*x*e/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*g^2*h*arcsin(c*x)*e/c^3 +
2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*h^3*arcsin(c*x)*e/c^5 - 3/8*b^2*d*g^2*h/c^2 + 3/4*(c^2*x^2 - 1)^2*
a^2*f*g^2*h/c^4 - 3/32*(c^2*x^2 - 1)^2*b^2*f*g^2*h/c^4 + 1/4*(c^2*x^2 - 1)^2*a^2*d*h^3/c^4 - 1/32*(c^2*x^2 - 1
)^2*b^2*d*h^3/c^4 - 76/375*(c^2*x^2 - 1)*b^2*f*g*h^2*x/c^4 + 3*(c^2*x^2 - 1)*a*b*f*g^2*h*arcsin(c*x)/c^4 + (c^
2*x^2 - 1)*a*b*d*h^3*arcsin(c*x)/c^4 + 1/3*(c^2*x^2 - 1)^3*a*b*f*h^3*arcsin(c*x)/c^6 + 6/5*a*b*f*g*h^2*x*arcsi
n(c*x)/c^4 + 15/32*b^2*f*g^2*h*arcsin(c*x)^2/c^4 + 5/32*b^2*d*h^3*arcsin(c*x)^2/c^4 + 1/2*(c^2*x^2 - 1)^2*b^2*
f*h^3*arcsin(c*x)^2/c^6 - 1/8*b^2*g^3*e/c^2 + 3/4*(c^2*x^2 - 1)^2*a^2*g*h^2*e/c^4 - 3/32*(c^2*x^2 - 1)^2*b^2*g
*h^2*e/c^4 - 76/1125*(c^2*x^2 - 1)*b^2*h^3*x*e/c^4 + 3*(c^2*x^2 - 1)*a*b*g*h^2*arcsin(c*x)*e/c^4 + 2/5*a*b*h^3
*x*arcsin(c*x)*e/c^4 + 15/32*b^2*g*h^2*arcsin(c*x)^2*e/c^4 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*f*g^3/c^3 + 2*sqrt(-c^
2*x^2 + 1)*a*b*d*g*h^2/c^3 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*f*g*h^2/c^5 - 13/72*(-c^2*x^2 + 1)^(3
/2)*a*b*f*h^3*x/c^5 - 4/5*(-c^2*x^2 + 1)^(3/2)*b^2*f*g*h^2*arcsin(c*x)/c^5 + 11/48*sqrt(-c^2*x^2 + 1)*b^2*f*h^
3*x*arcsin(c*x)/c^5 + 2*sqrt(-c^2*x^2 + 1)*a*b*g^2*h*e/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*h^3*e
/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b^2*h^3*arcsin(c*x)*e/c^5 + 3/2*(c^2*x^2 - 1)*a^2*f*g^2*h/c^4 - 15/32*(c^2*x^
2 - 1)*b^2*f*g^2*h/c^4 + 1/2*(c^2*x^2 - 1)*a^2*d*h^3/c^4 - 5/32*(c^2*x^2 - 1)*b^2*d*h^3/c^4 + 1/6*(c^2*x^2 - 1
)^3*a^2*f*h^3/c^6 - 1/108*(c^2*x^2 - 1)^3*b^2*f*h^3/c^6 - 298/375*b^2*f*g*h^2*x/c^4 + 15/16*a*b*f*g^2*h*arcsin
(c*x)/c^4 + 5/16*a*b*d*h^3*arcsin(c*x)/c^4 + (c^2*x^2 - 1)^2*a*b*f*h^3*arcsin(c*x)/c^6 + 1/2*(c^2*x^2 - 1)*b^2
*f*h^3*arcsin(c*x)^2/c^6 + 3/2*(c^2*x^2 - 1)*a^2*g*h^2*e/c^4 - 15/32*(c^2*x^2 - 1)*b^2*g*h^2*e/c^4 - 298/1125*
b^2*h^3*x*e/c^4 + 15/16*a*b*g*h^2*arcsin(c*x)*e/c^4 - 4/5*(-c^2*x^2 + 1)^(3/2)*a*b*f*g*h^2/c^5 + 11/48*sqrt(-c
^2*x^2 + 1)*a*b*f*h^3*x/c^5 + 6/5*sqrt(-c^2*x^2 + 1)*b^2*f*g*h^2*arcsin(c*x)/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*a
*b*h^3*e/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*b^2*h^3*arcsin(c*x)*e/c^5 - 51/256*b^2*f*g^2*h/c^4 - 17/256*b^2*d*h^3/c^
4 + 1/2*(c^2*x^2 - 1)^2*a^2*f*h^3/c^6 - 13/288*(c^2*x^2 - 1)^2*b^2*f*h^3/c^6 + (c^2*x^2 - 1)*a*b*f*h^3*arcsin(
c*x)/c^6 + 11/96*b^2*f*h^3*arcsin(c*x)^2/c^6 - 51/256*b^2*g*h^2*e/c^4 + 6/5*sqrt(-c^2*x^2 + 1)*a*b*f*g*h^2/c^5
+ 2/5*sqrt(-c^2*x^2 + 1)*a*b*h^3*e/c^5 + 1/2*(c^2*x^2 - 1)*a^2*f*h^3/c^6 - 11/96*(c^2*x^2 - 1)*b^2*f*h^3/c^6
+ 11/48*a*b*f*h^3*arcsin(c*x)/c^6 - 299/6912*b^2*f*h^3/c^6