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3.2.15 \(\int (g+h x)^3 (d+e x+f x^2) (a+b \text {ArcSin}(c x))^2 \, dx\) [115]

Optimal. Leaf size=1016 \[ -2 b^2 d g^3 x-\frac {16 b^2 h^2 (3 f g+e h) x}{75 c^4}-\frac {4 b^2 g \left (f g^2+3 h (e g+d h)\right ) x}{9 c^2}-\frac {5 b^2 f h^3 x^2}{96 c^4}-\frac {1}{4} b^2 g^2 (e g+3 d h) x^2-\frac {3 b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^2}{32 c^2}-\frac {8 b^2 h^2 (3 f g+e h) x^3}{225 c^2}-\frac {2}{27} b^2 g \left (f g^2+3 h (e g+d h)\right ) x^3-\frac {5 b^2 f h^3 x^4}{288 c^2}-\frac {1}{32} b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^4-\frac {2}{125} b^2 h^2 (3 f g+e h) x^5-\frac {1}{108} b^2 f h^3 x^6+\frac {2 b d g^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {16 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^5}+\frac {4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {5 b f h^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{48 c^5}+\frac {b g^2 (e g+3 d h) x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}+\frac {3 b h \left (3 f g^2+h (3 e g+d h)\right ) x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {8 b h^2 (3 f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^3}+\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {5 b f h^3 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{72 c^3}+\frac {b h \left (3 f g^2+h (3 e g+d h)\right ) x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {2 b h^2 (3 f g+e h) x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{25 c}+\frac {b f h^3 x^5 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{18 c}-\frac {5 f h^3 (a+b \text {ArcSin}(c x))^2}{96 c^6}-\frac {g^2 (e g+3 d h) (a+b \text {ArcSin}(c x))^2}{4 c^2}-\frac {3 h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \text {ArcSin}(c x))^2}{32 c^4}+d g^3 x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} g^2 (e g+3 d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 (a+b \text {ArcSin}(c x))^2+\frac {1}{5} h^2 (3 f g+e h) x^5 (a+b \text {ArcSin}(c x))^2+\frac {1}{6} f h^3 x^6 (a+b \text {ArcSin}(c x))^2 \]

[Out]

5/48*b*f*h^3*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5+1/2*b*g^2*(3*d*h+e*g)*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)
^(1/2)/c+3/16*b*h*(3*f*g^2+h*(d*h+3*e*g))*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+8/75*b*h^2*(e*h+3*f*g)*x^
2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+2/9*b*g*(f*g^2+3*h*(d*h+e*g))*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1
/2)/c+5/72*b*f*h^3*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+1/8*b*h*(3*f*g^2+h*(d*h+3*e*g))*x^3*(a+b*arcsi
n(c*x))*(-c^2*x^2+1)^(1/2)/c+2/25*b*h^2*(e*h+3*f*g)*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+1/18*b*f*h^3*x^
5*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c-5/288*b^2*f*h^3*x^4/c^2-16/75*b^2*h^2*(e*h+3*f*g)*x/c^4-4/9*b^2*g*(f*
g^2+3*h*(d*h+e*g))*x/c^2-5/96*b^2*f*h^3*x^2/c^4-3/32*b^2*h*(3*f*g^2+h*(d*h+3*e*g))*x^2/c^2-8/225*b^2*h^2*(e*h+
3*f*g)*x^3/c^2+d*g^3*x*(a+b*arcsin(c*x))^2-2*b^2*d*g^3*x-1/4*b^2*g^2*(3*d*h+e*g)*x^2-2/27*b^2*g*(f*g^2+3*h*(d*
h+e*g))*x^3-1/32*b^2*h*(3*f*g^2+h*(d*h+3*e*g))*x^4-2/125*b^2*h^2*(e*h+3*f*g)*x^5-1/108*b^2*f*h^3*x^6-5/96*f*h^
3*(a+b*arcsin(c*x))^2/c^6-1/4*g^2*(3*d*h+e*g)*(a+b*arcsin(c*x))^2/c^2-3/32*h*(3*f*g^2+h*(d*h+3*e*g))*(a+b*arcs
in(c*x))^2/c^4+1/2*g^2*(3*d*h+e*g)*x^2*(a+b*arcsin(c*x))^2+1/3*g*(f*g^2+3*h*(d*h+e*g))*x^3*(a+b*arcsin(c*x))^2
+1/4*h*(3*f*g^2+h*(d*h+3*e*g))*x^4*(a+b*arcsin(c*x))^2+1/5*h^2*(e*h+3*f*g)*x^5*(a+b*arcsin(c*x))^2+1/6*f*h^3*x
^6*(a+b*arcsin(c*x))^2+2*b*d*g^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+16/75*b*h^2*(e*h+3*f*g)*(a+b*arcsin(c*
x))*(-c^2*x^2+1)^(1/2)/c^5+4/9*b*g*(f*g^2+3*h*(d*h+e*g))*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3

________________________________________________________________________________________

Rubi [A]
time = 1.00, antiderivative size = 1016, normalized size of antiderivative = 1.00, 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 = {4835, 4715, 4767, 8, 4723, 4795, 4737, 30} \begin {gather*} -\frac {1}{108} b^2 f h^3 x^6+\frac {1}{6} f h^3 (a+b \text {ArcSin}(c x))^2 x^6+\frac {1}{5} h^2 (3 f g+e h) (a+b \text {ArcSin}(c x))^2 x^5-\frac {2}{125} b^2 h^2 (3 f g+e h) x^5+\frac {b f h^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^5}{18 c}-\frac {5 b^2 f h^3 x^4}{288 c^2}+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \text {ArcSin}(c x))^2 x^4-\frac {1}{32} b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^4+\frac {2 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^4}{25 c}+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) (a+b \text {ArcSin}(c x))^2 x^3-\frac {8 b^2 h^2 (3 f g+e h) x^3}{225 c^2}-\frac {2}{27} b^2 g \left (f g^2+3 h (e g+d h)\right ) x^3+\frac {5 b f h^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^3}{72 c^3}+\frac {b h \left (3 f g^2+h (3 e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^3}{8 c}-\frac {5 b^2 f h^3 x^2}{96 c^4}+\frac {1}{2} g^2 (e g+3 d h) (a+b \text {ArcSin}(c x))^2 x^2-\frac {1}{4} b^2 g^2 (e g+3 d h) x^2-\frac {3 b^2 h \left (3 f g^2+h (3 e g+d h)\right ) x^2}{32 c^2}+\frac {8 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^2}{75 c^3}+\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^2}{9 c}-2 b^2 d g^3 x+d g^3 (a+b \text {ArcSin}(c x))^2 x-\frac {16 b^2 h^2 (3 f g+e h) x}{75 c^4}-\frac {4 b^2 g \left (f g^2+3 h (e g+d h)\right ) x}{9 c^2}+\frac {5 b f h^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x}{48 c^5}+\frac {b g^2 (e g+3 d h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x}{2 c}+\frac {3 b h \left (3 f g^2+h (3 e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x}{16 c^3}-\frac {5 f h^3 (a+b \text {ArcSin}(c x))^2}{96 c^6}-\frac {g^2 (e g+3 d h) (a+b \text {ArcSin}(c x))^2}{4 c^2}-\frac {3 h \left (3 f g^2+h (3 e g+d h)\right ) (a+b \text {ArcSin}(c x))^2}{32 c^4}+\frac {2 b d g^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {16 b h^2 (3 f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^5}+\frac {4 b g \left (f g^2+3 h (e g+d h)\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.68, size = 734, normalized size = 0.72 \begin {gather*} d g^3 x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} g^2 (e g+3 d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} g \left (f g^2+3 h (e g+d h)\right ) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} h \left (3 f g^2+h (3 e g+d h)\right ) x^4 (a+b \text {ArcSin}(c x))^2+\frac {1}{5} h^2 (3 f g+e h) x^5 (a+b \text {ArcSin}(c x))^2+\frac {1}{6} f h^3 x^6 (a+b \text {ArcSin}(c x))^2-\frac {2 b g \left (f g^2+3 h (e g+d h)\right ) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \text {ArcSin}(c x)\right )}{27 c^3}-\frac {2 b h^2 (3 f g+e h) \left (-15 a \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+b c x \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{1125 c^5}-\frac {f h^3 \left (45 a^2-6 a b c x \sqrt {1-c^2 x^2} \left (15+10 c^2 x^2+8 c^4 x^4\right )+b^2 c^2 x^2 \left (45+15 c^2 x^2+8 c^4 x^4\right )-6 b \left (-15 a+b c x \sqrt {1-c^2 x^2} \left (15+10 c^2 x^2+8 c^4 x^4\right )\right ) \text {ArcSin}(c x)+45 b^2 \text {ArcSin}(c x)^2\right )}{864 c^6}-2 b d g^3 \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}\right )-\frac {1}{32} b h \left (3 f g^2+h (3 e g+d h)\right ) \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^3}-\frac {4 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {3 (a+b \text {ArcSin}(c x))^2}{b c^4}\right )-\frac {1}{4} b g^2 (e g+3 d h) \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {(a+b \text {ArcSin}(c x))^2}{b c^2}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(2599\) vs. \(2(932)=1864\).
time = 0.56, size = 2600, normalized size = 2.56

method result size
derivativedivides \(\text {Expression too large to display}\) \(2600\)
default \(\text {Expression too large to display}\) \(2600\)

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,method=_RETURNVERBOSE)

[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*(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*ar
csin(c*x)^2-54*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+3375*c*x*arcsin(c*x)^2+380*c^3*x^3+4470*arc
sin(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*c*x*arcs
in(c*x)^2+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^5-432*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)+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*c*x*arcsin(c*x)^2-2
*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*c*x*arcsin(c*x)^2-2*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*c*x*arcsin(c*x)^2-2*c^3*x^3-42*arcsin(c*
x)*(-c^2*x^2+1)^(1/2)+42*c*x)+1/128*c^2*d*h^3*(32*arcsin(c*x)^2*c^4*x^4+16*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*
x^3-64*arcsin(c*x)^2*c^2*x^2-4*c^4*x^4-40*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+20*arcsin(c*x)^2+20*c^2*x^2-25)+3
/128*c^2*e*g*h^2*(32*arcsin(c*x)^2*c^4*x^4+16*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-64*arcsin(c*x)^2*c^2*x^2-
4*c^4*x^4-40*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+20*arcsin(c*x)^2+20*c^2*x^2-25)+3/128*c^2*f*g^2*h*(32*arcsin(c
*x)^2*c^4*x^4+16*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-64*arcsin(c*x)^2*c^2*x^2-4*c^4*x^4-40*arcsin(c*x)*(-c^
2*x^2+1)^(1/2)*c*x+20*arcsin(c*x)^2+20*c^2*x^2-25)+c^5*g^3*d*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+
1)^(1/2))+3/4*c^4*d*g^2*h*(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*e*g^3*(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*c*x*arcsin(c*x)^2-2*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*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^
2-27*c*x*arcsin(c*x)^2-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+1/64*h^3*f*(32*arcsin(c*x)^2*c^4*x^
4+16*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3-64*arcsin(c*x)^2*c^2*x^2-4*c^4*x^4-40*arcsin(c*x)*(-c^2*x^2+1)^(1/
2)*c*x+20*arcsin(c*x)^2+20*c^2*x^2-25)+3*c^3*g*h^2*d*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
)+3*c^3*g^2*h*e*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+c^3*g^3*f*(c*x*arcsin(c*x)^2-2*c*x+
2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*c^2*d*h^3*(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*e*g*h^2*(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*f*g^2*h*(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)+h^3*c*e*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+3*c*g*h^2*f*(c*x*arcsin(c*x)^2-2*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-arcsi
n(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*e*h^3*x^5+3/5*arcsin(c*x)*c^6*
f*g*h^2*x^5+1/4*arcsin(c*x)*c^6*d*h^3*x^4+3/4*arcsin(c*x)*c^6*e*g*h^2*x^4+3/4*arcsin(c*x)*c^6*f*g^2*h*x^4+arcs
in(c*x)*c^6*d*g*h^2*x^3+arcsin(c*x)*c^6*e*g^2*h*x^3+1/3*arcsin(c*x)*c^6*f*g^3*x^3+3/2*arcsin(c*x)*c^6*d*g^2*h*
x^2+1/2*arcsin(c*x)*c^6*e*g^3*x^2+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*h^3*x^5*e + 3/4*a^2*f*g^2*h*x^4 + 1/4*a^2*d*h^3*x^4 + 3/4*a^
2*g*h^2*x^4*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 + 3/2*a^2*d*
g^2*h*x^2 + 1/2*a^2*g^3*x^2*e + 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/
c^4))*a*b*f*g^3 + 3/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d*g^2*h + 3/16*
(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*f*
g^2*h + 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d*g*h^2 + 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*ar
csin(c*x)/c^5)*c)*a*b*d*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*x)/c^7)*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 + 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
*g^3*e + 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*g^2*h*e + 3/1
6*(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*
g*h^2*e + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^
2*x^2 + 1)/c^6)*c)*a*b*h^3*e + 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*h^3*e)*x^5 + 15*(3*b^2*f*g^2*h + b^2*d*h^3 + 3*b^2*g*h^2*e)*x^4 + 20*(b^2*f*g^3 + 3*b
^2*d*g*h^2 + 3*b^2*g^2*h*e)*x^3 + 30*(3*b^2*d*g^2*h + b^2*g^3*e)*x^2)*arctan2(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*h^3*e)*x^5 + 15*(3*b^2*c*f*g^2*h + b^2
*c*d*h^3 + 3*b^2*c*g*h^2*e)*x^4 + 20*(b^2*c*f*g^3 + 3*b^2*c*d*g*h^2 + 3*b^2*c*g^2*h*e)*x^3 + 30*(3*b^2*c*d*g^2
*h + b^2*c*g^3*e)*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 = 3.17, size = 1590, normalized size = 1.56 \begin {gather*} \frac {1000 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} f h^{3} x^{6} + 2592 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} f g h^{2} x^{5} + 375 \, {\left (27 \, {\left (8 \, a^{2} - b^{2}\right )} c^{6} f g^{2} h + {\left (9 \, {\left (8 \, a^{2} - b^{2}\right )} c^{6} d - 5 \, b^{2} c^{4} f\right )} h^{3}\right )} x^{4} + 160 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{6} f g^{3} + 3 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{6} d - 24 \, b^{2} c^{4} f\right )} g h^{2}\right )} x^{3} + 1125 \, {\left (9 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{6} d - 3 \, b^{2} c^{4} f\right )} g^{2} h - {\left (9 \, b^{2} c^{4} d + 5 \, b^{2} c^{2} f\right )} h^{3}\right )} x^{2} + 225 \, {\left (80 \, b^{2} c^{6} f h^{3} x^{6} + 288 \, b^{2} c^{6} f g h^{2} x^{5} + 720 \, b^{2} c^{6} d g^{2} h x^{2} + 480 \, b^{2} c^{6} d g^{3} x + 120 \, {\left (3 \, b^{2} c^{6} f g^{2} h + b^{2} c^{6} d h^{3}\right )} x^{4} - 45 \, {\left (8 \, b^{2} c^{4} d + 3 \, b^{2} c^{2} f\right )} g^{2} h - 5 \, {\left (9 \, b^{2} c^{2} d + 5 \, b^{2} f\right )} h^{3} + 160 \, {\left (b^{2} c^{6} f g^{3} + 3 \, b^{2} c^{6} d g h^{2}\right )} x^{3} + 3 \, {\left (32 \, b^{2} c^{6} h^{3} x^{5} + 120 \, b^{2} c^{6} g h^{2} x^{4} + 160 \, b^{2} c^{6} g^{2} h x^{3} + 80 \, b^{2} c^{6} g^{3} x^{2} - 40 \, b^{2} c^{4} g^{3} - 45 \, b^{2} c^{2} g h^{2}\right )} e\right )} \arcsin \left (c x\right )^{2} + 480 \, {\left (25 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{6} d - 4 \, b^{2} c^{4} f\right )} g^{3} - 12 \, {\left (25 \, b^{2} c^{4} d + 12 \, b^{2} c^{2} f\right )} g h^{2}\right )} x + 450 \, {\left (80 \, a b c^{6} f h^{3} x^{6} + 288 \, a b c^{6} f g h^{2} x^{5} + 720 \, a b c^{6} d g^{2} h x^{2} + 480 \, a b c^{6} d g^{3} x + 120 \, {\left (3 \, a b c^{6} f g^{2} h + a b c^{6} d h^{3}\right )} x^{4} - 45 \, {\left (8 \, a b c^{4} d + 3 \, a b c^{2} f\right )} g^{2} h - 5 \, {\left (9 \, a b c^{2} d + 5 \, a b f\right )} h^{3} + 160 \, {\left (a b c^{6} f g^{3} + 3 \, a b c^{6} d g h^{2}\right )} x^{3} + 3 \, {\left (32 \, a b c^{6} h^{3} x^{5} + 120 \, a b c^{6} g h^{2} x^{4} + 160 \, a b c^{6} g^{2} h x^{3} + 80 \, a b c^{6} g^{3} x^{2} - 40 \, a b c^{4} g^{3} - 45 \, a b c^{2} g h^{2}\right )} e\right )} \arcsin \left (c x\right ) + 3 \, {\left (288 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} h^{3} x^{5} + 3375 \, {\left (8 \, a^{2} - b^{2}\right )} c^{6} g h^{2} x^{4} + 160 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{6} g^{2} h - 8 \, b^{2} c^{4} h^{3}\right )} x^{3} + 1125 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{6} g^{3} - 9 \, b^{2} c^{4} g h^{2}\right )} x^{2} - 1920 \, {\left (25 \, b^{2} c^{4} g^{2} h + 4 \, b^{2} c^{2} h^{3}\right )} x\right )} e + 30 \, {\left (200 \, a b c^{5} f h^{3} x^{5} + 864 \, a b c^{5} f g h^{2} x^{4} + 800 \, {\left (9 \, a b c^{5} d + 2 \, a b c^{3} f\right )} g^{3} + 192 \, {\left (25 \, a b c^{3} d + 12 \, a b c f\right )} g h^{2} + 50 \, {\left (27 \, a b c^{5} f g^{2} h + {\left (9 \, a b c^{5} d + 5 \, a b c^{3} f\right )} h^{3}\right )} x^{3} + 32 \, {\left (25 \, a b c^{5} f g^{3} + 3 \, {\left (25 \, a b c^{5} d + 12 \, a b c^{3} f\right )} g h^{2}\right )} x^{2} + 75 \, {\left (9 \, {\left (8 \, a b c^{5} d + 3 \, a b c^{3} f\right )} g^{2} h + {\left (9 \, a b c^{3} d + 5 \, a b c f\right )} h^{3}\right )} x + {\left (200 \, b^{2} c^{5} f h^{3} x^{5} + 864 \, b^{2} c^{5} f g h^{2} x^{4} + 800 \, {\left (9 \, b^{2} c^{5} d + 2 \, b^{2} c^{3} f\right )} g^{3} + 192 \, {\left (25 \, b^{2} c^{3} d + 12 \, b^{2} c f\right )} g h^{2} + 50 \, {\left (27 \, b^{2} c^{5} f g^{2} h + {\left (9 \, b^{2} c^{5} d + 5 \, b^{2} c^{3} f\right )} h^{3}\right )} x^{3} + 32 \, {\left (25 \, b^{2} c^{5} f g^{3} + 3 \, {\left (25 \, b^{2} c^{5} d + 12 \, b^{2} c^{3} f\right )} g h^{2}\right )} x^{2} + 75 \, {\left (9 \, {\left (8 \, b^{2} c^{5} d + 3 \, b^{2} c^{3} f\right )} g^{2} h + {\left (9 \, b^{2} c^{3} d + 5 \, b^{2} c f\right )} h^{3}\right )} x + 3 \, {\left (96 \, b^{2} c^{5} h^{3} x^{4} + 450 \, b^{2} c^{5} g h^{2} x^{3} + 1600 \, b^{2} c^{3} g^{2} h + 256 \, b^{2} c h^{3} + 32 \, {\left (25 \, b^{2} c^{5} g^{2} h + 4 \, b^{2} c^{3} h^{3}\right )} x^{2} + 75 \, {\left (8 \, b^{2} c^{5} g^{3} + 9 \, b^{2} c^{3} g h^{2}\right )} x\right )} e\right )} \arcsin \left (c x\right ) + 3 \, {\left (96 \, a b c^{5} h^{3} x^{4} + 450 \, a b c^{5} g h^{2} x^{3} + 1600 \, a b c^{3} g^{2} h + 256 \, a b c h^{3} + 32 \, {\left (25 \, a b c^{5} g^{2} h + 4 \, a b c^{3} h^{3}\right )} x^{2} + 75 \, {\left (8 \, a b c^{5} g^{3} + 9 \, a b c^{3} g h^{2}\right )} x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{108000 \, c^{6}} \end {gather*}

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 + 2592*(25*a^2 - 2*b^2)*c^6*f*g*h^2*x^5 + 375*(27*(8*a^2 - b^2)*c^
6*f*g^2*h + (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 + 3*(25*(9*a^2
- 2*b^2)*c^6*d - 24*b^2*c^4*f)*g*h^2)*x^3 + 1125*(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 + 288*b^2*c^6*f*g*h^2*x^5 + 720*b^2*c^6*d*g^2*h*x^2 + 480
*b^2*c^6*d*g^3*x + 120*(3*b^2*c^6*f*g^2*h + 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*d*g*h^2)*x^3 + 3*(32*b^2*c^6*h^3*x^5 + 120*b^2*c^6*g*
h^2*x^4 + 160*b^2*c^6*g^2*h*x^3 + 80*b^2*c^6*g^3*x^2 - 40*b^2*c^4*g^3 - 45*b^2*c^2*g*h^2)*e)*arcsin(c*x)^2 + 4
80*(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 + 288*a*b*c^6*f*g*h^2*x^5 + 720*a*b*c^6*d*g^2*h*x^2 + 480*a*b*c^6*d*g^3*x + 120*(3*a*b*c^6*f*g^2*h
+ 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*d*g*h^2)*x^3 + 3*(32*a*b*c^6*h^3*x^5 + 120*a*b*c^6*g*h^2*x^4 + 160*a*b*c^6*g^2*h*x^3 + 80*a*b*c
^6*g^3*x^2 - 40*a*b*c^4*g^3 - 45*a*b*c^2*g*h^2)*e)*arcsin(c*x) + 3*(288*(25*a^2 - 2*b^2)*c^6*h^3*x^5 + 3375*(8
*a^2 - b^2)*c^6*g*h^2*x^4 + 160*(25*(9*a^2 - 2*b^2)*c^6*g^2*h - 8*b^2*c^4*h^3)*x^3 + 1125*(8*(2*a^2 - b^2)*c^6
*g^3 - 9*b^2*c^4*g*h^2)*x^2 - 1920*(25*b^2*c^4*g^2*h + 4*b^2*c^2*h^3)*x)*e + 30*(200*a*b*c^5*f*h^3*x^5 + 864*a
*b*c^5*f*g*h^2*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 + (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 + 3*(25*a*b*c^5*d + 12*a*b*c^3*f)*g*
h^2)*x^2 + 75*(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
 + 864*b^2*c^5*f*g*h^2*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 + (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 + 3*(25*b^2*c^5*d + 12*b^2*c
^3*f)*g*h^2)*x^2 + 75*(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 + 3*(96*b^2*c^5*
h^3*x^4 + 450*b^2*c^5*g*h^2*x^3 + 1600*b^2*c^3*g^2*h + 256*b^2*c*h^3 + 32*(25*b^2*c^5*g^2*h + 4*b^2*c^3*h^3)*x
^2 + 75*(8*b^2*c^5*g^3 + 9*b^2*c^3*g*h^2)*x)*e)*arcsin(c*x) + 3*(96*a*b*c^5*h^3*x^4 + 450*a*b*c^5*g*h^2*x^3 +
1600*a*b*c^3*g^2*h + 256*a*b*c*h^3 + 32*(25*a*b*c^5*g^2*h + 4*a*b*c^3*h^3)*x^2 + 75*(8*a*b*c^5*g^3 + 9*a*b*c^3
*g*h^2)*x)*e)*sqrt(-c^2*x^2 + 1))/c^6

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2992 vs. \(2 (1006) = 2012\).
time = 1.44, size = 2992, normalized size = 2.94 \begin {gather*} \text {Too large to display} \end {gather*}

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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3444 vs. \(2 (932) = 1864\).
time = 0.50, size = 3444, normalized size = 3.39 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/6*a^2*f*h^3*x^6 + 3/5*a^2*f*g*h^2*x^5 + 1/5*a^2*e*h^3*x^5 + 3/4*a^2*f*g^2*h*x^4 + 3/4*a^2*e*g*h^2*x^4 + 1/4*
a^2*d*h^3*x^4 + 1/3*a^2*f*g^3*x^3 + a^2*e*g^2*h*x^3 + a^2*d*g*h^2*x^3 + b^2*d*g^3*x*arcsin(c*x)^2 + 2*a*b*d*g^
3*x*arcsin(c*x) + 1/3*(c^2*x^2 - 1)*b^2*f*g^3*x*arcsin(c*x)^2/c^2 + (c^2*x^2 - 1)*b^2*e*g^2*h*x*arcsin(c*x)^2/
c^2 + (c^2*x^2 - 1)*b^2*d*g*h^2*x*arcsin(c*x)^2/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*b^2*e*g^3*x*arcsin(c*x)/c + 3/2*s
qrt(-c^2*x^2 + 1)*b^2*d*g^2*h*x*arcsin(c*x)/c + a^2*d*g^3*x - 2*b^2*d*g^3*x + 2/3*(c^2*x^2 - 1)*a*b*f*g^3*x*ar
csin(c*x)/c^2 + 2*(c^2*x^2 - 1)*a*b*e*g^2*h*x*arcsin(c*x)/c^2 + 2*(c^2*x^2 - 1)*a*b*d*g*h^2*x*arcsin(c*x)/c^2
+ 1/2*(c^2*x^2 - 1)*b^2*e*g^3*arcsin(c*x)^2/c^2 + 3/2*(c^2*x^2 - 1)*b^2*d*g^2*h*arcsin(c*x)^2/c^2 + 1/3*b^2*f*
g^3*x*arcsin(c*x)^2/c^2 + b^2*e*g^2*h*x*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 + 1/5*(c^2*x^2 - 1)^2*b^2*e*h^3*x*arcsin(c*x)^2/c^4 + 1/2*sqrt(-c^2*x^2 +
1)*a*b*e*g^3*x/c + 3/2*sqrt(-c^2*x^2 + 1)*a*b*d*g^2*h*x/c + 2*sqrt(-c^2*x^2 + 1)*b^2*d*g^3*arcsin(c*x)/c - 3/8
*(-c^2*x^2 + 1)^(3/2)*b^2*f*g^2*h*x*arcsin(c*x)/c^3 - 3/8*(-c^2*x^2 + 1)^(3/2)*b^2*e*g*h^2*x*arcsin(c*x)/c^3 -
 1/8*(-c^2*x^2 + 1)^(3/2)*b^2*d*h^3*x*arcsin(c*x)/c^3 - 2/27*(c^2*x^2 - 1)*b^2*f*g^3*x/c^2 - 2/9*(c^2*x^2 - 1)
*b^2*e*g^2*h*x/c^2 - 2/9*(c^2*x^2 - 1)*b^2*d*g*h^2*x/c^2 + (c^2*x^2 - 1)*a*b*e*g^3*arcsin(c*x)/c^2 + 3*(c^2*x^
2 - 1)*a*b*d*g^2*h*arcsin(c*x)/c^2 + 2/3*a*b*f*g^3*x*arcsin(c*x)/c^2 + 2*a*b*e*g^2*h*x*arcsin(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 + 2/5*(c^2*x^2 - 1)^2*a*b*e*h^3
*x*arcsin(c*x)/c^4 + 1/4*b^2*e*g^3*arcsin(c*x)^2/c^2 + 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 + 3/4*(c^2*x^2 - 1)^2*b^2*e*g*h^2*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 + 2/5*(c^2*x^2 - 1)*b^2*e*h^3*x*arc
sin(c*x)^2/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 - 3/8*(-c^2*x^2
 + 1)^(3/2)*a*b*e*g*h^2*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*e*g^2*h*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 + 15/16*sqrt(-c^2*x^2 + 1)*b^2*e*g*h^
2*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*arcsin(c*x)/c^5 + 1/2*(c^2*x^2 - 1)*a^2*e*g^3/c^2 - 1/4*(c^2*x^2 - 1)*b^2*e*g^3/c^2 + 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*e*g^2*h*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 - 2/125*(c^2*x^2 - 1)^2*b^2*e*h^3*x/c^4
+ 1/2*a*b*e*g^3*arcsin(c*x)/c^2 + 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 + 3/2*(c^2*x^2 - 1)^2*a*b*e*g*h^2*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 + 4/5*(c^2*x^2 - 1)*a*b*e*h^3*x*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 -
1)*b^2*f*g^2*h*arcsin(c*x)^2/c^4 + 3/2*(c^2*x^2 - 1)*b^2*e*g*h^2*arcsin(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 +
 1/5*b^2*e*h^3*x*arcsin(c*x)^2/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*e*g
^2*h/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 + 15/16*sqrt(
-c^2*x^2 + 1)*a*b*e*g*h^2*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*arcsin(c*x)/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*e*g^2*h*ar
csin(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 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*e*h^3*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 - 1/8*b^2*e*g^3/c^2 - 3/8*b^2*d*g^2*h/c^2 - 3/32*(c^2*x^2 - 1)^2*b^2*f
*g^2*h/c^4 - 3/32*(c^2*x^2 - 1)^2*b^2*e*g*h^2/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 - 76/1125*(c^2*x^2 - 1)*b^2*e*h^3*x/c^4 + 3*(c^2*x^2 - 1)*a*b*f*g^2*h*arcsin(c*x)/c^4 + 3*(c
^2*x^2 - 1)*a*b*e*g*h^2*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*arcsin(c*x)/c^4 + 2/5*a*b*e*h^3*x*arcsin(c*x)/c^4 + 15/32*b^2*f*g^2*h*
arcsin(c*x)^2/c^4 + 15/32*b^2*e*g*h^2*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 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*f*g^3/c^3 + 2*sqrt(-c^2*x^2 + 1)*a*b*e*g^2*h/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 + 2/25*(c^2*x^
2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*e*h^3/c^5 - 13/...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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