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

Optimal. Leaf size=335 \[ \frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{8 b d e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{16 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8 b^2 d e x}{9 c^2}-\frac{8 b^2 e^2 x^3}{225 c^2}-\frac{16 b^2 e^2 x}{75 c^4}-2 b^2 d^2 x-\frac{4}{27} b^2 d e x^3-\frac{2}{125} b^2 e^2 x^5 \]

[Out]

-2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4*b^2*d*e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2
) - (2*b^2*e^2*x^5)/125 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (8*b*d*e*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/(9*c^3) + (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*d*e*x^2*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*e^2*x^
4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + d^2*x*(a + b*ArcSin[c*x])^2 + (2*d*e*x^3*(a + b*ArcSin[c*x])
^2)/3 + (e^2*x^5*(a + b*ArcSin[c*x])^2)/5

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Rubi [A]  time = 0.556517, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4667, 4619, 4677, 8, 4627, 4707, 30} \[ \frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{8 b d e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{16 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8 b^2 d e x}{9 c^2}-\frac{8 b^2 e^2 x^3}{225 c^2}-\frac{16 b^2 e^2 x}{75 c^4}-2 b^2 d^2 x-\frac{4}{27} b^2 d e x^3-\frac{2}{125} b^2 e^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4*b^2*d*e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2
) - (2*b^2*e^2*x^5)/125 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (8*b*d*e*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/(9*c^3) + (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*d*e*x^2*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*e^2*x^
4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + d^2*x*(a + b*ArcSin[c*x])^2 + (2*d*e*x^3*(a + b*ArcSin[c*x])
^2)/3 + (e^2*x^5*(a + b*ArcSin[c*x])^2)/5

Rule 4667

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

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 30

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

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \sin ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \sin ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} (4 b c d e) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{5} \left (2 b c e^2\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d^2\right ) \int 1 \, dx-\frac{1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx-\frac{(8 b d e) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{9 c}-\frac{1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx-\frac{\left (8 b e^2\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 c}\\ &=-2 b^2 d^2 x-\frac{4}{27} b^2 d e x^3-\frac{2}{125} b^2 e^2 x^5+\frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{8 b d e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac{\left (16 b e^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 c^3}-\frac{\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=-2 b^2 d^2 x-\frac{8 b^2 d e x}{9 c^2}-\frac{4}{27} b^2 d e x^3-\frac{8 b^2 e^2 x^3}{225 c^2}-\frac{2}{125} b^2 e^2 x^5+\frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{8 b d e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{16 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=-2 b^2 d^2 x-\frac{8 b^2 d e x}{9 c^2}-\frac{16 b^2 e^2 x}{75 c^4}-\frac{4}{27} b^2 d e x^3-\frac{8 b^2 e^2 x^3}{225 c^2}-\frac{2}{125} b^2 e^2 x^5+\frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{8 b d e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{16 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.321287, size = 291, normalized size = 0.87 \[ \frac{225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+30 a b \sqrt{1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )+30 b \sin ^{-1}(c x) \left (15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt{1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )\right )-2 b^2 c x \left (c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )+60 c^2 e \left (25 d+e x^2\right )+360 e^2\right )+225 b^2 c^5 x \sin ^{-1}(c x)^2 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{3375 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + 30*a*b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2)
+ c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)) - 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 25
0*d*e*x^2 + 27*e^2*x^4)) + 30*b*(15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(24*e^2 +
4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSin[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d
*e*x^2 + 3*e^2*x^4)*ArcSin[c*x]^2)/(3375*c^5)

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Maple [B]  time = 0.075, size = 635, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^4*(1/5*e^2*c^5*x^5+2/3*c^5*e*d*x^3+d^2*c^5*x)+b^2/c^4*(1/3375*e^2*(675*arcsin(c*x)^2*c^5*x^5+270*ar
csin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-2250*c^3*x^3*arcsin(c*x)^2-54*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^2*x^2+3375*arcsin(c*x)^2*c*x+380*c^3*x^3+4470*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4470*c*x)+2/27*c^2*e*d*(9*c^3*
x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2
*x^2+1)^(1/2)+42*c*x)+2/27*e^2*(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)+d^2*c^4*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c
^2*x^2+1)^(1/2))+2*c^2*e*d*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+e^2*(arcsin(c*x)^2*c*x-2
*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c^4*(1/5*arcsin(c*x)*e^2*c^5*x^5+2/3*arcsin(c*x)*c^5*e*d*x^3+arc
sin(c*x)*d^2*c^5*x-1/5*e^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))-2/3*c^2*e*d*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+d^2*c^4*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.48144, size = 590, normalized size = 1.76 \begin{align*} \frac{1}{5} \, b^{2} e^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac{1}{5} \, a^{2} e^{2} x^{5} + \frac{2}{3} \, b^{2} d e x^{3} \arcsin \left (c x\right )^{2} + \frac{2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac{4}{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 d e + \frac{4}{27} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac{2}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} + \frac{2}{1125} \,{\left (15 \,{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} - 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*e^2*x^5*arcsin(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arcsin(c*x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*
arcsin(c*x)^2 + 4/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d*e +
4/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*d*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*e^2 + 2/1125*(15*(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*arcsin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*e^2 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*ar
csin(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c

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Fricas [A]  time = 2.18965, size = 790, normalized size = 2.36 \begin{align*} \frac{27 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \,{\left (25 \,{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \,{\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \arcsin \left (c x\right )^{2} + 15 \,{\left (225 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e - 48 \, b^{2} c e^{2}\right )} x + 450 \,{\left (3 \, a b c^{5} e^{2} x^{5} + 10 \, a b c^{5} d e x^{3} + 15 \, a b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) + 30 \,{\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \,{\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2} +{\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \,{\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{3375 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(27*(25*a^2 - 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 - 2*b^2)*c^5*d*e - 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^
5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^2*c^5*d^2*x)*arcsin(c*x)^2 + 15*(225*(a^2 - 2*b^2)*c^5*d^2 - 200*b^2*c^3
*d*e - 48*b^2*c*e^2)*x + 450*(3*a*b*c^5*e^2*x^5 + 10*a*b*c^5*d*e*x^3 + 15*a*b*c^5*d^2*x)*arcsin(c*x) + 30*(9*a
*b*c^4*e^2*x^4 + 225*a*b*c^4*d^2 + 100*a*b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e + 6*a*b*c^2*e^2)*x^2 + (9*
b^2*c^4*e^2*x^4 + 225*b^2*c^4*d^2 + 100*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c^4*d*e + 6*b^2*c^2*e^2)*x^2)*arc
sin(c*x))*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 6.24232, size = 595, normalized size = 1.78 \begin{align*} \begin{cases} a^{2} d^{2} x + \frac{2 a^{2} d e x^{3}}{3} + \frac{a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname{asin}{\left (c x \right )} + \frac{4 a b d e x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{2 a b e^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{2 a b d^{2} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{4 a b d e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{2 a b e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{8 a b d e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{8 a b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} + \frac{16 a b e^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname{asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + \frac{2 b^{2} d e x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{4 b^{2} d e x^{3}}{27} + \frac{b^{2} e^{2} x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} - \frac{2 b^{2} e^{2} x^{5}}{125} + \frac{2 b^{2} d^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{c} + \frac{4 b^{2} d e x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c} + \frac{2 b^{2} e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{25 c} - \frac{8 b^{2} d e x}{9 c^{2}} - \frac{8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac{8 b^{2} d e \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c^{3}} + \frac{8 b^{2} e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{75 c^{3}} - \frac{16 b^{2} e^{2} x}{75 c^{4}} + \frac{16 b^{2} e^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{75 c^{5}} & \text{for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac{2 d e x^{3}}{3} + \frac{e^{2} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2*x*asin(c*x) + 4*a*b*d*e*x**3*asin(c
*x)/3 + 2*a*b*e**2*x**5*asin(c*x)/5 + 2*a*b*d**2*sqrt(-c**2*x**2 + 1)/c + 4*a*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/
(9*c) + 2*a*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 8*a*b*d*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*a*b*e**2*x**
2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 16*a*b*e**2*sqrt(-c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*asin(c*x)**2 - 2*b
**2*d**2*x + 2*b**2*d*e*x**3*asin(c*x)**2/3 - 4*b**2*d*e*x**3/27 + b**2*e**2*x**5*asin(c*x)**2/5 - 2*b**2*e**2
*x**5/125 + 2*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 4*b**2*d*e*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c
) + 2*b**2*e**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) - 8*b**2*d*e*x/(9*c**2) - 8*b**2*e**2*x**3/(225*c**
2) + 8*b**2*d*e*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 8*b**2*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*
c**3) - 16*b**2*e**2*x/(75*c**4) + 16*b**2*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**5), Ne(c, 0)), (a**2*(d*
*2*x + 2*d*e*x**3/3 + e**2*x**5/5), True))

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Giac [B]  time = 1.33001, size = 915, normalized size = 2.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5*e^2 + b^2*d^2*x*arcsin(c*x)^2 + 2/3*a^2*d*x^3*e + 2*a*b*d^2*x*arcsin(c*x) + 2/3*(c^2*x^2 - 1)*b^2*
d*x*arcsin(c*x)^2*e/c^2 + a^2*d^2*x - 2*b^2*d^2*x + 4/3*(c^2*x^2 - 1)*a*b*d*x*arcsin(c*x)*e/c^2 + 2/3*b^2*d*x*
arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c + 1/5*(c^2*x^2 - 1)^2*b^2*x*arcsin(c*x)^2*e^2
/c^4 - 4/27*(c^2*x^2 - 1)*b^2*d*x*e/c^2 + 4/3*a*b*d*x*arcsin(c*x)*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 4/9
*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin(c*x)*e/c^3 + 2/5*(c^2*x^2 - 1)^2*a*b*x*arcsin(c*x)*e^2/c^4 + 2/5*(c^2*x^2 -
1)*b^2*x*arcsin(c*x)^2*e^2/c^4 - 28/27*b^2*d*x*e/c^2 - 4/9*(-c^2*x^2 + 1)^(3/2)*a*b*d*e/c^3 + 4/3*sqrt(-c^2*x
^2 + 1)*b^2*d*arcsin(c*x)*e/c^3 - 2/125*(c^2*x^2 - 1)^2*b^2*x*e^2/c^4 + 4/5*(c^2*x^2 - 1)*a*b*x*arcsin(c*x)*e^
2/c^4 + 1/5*b^2*x*arcsin(c*x)^2*e^2/c^4 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^5 + 4/
3*sqrt(-c^2*x^2 + 1)*a*b*d*e/c^3 - 76/1125*(c^2*x^2 - 1)*b^2*x*e^2/c^4 + 2/5*a*b*x*arcsin(c*x)*e^2/c^4 + 2/25*
(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b^2*arcsin(c*x)*e^2/c^5 - 298/1125*
b^2*x*e^2/c^4 - 4/15*(-c^2*x^2 + 1)^(3/2)*a*b*e^2/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^5 + 2/5*s
qrt(-c^2*x^2 + 1)*a*b*e^2/c^5