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

Optimal. Leaf size=569 \[ \frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^5}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^5}+\frac{32 b e^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^7}+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{16 b^2 d e^2 x}{25 c^4}-\frac{12 b^2 e^3 x^5}{1225 c^2}-\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{32 b^2 e^3 x}{245 c^6}-\frac{2}{9} b^2 d^2 e x^3-2 b^2 d^3 x-\frac{6}{125} b^2 d e^2 x^5-\frac{2}{343} b^2 e^3 x^7 \]

[Out]

-2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) - (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) - (2*b^2*d^2*e*x
^3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) - (16*b^2*e^3*x^3)/(735*c^4) - (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(12
25*c^2) - (2*b^2*e^3*x^7)/343 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d^2*e*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/(3*c^3) + (16*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^5) + (32*b*e^3*Sqrt
[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^7) + (2*b*d^2*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (
8*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^3) + (16*b*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/(245*c^5) + (6*b*d*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1 - c^2*x
^2]*(a + b*ArcSin[c*x]))/(245*c^3) + (2*b*e^3*x^6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(49*c) + d^3*x*(a + b
*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSin[c*x])^2)/5 + (e^3*x^7*(a + b*Ar
cSin[c*x])^2)/7

________________________________________________________________________________________

Rubi [A]  time = 0.96268, antiderivative size = 569, normalized size of antiderivative = 1., number of steps used = 26, 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} \[ \frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^5}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^5}+\frac{32 b e^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^7}+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{16 b^2 d e^2 x}{25 c^4}-\frac{12 b^2 e^3 x^5}{1225 c^2}-\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{32 b^2 e^3 x}{245 c^6}-\frac{2}{9} b^2 d^2 e x^3-2 b^2 d^3 x-\frac{6}{125} b^2 d e^2 x^5-\frac{2}{343} b^2 e^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) - (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) - (2*b^2*d^2*e*x
^3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) - (16*b^2*e^3*x^3)/(735*c^4) - (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(12
25*c^2) - (2*b^2*e^3*x^7)/343 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d^2*e*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/(3*c^3) + (16*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^5) + (32*b*e^3*Sqrt
[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^7) + (2*b*d^2*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (
8*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^3) + (16*b*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/(245*c^5) + (6*b*d*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1 - c^2*x
^2]*(a + b*ArcSin[c*x]))/(245*c^3) + (2*b*e^3*x^6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(49*c) + d^3*x*(a + b
*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSin[c*x])^2)/5 + (e^3*x^7*(a + b*Ar
cSin[c*x])^2)/7

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 )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 \left (a+b \sin ^{-1}(c x)\right )^2+3 d^2 e x^2 \left (a+b \sin ^{-1}(c x)\right )^2+3 d e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+e^3 x^6 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (3 d^2 e\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (3 d e^2\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+e^3 \int x^6 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\left (2 b c d^2 e\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{5} \left (6 b c d e^2\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{7} \left (2 b c e^3\right ) \int \frac{x^7 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d^3\right ) \int 1 \, dx-\frac{1}{3} \left (2 b^2 d^2 e\right ) \int x^2 \, dx-\frac{\left (4 b d^2 e\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 c}-\frac{1}{25} \left (6 b^2 d e^2\right ) \int x^4 \, dx-\frac{\left (24 b d e^2\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 c}-\frac{1}{49} \left (2 b^2 e^3\right ) \int x^6 \, dx-\frac{\left (12 b e^3\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{49 c}\\ &=-2 b^2 d^3 x-\frac{2}{9} b^2 d^2 e x^3-\frac{6}{125} b^2 d e^2 x^5-\frac{2}{343} b^2 e^3 x^7+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (4 b^2 d^2 e\right ) \int 1 \, dx}{3 c^2}-\frac{\left (16 b d e^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 c^3}-\frac{\left (8 b^2 d e^2\right ) \int x^2 \, dx}{25 c^2}-\frac{\left (48 b e^3\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{245 c^3}-\frac{\left (12 b^2 e^3\right ) \int x^4 \, dx}{245 c^2}\\ &=-2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}-\frac{2}{343} b^2 e^3 x^7+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^5}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^5}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (16 b^2 d e^2\right ) \int 1 \, dx}{25 c^4}-\frac{\left (32 b e^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{245 c^5}-\frac{\left (16 b^2 e^3\right ) \int x^2 \, dx}{245 c^4}\\ &=-2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{16 b^2 d e^2 x}{25 c^4}-\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}-\frac{2}{343} b^2 e^3 x^7+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^5}+\frac{32 b e^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^7}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^5}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (32 b^2 e^3\right ) \int 1 \, dx}{245 c^6}\\ &=-2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{16 b^2 d e^2 x}{25 c^4}-\frac{32 b^2 e^3 x}{245 c^6}-\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}-\frac{2}{343} b^2 e^3 x^7+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^5}+\frac{32 b e^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^7}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c^3}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^5}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{245 c^3}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.493539, size = 435, normalized size = 0.76 \[ -\frac{2 b d^2 e \left (-3 a \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right )}{9 c^3}-2 b d^3 \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{2 b d e^2 \left (-15 a \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )+b c x \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 b \sqrt{1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right ) \sin ^{-1}(c x)\right )}{375 c^5}-\frac{2 b e^3 \left (-105 a \sqrt{1-c^2 x^2} \left (5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right )+b c x \left (75 c^6 x^6+126 c^4 x^4+280 c^2 x^2+1680\right )-105 b \sqrt{1-c^2 x^2} \left (5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right ) \sin ^{-1}(c x)\right )}{25725 c^7}+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

d^3*x*(a + b*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSin[c*x])^2)/5 + (e^3*x
^7*(a + b*ArcSin[c*x])^2)/7 - (2*b*d^2*e*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqr
t[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(9*c^3) - (2*b*d*e^2*(-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]
))/(375*c^5) - (2*b*e^3*(-105*a*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6) + b*c*x*(1680 + 280
*c^2*x^2 + 126*c^4*x^4 + 75*c^6*x^6) - 105*b*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6)*ArcSin
[c*x]))/(25725*c^7) - 2*b*d^3*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c)

________________________________________________________________________________________

Maple [B]  time = 0.125, size = 1194, 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((e*x^2+d)^3*(a+b*arcsin(c*x))^2,x)

[Out]

1/c*(a^2/c^6*(1/7*e^3*c^7*x^7+3/5*c^7*d*e^2*x^5+c^7*d^2*e*x^3+d^3*c^7*x)+b^2/c^6*(1/385875*e^3*(55125*arcsin(c
*x)^2*c^7*x^7+15750*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^6*x^6-231525*arcsin(c*x)^2*c^5*x^5-2250*c^7*x^7-73710*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4+385875*c^3*x^3*arcsin(c*x)^2+14742*c^5*x^5+158970*arcsin(c*x)*(-c^2*x^2+1)
^(1/2)*c^2*x^2-385875*arcsin(c*x)^2*c*x-52990*c^3*x^3-453810*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+453810*c*x)+1/1125
*c^2*d*e^2*(675*arcsin(c*x)^2*c^5*x^5+270*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-2250*c^3*x^3*arcsin(c*x)^2-54
*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+3375*arcsin(c*x)^2*c*x+380*c^3*x^3+4470*arcsin(c*x)*(-c^2
*x^2+1)^(1/2)-4470*c*x)+1/1125*e^3*(675*arcsin(c*x)^2*c^5*x^5+270*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-2250*
c^3*x^3*arcsin(c*x)^2-54*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+3375*arcsin(c*x)^2*c*x+380*c^3*x^
3+4470*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4470*c*x)+1/9*c^4*d^2*e*(9*c^3*x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2
+1)^(1/2)*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+2/9*c^2*d*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)+1/9*e^3*(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^3*c^6*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-
c^2*x^2+1)^(1/2))+3*c^4*d^2*e*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+3*c^2*d*e^2*(arcsin(c
*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+e^3*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2
)))+2*a*b/c^6*(1/7*arcsin(c*x)*e^3*c^7*x^7+3/5*arcsin(c*x)*c^7*d*e^2*x^5+arcsin(c*x)*c^7*d^2*e*x^3+arcsin(c*x)
*d^3*c^7*x-1/7*e^3*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^
(1/2)-16/35*(-c^2*x^2+1)^(1/2))-3/5*c^2*d*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))-c^4*d^2*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+d^3*c^6*(-c^2*x^2
+1)^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.52988, size = 944, normalized size = 1.66 \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)^3*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/7*b^2*e^3*x^7*arcsin(c*x)^2 + 1/7*a^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arcsin(c*x)^2 + 3/5*a^2*d*e^2*x^5 + b^2*d^
2*e*x^3*arcsin(c*x)^2 + a^2*d^2*e*x^3 + b^2*d^3*x*arcsin(c*x)^2 + 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^2*e + 2/9*(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^2*e + 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*d*e^2 + 2/375*(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*d*e^2 + 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 +
8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*e^3 + 2/25725*(105*(5*sqrt(-c^2*x^2 + 1)*x^6
/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcsin(c*x)
- (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*e^3 - 2*b^2*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*
x)/c) + a^2*d^3*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^3/c

________________________________________________________________________________________

Fricas [A]  time = 2.25981, size = 1277, normalized size = 2.24 \begin{align*} \frac{1125 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \,{\left (49 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{7} d e^{2} - 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \,{\left (1225 \,{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} e - 1176 \, b^{2} c^{5} d e^{2} - 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \,{\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \arcsin \left (c x\right )^{2} + 105 \,{\left (3675 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} - 4900 \, b^{2} c^{5} d^{2} e - 2352 \, b^{2} c^{3} d e^{2} - 480 \, b^{2} c e^{3}\right )} x + 22050 \,{\left (5 \, a b c^{7} e^{3} x^{7} + 21 \, a b c^{7} d e^{2} x^{5} + 35 \, a b c^{7} d^{2} e x^{3} + 35 \, a b c^{7} d^{3} x\right )} \arcsin \left (c x\right ) + 210 \,{\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \,{\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} +{\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2} +{\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \,{\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} +{\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{385875 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

1/385875*(1125*(49*a^2 - 2*b^2)*c^7*e^3*x^7 + 189*(49*(25*a^2 - 2*b^2)*c^7*d*e^2 - 20*b^2*c^5*e^3)*x^5 + 35*(1
225*(9*a^2 - 2*b^2)*c^7*d^2*e - 1176*b^2*c^5*d*e^2 - 240*b^2*c^3*e^3)*x^3 + 11025*(5*b^2*c^7*e^3*x^7 + 21*b^2*
c^7*d*e^2*x^5 + 35*b^2*c^7*d^2*e*x^3 + 35*b^2*c^7*d^3*x)*arcsin(c*x)^2 + 105*(3675*(a^2 - 2*b^2)*c^7*d^3 - 490
0*b^2*c^5*d^2*e - 2352*b^2*c^3*d*e^2 - 480*b^2*c*e^3)*x + 22050*(5*a*b*c^7*e^3*x^7 + 21*a*b*c^7*d*e^2*x^5 + 35
*a*b*c^7*d^2*e*x^3 + 35*a*b*c^7*d^3*x)*arcsin(c*x) + 210*(75*a*b*c^6*e^3*x^6 + 3675*a*b*c^6*d^3 + 2450*a*b*c^4
*d^2*e + 1176*a*b*c^2*d*e^2 + 240*a*b*e^3 + 9*(49*a*b*c^6*d*e^2 + 10*a*b*c^4*e^3)*x^4 + (1225*a*b*c^6*d^2*e +
588*a*b*c^4*d*e^2 + 120*a*b*c^2*e^3)*x^2 + (75*b^2*c^6*e^3*x^6 + 3675*b^2*c^6*d^3 + 2450*b^2*c^4*d^2*e + 1176*
b^2*c^2*d*e^2 + 240*b^2*e^3 + 9*(49*b^2*c^6*d*e^2 + 10*b^2*c^4*e^3)*x^4 + (1225*b^2*c^6*d^2*e + 588*b^2*c^4*d*
e^2 + 120*b^2*c^2*e^3)*x^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^7

________________________________________________________________________________________

Sympy [A]  time = 17.9512, size = 989, normalized size = 1.74 \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)**3*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((a**2*d**3*x + a**2*d**2*e*x**3 + 3*a**2*d*e**2*x**5/5 + a**2*e**3*x**7/7 + 2*a*b*d**3*x*asin(c*x) +
2*a*b*d**2*e*x**3*asin(c*x) + 6*a*b*d*e**2*x**5*asin(c*x)/5 + 2*a*b*e**3*x**7*asin(c*x)/7 + 2*a*b*d**3*sqrt(-
c**2*x**2 + 1)/c + 2*a*b*d**2*e*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 6*a*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c
) + 2*a*b*e**3*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + 4*a*b*d**2*e*sqrt(-c**2*x**2 + 1)/(3*c**3) + 8*a*b*d*e**2*x*
*2*sqrt(-c**2*x**2 + 1)/(25*c**3) + 12*a*b*e**3*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 16*a*b*d*e**2*sqrt(-c**
2*x**2 + 1)/(25*c**5) + 16*a*b*e**3*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 32*a*b*e**3*sqrt(-c**2*x**2 + 1)/(2
45*c**7) + b**2*d**3*x*asin(c*x)**2 - 2*b**2*d**3*x + b**2*d**2*e*x**3*asin(c*x)**2 - 2*b**2*d**2*e*x**3/9 + 3
*b**2*d*e**2*x**5*asin(c*x)**2/5 - 6*b**2*d*e**2*x**5/125 + b**2*e**3*x**7*asin(c*x)**2/7 - 2*b**2*e**3*x**7/3
43 + 2*b**2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 2*b**2*d**2*e*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c) +
6*b**2*d*e**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) + 2*b**2*e**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/(49
*c) - 4*b**2*d**2*e*x/(3*c**2) - 8*b**2*d*e**2*x**3/(75*c**2) - 12*b**2*e**3*x**5/(1225*c**2) + 4*b**2*d**2*e*
sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + 8*b**2*d*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c**3) + 12*b**
2*e**3*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(245*c**3) - 16*b**2*d*e**2*x/(25*c**4) - 16*b**2*e**3*x**3/(735*c*
*4) + 16*b**2*d*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c**5) + 16*b**2*e**3*x**2*sqrt(-c**2*x**2 + 1)*asin(c*
x)/(245*c**5) - 32*b**2*e**3*x/(245*c**6) + 32*b**2*e**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(245*c**7), Ne(c, 0)),
(a**2*(d**3*x + d**2*e*x**3 + 3*d*e**2*x**5/5 + e**3*x**7/7), True))

________________________________________________________________________________________

Giac [B]  time = 1.38743, size = 1642, normalized size = 2.89 \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)^3*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

1/7*a^2*x^7*e^3 + 3/5*a^2*d*x^5*e^2 + b^2*d^3*x*arcsin(c*x)^2 + a^2*d^2*x^3*e + 2*a*b*d^3*x*arcsin(c*x) + (c^2
*x^2 - 1)*b^2*d^2*x*arcsin(c*x)^2*e/c^2 + a^2*d^3*x - 2*b^2*d^3*x + 2*(c^2*x^2 - 1)*a*b*d^2*x*arcsin(c*x)*e/c^
2 + b^2*d^2*x*arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c + 3/5*(c^2*x^2 - 1)^2*b^2*d*x*a
rcsin(c*x)^2*e^2/c^4 - 2/9*(c^2*x^2 - 1)*b^2*d^2*x*e/c^2 + 2*a*b*d^2*x*arcsin(c*x)*e/c^2 + 2*sqrt(-c^2*x^2 + 1
)*a*b*d^3/c - 2/3*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*arcsin(c*x)*e/c^3 + 6/5*(c^2*x^2 - 1)^2*a*b*d*x*arcsin(c*x)*e^2
/c^4 + 6/5*(c^2*x^2 - 1)*b^2*d*x*arcsin(c*x)^2*e^2/c^4 - 14/9*b^2*d^2*x*e/c^2 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b*d
^2*e/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)*e/c^3 + 1/7*(c^2*x^2 - 1)^3*b^2*x*arcsin(c*x)^2*e^3/c^6 -
6/125*(c^2*x^2 - 1)^2*b^2*d*x*e^2/c^4 + 12/5*(c^2*x^2 - 1)*a*b*d*x*arcsin(c*x)*e^2/c^4 + 3/5*b^2*d*x*arcsin(c*
x)^2*e^2/c^4 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)*e^2/c^5 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^
2*e/c^3 + 2/7*(c^2*x^2 - 1)^3*a*b*x*arcsin(c*x)*e^3/c^6 + 3/7*(c^2*x^2 - 1)^2*b^2*x*arcsin(c*x)^2*e^3/c^6 - 76
/375*(c^2*x^2 - 1)*b^2*d*x*e^2/c^4 + 6/5*a*b*d*x*arcsin(c*x)*e^2/c^4 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)
*a*b*d*e^2/c^5 - 4/5*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin(c*x)*e^2/c^5 - 2/343*(c^2*x^2 - 1)^3*b^2*x*e^3/c^6 + 6/
7*(c^2*x^2 - 1)^2*a*b*x*arcsin(c*x)*e^3/c^6 + 3/7*(c^2*x^2 - 1)*b^2*x*arcsin(c*x)^2*e^3/c^6 - 298/375*b^2*d*x*
e^2/c^4 + 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^3/c^7 - 4/5*(-c^2*x^2 + 1)^(3/2)*a*b*d*e^2
/c^5 + 6/5*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)*e^2/c^5 - 234/8575*(c^2*x^2 - 1)^2*b^2*x*e^3/c^6 + 6/7*(c^2*x^
2 - 1)*a*b*x*arcsin(c*x)*e^3/c^6 + 1/7*b^2*x*arcsin(c*x)^2*e^3/c^6 + 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a
*b*e^3/c^7 + 6/35*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^3/c^7 + 6/5*sqrt(-c^2*x^2 + 1)*a*b*d*e^
2/c^5 - 1514/25725*(c^2*x^2 - 1)*b^2*x*e^3/c^6 + 2/7*a*b*x*arcsin(c*x)*e^3/c^6 + 6/35*(c^2*x^2 - 1)^2*sqrt(-c^
2*x^2 + 1)*a*b*e^3/c^7 - 2/7*(-c^2*x^2 + 1)^(3/2)*b^2*arcsin(c*x)*e^3/c^7 - 4322/25725*b^2*x*e^3/c^6 - 2/7*(-c
^2*x^2 + 1)^(3/2)*a*b*e^3/c^7 + 2/7*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^3/c^7 + 2/7*sqrt(-c^2*x^2 + 1)*a*b*e^
3/c^7