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

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

[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

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Rubi [A]  time = 0.698374, antiderivative size = 425, normalized size of antiderivative = 1., 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 = {4751, 4619, 4677, 8, 4627, 4707, 4641, 30} \[ \frac{b x \sqrt{1-c^2 x^2} (d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac{(d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{2 b d g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{2 b x^2 \sqrt{1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{4 b \sqrt{1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{b f h x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b f h x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac{3 f h \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac{1}{2} x^2 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} x^3 (e h+f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 x (e h+f g)}{9 c^2}-\frac{3 b^2 f h x^2}{32 c^2}-\frac{1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac{2}{27} b^2 x^3 (e h+f g)-\frac{1}{32} b^2 f h x^4 \]

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 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) \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*}

Mathematica [A]  time = 0.332326, size = 364, normalized size = 0.86 \[ -\frac{1}{4} b (d h+e g) \left (-\frac{2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-2 b d g \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{2 b (e h+f g) \left (-3 a \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right )}{27 c^3}-\frac{1}{32} b f h \left (-\frac{4 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )^2}{b c^4}+\frac{3 b x^2}{c^2}+b x^4\right )+\frac{1}{2} x^2 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} x^3 (e h+f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \]

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]  time = 0.114, size = 870, normalized size = 2.1 \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)*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x)

[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*(1/32
*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*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+5*arcsin(c*x)^2+5*c^2*x^2-4)+1/4*h*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)+1/4*c^2*g*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)+1/27*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*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*f*g*(9*c^3*x^3*arcsin(c*x)^2
+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*
c*x)+c^3*g*d*(arcsin(c*x)^2*c*x-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*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+h*c*e*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+
1)^(1/2))+c*f*g*(arcsin(c*x)^2*c*x-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*x^3*e*h+1/3*arcsin(c*x)*c^4*x^3*f*g+1/2*arcsin(c*x)*c^4*x^2*d*h+1/2*arcsin(c*x)*c^4*x
^2*e*g+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*arcsin(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., size = 0, normalized size = 0. \begin{align*} \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} + \frac{1}{2} \, a^{2} e g x^{2} + \frac{1}{2} \, a^{2} d h x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b e g + \frac{2}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b f g + \frac{1}{2} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d h + \frac{2}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e h + \frac{1}{16} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b f h - 2 \, b^{2} d g{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d g x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d g}{c} + \frac{1}{12} \,{\left (3 \, b^{2} f h x^{4} + 4 \,{\left (b^{2} f g + b^{2} e h\right )} x^{3} + 6 \,{\left (b^{2} e g + b^{2} d h\right )} x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + \int \frac{{\left (3 \, b^{2} c f h x^{4} + 4 \,{\left (b^{2} c f g + b^{2} c e h\right )} x^{3} + 6 \,{\left (b^{2} c e g + b^{2} c d h\right )} x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{6 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

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 + 1/3*a^2*e*h*x^3 + b^2*d*g*x*arcsin(c*x)^2 + 1/2*a^2*e*g*x^2 + 1/2*a^2*d*h*
x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*e*g
+ 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*f*g + 1/2*(2*x^2*ar
csin(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*h + 2/9*(3*x^3*arcsi
n(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*e*h + 1/16*(8*x^4*arcsin(c*x) + (2*sqr
t(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*f*h -
2*b^2*d*g*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*g*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d
*g/c + 1/12*(3*b^2*f*h*x^4 + 4*(b^2*f*g + b^2*e*h)*x^3 + 6*(b^2*e*g + b^2*d*h)*x^2)*arctan2(c*x, sqrt(c*x + 1)
*sqrt(-c*x + 1))^2 + integrate(1/6*(3*b^2*c*f*h*x^4 + 4*(b^2*c*f*g + b^2*c*e*h)*x^3 + 6*(b^2*c*e*g + b^2*c*d*h
)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 3.19133, size = 1289, normalized size = 3.03 \begin{align*} \frac{27 \,{\left (8 \, a^{2} - b^{2}\right )} c^{4} f h x^{4} + 32 \,{\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} f g +{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} e h\right )} x^{3} + 27 \,{\left (8 \,{\left (2 \, a^{2} - b^{2}\right )} c^{4} e g +{\left (8 \,{\left (2 \, a^{2} - b^{2}\right )} c^{4} d - 3 \, b^{2} c^{2} f\right )} h\right )} x^{2} + 9 \,{\left (24 \, b^{2} c^{4} f h x^{4} + 96 \, b^{2} c^{4} d g x - 24 \, b^{2} c^{2} e g + 32 \,{\left (b^{2} c^{4} f g + b^{2} c^{4} e h\right )} x^{3} + 48 \,{\left (b^{2} c^{4} e g + b^{2} c^{4} d h\right )} x^{2} - 3 \,{\left (8 \, b^{2} c^{2} d + 3 \, b^{2} f\right )} h\right )} \arcsin \left (c x\right )^{2} - 96 \,{\left (4 \, b^{2} c^{2} e h -{\left (9 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{4} d - 4 \, b^{2} c^{2} f\right )} g\right )} x + 18 \,{\left (24 \, a b c^{4} f h x^{4} + 96 \, a b c^{4} d g x - 24 \, a b c^{2} e g + 32 \,{\left (a b c^{4} f g + a b c^{4} e h\right )} x^{3} + 48 \,{\left (a b c^{4} e g + a b c^{4} d h\right )} x^{2} - 3 \,{\left (8 \, a b c^{2} d + 3 \, a b f\right )} h\right )} \arcsin \left (c x\right ) + 6 \,{\left (18 \, a b c^{3} f h x^{3} + 64 \, a b c e h + 32 \,{\left (a b c^{3} f g + a b c^{3} e h\right )} x^{2} + 32 \,{\left (9 \, a b c^{3} d + 2 \, a b c f\right )} g + 9 \,{\left (8 \, a b c^{3} e g +{\left (8 \, a b c^{3} d + 3 \, a b c f\right )} h\right )} x +{\left (18 \, b^{2} c^{3} f h x^{3} + 64 \, b^{2} c e h + 32 \,{\left (b^{2} c^{3} f g + b^{2} c^{3} e h\right )} x^{2} + 32 \,{\left (9 \, b^{2} c^{3} d + 2 \, b^{2} c f\right )} g + 9 \,{\left (8 \, b^{2} c^{3} e g +{\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} h\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{864 \, c^{4}} \end{align*}

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

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Sympy [A]  time = 6.11447, size = 1059, normalized size = 2.49 \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)*(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]  time = 1.32244, size = 1613, normalized size = 3.8 \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)*(f*x^2+e*x+d)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

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