3.2.17 \(\int (g+h x) (d+e x+f x^2) (a+b \text {ArcSin}(c x))^2 \, dx\) [117]

Optimal. Leaf size=425 \[ -2 b^2 d g x-\frac {4 b^2 (f g+e h) x}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}-\frac {3 f h (a+b \text {ArcSin}(c x))^2}{32 c^4}-\frac {(e g+d h) (a+b \text {ArcSin}(c x))^2}{4 c^2}+d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^2 \]

[Out]

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

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Rubi [A]
time = 0.48, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4835, 4715, 4767, 8, 4723, 4795, 4737, 30} \begin {gather*} -\frac {3 f h (a+b \text {ArcSin}(c x))^2}{32 c^4}+\frac {b x \sqrt {1-c^2 x^2} (d h+e g) (a+b \text {ArcSin}(c x))}{2 c}-\frac {(d h+e g) (a+b \text {ArcSin}(c x))^2}{4 c^2}+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (e h+f g) (a+b \text {ArcSin}(c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {4 b \sqrt {1-c^2 x^2} (e h+f g) (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {1}{2} x^2 (d h+e g) (a+b \text {ArcSin}(c x))^2+d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{3} x^3 (e h+f g) (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^2-\frac {4 b^2 x (e h+f g)}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac {2}{27} b^2 x^3 (e h+f g)-\frac {1}{32} b^2 f h x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 30

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

Rule 4715

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

Rule 4723

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

Rule 4737

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

Rule 4767

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

Rule 4795

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

Rule 4835

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

Rubi steps

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

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Mathematica [A]
time = 0.25, size = 364, normalized size = 0.86 \begin {gather*} d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^2-\frac {2 b (f g+e h) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \text {ArcSin}(c x)\right )}{27 c^3}-2 b d g \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}\right )-\frac {1}{32} b f h \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \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 (e g+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)*(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2,x]

[Out]

d*g*x*(a + b*ArcSin[c*x])^2 + ((e*g + d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + ((f*g + e*h)*x^3*(a + b*ArcSin[c*x])
^2)/3 + (f*h*x^4*(a + b*ArcSin[c*x])^2)/4 - (2*b*(f*g + e*h)*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6
+ c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - 2*b*d*g*(b*x - (Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x]))/c) - (b*f*h*((3*b*x^2)/c^2 + b*x^4 - (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^3 - (4*x
^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (3*(a + b*ArcSin[c*x])^2)/(b*c^4)))/32 - (b*(e*g + d*h)*(b*x^2 -
(2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(869\) vs. \(2(383)=766\).
time = 0.25, size = 870, normalized size = 2.05 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^3*(1/4*h*f*c^4*x^4+1/3*(c*e*h+c*f*g)*c^3*x^3+1/2*(c^2*d*h+c^2*e*g)*c^2*x^2+c^4*g*d*x)+b^2/c^3*(1/12
8*h*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^4-4
0*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+20*arcsin(c*x)^2+20*c^2*x^2-25)+1/4*c^2*d*h*(2*arcsin(c*x)^2*c^2*x^2+2*ar
csin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/4*c^2*e*g*(2*arcsin(c*x)^2*c^2*x^2+2*arcsin(c*x)*(-c
^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/27*h*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)+1/27*c*g*f*(9*c^3*x^3*arcsi
n(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)+c^3*g*d*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*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)+h*c*e*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-
c^2*x^2+1)^(1/2))+c*g*f*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c^3*(1/4*arcsin(c*x)
*h*f*c^4*x^4+1/3*arcsin(c*x)*c^4*e*h*x^3+1/3*arcsin(c*x)*c^4*f*g*x^3+1/2*arcsin(c*x)*c^4*d*h*x^2+1/2*arcsin(c*
x)*c^4*e*g*x^2+arcsin(c*x)*c^4*g*d*x-1/4*h*f*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*a
rcsin(c*x))-1/12*(4*c*e*h+4*c*f*g)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-1/12*(6*c^2*d*h+6*
c^2*e*g)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^3*g*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)*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/4*a^2*f*h*x^4 + 1/3*a^2*f*g*x^3 + b^2*d*g*x*arcsin(c*x)^2 + 1/3*a^2*h*x^3*e + 1/2*a^2*d*h*x^2 + 1/2*a^2*g*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 + 1/2*(2*x^2
*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d*h + 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^
2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*f*h - 2*b^2*d*g*(x - sqrt(-c^2*x^2
+ 1)*arcsin(c*x)/c) + a^2*d*g*x + 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*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*h*e + 2*(c*x*a
rcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d*g/c + 1/12*(3*b^2*f*h*x^4 + 4*(b^2*f*g + b^2*h*e)*x^3 + 6*(b^2*d*h + b^
2*g*e)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/6*(3*b^2*c*f*h*x^4 + 4*(b^2*c*f*g + b^2
*c*h*e)*x^3 + 6*(b^2*c*d*h + b^2*c*g*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 = 2.07, size = 600, normalized size = 1.41 \begin {gather*} \frac {27 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} f h x^{4} + 32 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} f g x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d - 3 \, b^{2} c^{2} f\right )} h x^{2} + 96 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d - 4 \, b^{2} c^{2} f\right )} g x + 9 \, {\left (24 \, b^{2} c^{4} f h x^{4} + 32 \, b^{2} c^{4} f g x^{3} + 48 \, b^{2} c^{4} d h x^{2} + 96 \, b^{2} c^{4} d g x - 3 \, {\left (8 \, b^{2} c^{2} d + 3 \, b^{2} f\right )} h + 8 \, {\left (4 \, b^{2} c^{4} h x^{3} + 6 \, b^{2} c^{4} g x^{2} - 3 \, b^{2} c^{2} g\right )} e\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (24 \, a b c^{4} f h x^{4} + 32 \, a b c^{4} f g x^{3} + 48 \, a b c^{4} d h x^{2} + 96 \, a b c^{4} d g x - 3 \, {\left (8 \, a b c^{2} d + 3 \, a b f\right )} h + 8 \, {\left (4 \, a b c^{4} h x^{3} + 6 \, a b c^{4} g x^{2} - 3 \, a b c^{2} g\right )} e\right )} \arcsin \left (c x\right ) + 8 \, {\left (4 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} h x^{3} + 27 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} g x^{2} - 48 \, b^{2} c^{2} h x\right )} e + 6 \, {\left (18 \, a b c^{3} f h x^{3} + 32 \, a b c^{3} f g x^{2} + 9 \, {\left (8 \, a b c^{3} d + 3 \, a b c f\right )} h x + 32 \, {\left (9 \, a b c^{3} d + 2 \, a b c f\right )} g + {\left (18 \, b^{2} c^{3} f h x^{3} + 32 \, b^{2} c^{3} f g x^{2} + 9 \, {\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} h x + 32 \, {\left (9 \, b^{2} c^{3} d + 2 \, b^{2} c f\right )} g + 8 \, {\left (4 \, b^{2} c^{3} h x^{2} + 9 \, b^{2} c^{3} g x + 8 \, b^{2} c h\right )} e\right )} \arcsin \left (c x\right ) + 8 \, {\left (4 \, a b c^{3} h x^{2} + 9 \, a b c^{3} g x + 8 \, a b c h\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (416) = 832\).
time = 0.61, size = 1059, normalized size = 2.49 \begin {gather*} \begin {cases} a^{2} d g x + \frac {a^{2} d h x^{2}}{2} + \frac {a^{2} e g x^{2}}{2} + \frac {a^{2} e h x^{3}}{3} + \frac {a^{2} f g x^{3}}{3} + \frac {a^{2} f h x^{4}}{4} + 2 a b d g x \operatorname {asin}{\left (c x \right )} + a b d h x^{2} \operatorname {asin}{\left (c x \right )} + a b e g x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b e h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b f g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {a b f h x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {2 a b d g \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b d h x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {a b e g x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {2 a b e h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 a b f g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {a b f h x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {a b d h \operatorname {asin}{\left (c x \right )}}{2 c^{2}} - \frac {a b e g \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b e h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 a b f g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 a b f h x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b f h \operatorname {asin}{\left (c x \right )}}{16 c^{4}} + b^{2} d g x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d g x + \frac {b^{2} d h x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} d h x^{2}}{4} + \frac {b^{2} e g x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e g x^{2}}{4} + \frac {b^{2} e h x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e h x^{3}}{27} + \frac {b^{2} f g x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} f g x^{3}}{27} + \frac {b^{2} f h x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {b^{2} f h x^{4}}{32} + \frac {2 b^{2} d g \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} d h x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} + \frac {b^{2} e g x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} + \frac {2 b^{2} e h x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {2 b^{2} f g x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {b^{2} f h x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{8 c} - \frac {b^{2} d h \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {b^{2} e g \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} e h x}{9 c^{2}} - \frac {4 b^{2} f g x}{9 c^{2}} - \frac {3 b^{2} f h x^{2}}{32 c^{2}} + \frac {4 b^{2} e h \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {4 b^{2} f g \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {3 b^{2} f h x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} f h \operatorname {asin}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d g x + \frac {d h x^{2}}{2} + \frac {e g x^{2}}{2} + \frac {e h x^{3}}{3} + \frac {f g x^{3}}{3} + \frac {f h x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (383) = 766\).
time = 0.45, size = 1145, normalized size = 2.69 \begin {gather*} \frac {1}{4} \, a^{2} f h x^{4} + \frac {1}{3} \, a^{2} f g x^{3} + \frac {1}{3} \, a^{2} e h x^{3} + b^{2} d g x \arcsin \left (c x\right )^{2} + 2 \, a b d g x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} f g x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e h x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e g x \arcsin \left (c x\right )}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d h x \arcsin \left (c x\right )}{2 \, c} + a^{2} d g x - 2 \, b^{2} d g x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b f g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b e h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e g \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d h \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {b^{2} f g x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {b^{2} e h x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e g x}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d h x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d g \arcsin \left (c x\right )}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} f h x \arcsin \left (c x\right )}{8 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} f g x}{27 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e h x}{27 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e g \arcsin \left (c x\right )}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b d h \arcsin \left (c x\right )}{c^{2}} + \frac {2 \, a b f g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, a b e h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} e g \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {b^{2} d h \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} f h \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d g}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b f h x}{8 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} f g \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e h \arcsin \left (c x\right )}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} f h x \arcsin \left (c x\right )}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e g}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e g}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d h}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d h}{4 \, c^{2}} - \frac {14 \, b^{2} f g x}{27 \, c^{2}} - \frac {14 \, b^{2} e h x}{27 \, c^{2}} + \frac {a b e g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {a b d h \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b f h \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} f h \arcsin \left (c x\right )^{2}}{2 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b f g}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b f h x}{16 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} f g \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e h \arcsin \left (c x\right )}{3 \, c^{3}} - \frac {b^{2} e g}{8 \, c^{2}} - \frac {b^{2} d h}{8 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} f h}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b f h \arcsin \left (c x\right )}{c^{4}} + \frac {5 \, b^{2} f h \arcsin \left (c x\right )^{2}}{32 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b f g}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e h}{3 \, c^{3}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} f h}{32 \, c^{4}} + \frac {5 \, a b f h \arcsin \left (c x\right )}{16 \, c^{4}} - \frac {17 \, b^{2} f h}{256 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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