3.174 \(\int x^4 (d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=476 \[ \frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}+\frac{256 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}-\frac{100976 b^2 d^3 x}{4002075 c^4}-\frac{12622 b^2 d^3 x^5}{6670125} \]

[Out]

(-100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) - (12622*b^2*d^3*x^5)/6670125 + (9410*b
^2*c^2*d^3*x^7)/1120581 - (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 + (256*b*d^3*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(17325*c^3) + (32*
b*d^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(5775*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))
/(693*c^5) - (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(1155*c^5) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a +
b*ArcSin[c*x]))/(1617*c^5) - (8*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(297*c^5) + (2*b*d^3*(1 - c^2*x
^2)^(11/2)*(a + b*ArcSin[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSin[c*x])^2)/1155 + (8*d^3*x^5*(1 - c^2*x^2)
*(a + b*ArcSin[c*x])^2)/231 + (2*d^3*x^5*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/33 + (d^3*x^5*(1 - c^2*x^2)^3*
(a + b*ArcSin[c*x])^2)/11

________________________________________________________________________________________

Rubi [A]  time = 1.01765, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4699, 4627, 4707, 4677, 8, 30, 266, 43, 4689, 12, 1153} \[ \frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}+\frac{256 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}-\frac{100976 b^2 d^3 x}{4002075 c^4}-\frac{12622 b^2 d^3 x^5}{6670125} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

(-100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) - (12622*b^2*d^3*x^5)/6670125 + (9410*b
^2*c^2*d^3*x^7)/1120581 - (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 + (256*b*d^3*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(17325*c^3) + (32*
b*d^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(5775*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))
/(693*c^5) - (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(1155*c^5) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a +
b*ArcSin[c*x]))/(1617*c^5) - (8*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(297*c^5) + (2*b*d^3*(1 - c^2*x
^2)^(11/2)*(a + b*ArcSin[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSin[c*x])^2)/1155 + (8*d^3*x^5*(1 - c^2*x^2)
*(a + b*ArcSin[c*x])^2)/231 + (2*d^3*x^5*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/33 + (d^3*x^5*(1 - c^2*x^2)^3*
(a + b*ArcSin[c*x])^2)/11

Rule 4699

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

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 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 30

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4689

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} (6 d) \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{11} \left (2 b c d^3\right ) \int x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{77 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{99 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{33} \left (8 d^2\right ) \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{33} \left (4 b c d^3\right ) \int x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{11} \left (2 b^2 c^2 d^3\right ) \int \frac{\left (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right )}{693 c^6} \, dx\\ &=\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{165 c^5}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{231 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{231} \left (16 d^3\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b^2 d^3\right ) \int \left (1-c^2 x^2\right )^3 \left (-8-28 c^2 x^2-63 c^4 x^4\right ) \, dx}{7623 c^4}-\frac{1}{231} \left (16 b c d^3\right ) \int x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{33} \left (4 b^2 c^2 d^3\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right )}{315 c^6} \, dx\\ &=\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^3\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+113 c^6 x^6-161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{7623 c^4}+\frac{\left (4 b^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right ) \, dx}{10395 c^4}-\frac{\left (32 b c d^3\right ) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{1155}+\frac{1}{231} \left (16 b^2 c^2 d^3\right ) \int \frac{-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=-\frac{16 b^2 d^3 x}{7623 c^4}-\frac{8 b^2 d^3 x^3}{22869 c^2}-\frac{2 b^2 d^3 x^5}{12705}+\frac{226 b^2 c^2 d^3 x^7}{53361}-\frac{46 b^2 c^4 d^3 x^9}{9801}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (32 b^2 d^3\right ) \int x^4 \, dx}{5775}+\frac{\left (4 b^2 d^3\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+50 c^6 x^6-35 c^8 x^8\right ) \, dx}{10395 c^4}+\frac{\left (16 b^2 d^3\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{24255 c^4}-\frac{\left (128 b d^3\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{5775 c}\\ &=-\frac{8368 b^2 d^3 x}{800415 c^4}-\frac{4184 b^2 d^3 x^3}{2401245 c^2}-\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^3}+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (256 b d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{17325 c^3}-\frac{\left (128 b^2 d^3\right ) \int x^2 \, dx}{17325 c^2}\\ &=-\frac{8368 b^2 d^3 x}{800415 c^4}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}-\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{256 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^3}+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (256 b^2 d^3\right ) \int 1 \, dx}{17325 c^4}\\ &=-\frac{100976 b^2 d^3 x}{4002075 c^4}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}-\frac{12622 b^2 d^3 x^5}{6670125}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{256 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{17325 c^3}+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{5775 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{693 c^5}-\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{1617 c^5}-\frac{8 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{297 c^5}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{11/2} \left (a+b \sin ^{-1}(c x)\right )}{121 c^5}+\frac{16 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{1155}+\frac{8}{231} d^3 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.434573, size = 301, normalized size = 0.63 \[ -\frac{d^3 \left (12006225 a^2 c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right )+6930 a b \sqrt{1-c^2 x^2} \left (33075 c^{10} x^{10}-111475 c^8 x^8+117625 c^6 x^6-18933 c^4 x^4-25244 c^2 x^2-50488\right )+6930 b \sin ^{-1}(c x) \left (3465 a c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right )+b \sqrt{1-c^2 x^2} \left (33075 c^{10} x^{10}-111475 c^8 x^8+117625 c^6 x^6-18933 c^4 x^4-25244 c^2 x^2-50488\right )\right )+b^2 \left (-20837250 c^{11} x^{11}[phone] c^9 x^9-116448750 c^7 x^7+26241138 c^5 x^5+58313640 c^3 x^3+349881840 c x\right )[phone] b^2 c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right ) \sin ^{-1}(c x)^2\right )}{13867189875 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

-(d^3*(12006225*a^2*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + 6930*a*b*Sqrt[1 - c^2*x^2]*(-50
488 - 25244*c^2*x^2 - 18933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^10) + b^2*(349881840*c*x
+ 58313640*c^3*x^3 + 26241138*c^5*x^5 - 116448750*c^7*x^7 + 85835750*c^9*x^9 - 20837250*c^11*x^11) + 6930*b*(3
465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-50488 - 25244*c^2*x^2 -
18933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^10))*ArcSin[c*x] + 12006225*b^2*c^5*x^5*(-231
+ 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]^2))/(13867189875*c^5)

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Maple [A]  time = 0.116, size = 672, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x)

[Out]

1/c^5*(-d^3*a^2*(1/11*c^11*x^11-1/3*c^9*x^9+3/7*c^7*x^7-1/5*c^5*x^5)-d^3*b^2*(-32/1155*arcsin(c*x)*(-c^2*x^2+1
)^(1/2)+2/315*arcsin(c*x)^2*(35*c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x-2/56595*(5*c^6*x^6-21*c^4
*x^4+35*c^2*x^2-35)*c*x+16/3465*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+8/297*arcsin(c*x)*(c^2*x^2-1)^4*(-c
^2*x^2+1)^(1/2)+32/1155*c*x-8/93555*(35*c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x-16/10395*(c^2*x^2
-3)*c*x+2/121*arcsin(c*x)*(c^2*x^2-1)^5*(-c^2*x^2+1)^(1/2)-4/1925*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)
+2/1617*arcsin(c*x)*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)+4/28875*(3*c^4*x^4-10*c^2*x^2+15)*c*x-2/83853*(63*c^10*x^
10-385*c^8*x^8+990*c^6*x^6-1386*c^4*x^4+1155*c^2*x^2-693)*c*x+1/35*arcsin(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*
x^2-35)*c*x+1/693*arcsin(c*x)^2*(63*c^10*x^10-385*c^8*x^8+990*c^6*x^6-1386*c^4*x^4+1155*c^2*x^2-693)*c*x)-2*d^
3*a*b*(1/11*arcsin(c*x)*c^11*x^11-1/3*arcsin(c*x)*c^9*x^9+3/7*arcsin(c*x)*c^7*x^7-1/5*arcsin(c*x)*c^5*x^5+1/12
1*c^10*x^10*(-c^2*x^2+1)^(1/2)-91/3267*c^8*x^8*(-c^2*x^2+1)^(1/2)+4705/160083*c^6*x^6*(-c^2*x^2+1)^(1/2)-6311/
1334025*c^4*x^4*(-c^2*x^2+1)^(1/2)-25244/4002075*c^2*x^2*(-c^2*x^2+1)^(1/2)-50488/4002075*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.94875, size = 1540, normalized size = 3.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/11*b^2*c^6*d^3*x^11*arcsin(c*x)^2 - 1/11*a^2*c^6*d^3*x^11 + 1/3*b^2*c^4*d^3*x^9*arcsin(c*x)^2 + 1/3*a^2*c^4
*d^3*x^9 - 3/7*b^2*c^2*d^3*x^7*arcsin(c*x)^2 - 3/7*a^2*c^2*d^3*x^7 - 2/7623*(693*x^11*arcsin(c*x) + (63*sqrt(-
c^2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(-c^2*x^2 + 1)*x^6/c^6 + 96*sqrt(-c^2*x^2 + 1)*
x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^2/c^10 + 256*sqrt(-c^2*x^2 + 1)/c^12)*c)*a*b*c^6*d^3 - 2/26413695*(3465*(63
*sqrt(-c^2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(-c^2*x^2 + 1)*x^6/c^6 + 96*sqrt(-c^2*x^
2 + 1)*x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^2/c^10 + 256*sqrt(-c^2*x^2 + 1)/c^12)*c*arcsin(c*x) - (19845*c^10*x^
11 + 26950*c^8*x^9 + 39600*c^6*x^7 + 66528*c^4*x^5 + 147840*c^2*x^3 + 887040*x)/c^10)*b^2*c^6*d^3 + 1/5*b^2*d^
3*x^5*arcsin(c*x)^2 + 2/945*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/
c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c)*a*b*c^4*
d^3 + 2/297675*(315*(35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4
/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c*arcsin(c*x) - (1225*c^8*x^9 + 1800*c^6*x
^7 + 3024*c^4*x^5 + 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^4*d^3 + 1/5*a^2*d^3*x^5 - 6/245*(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*c^2*d^3 - 2/8575*(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 + 16
80*x)/c^6)*b^2*c^2*d^3 + 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*d^3 + 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*d^3

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Fricas [A]  time = 1.98682, size = 1071, normalized size = 2.25 \begin{align*} -\frac{10418625 \,{\left (121 \, a^{2} - 2 \, b^{2}\right )} c^{11} d^{3} x^{11} - 471625 \,{\left (9801 \, a^{2} - 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \,{\left (480249 \, a^{2} - 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} - 2079 \,{\left (1334025 \, a^{2} - 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} + 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \,{\left (105 \, b^{2} c^{11} d^{3} x^{11} - 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} - 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \arcsin \left (c x\right )^{2} + 24012450 \,{\left (105 \, a b c^{11} d^{3} x^{11} - 385 \, a b c^{9} d^{3} x^{9} + 495 \, a b c^{7} d^{3} x^{7} - 231 \, a b c^{5} d^{3} x^{5}\right )} \arcsin \left (c x\right ) + 6930 \,{\left (33075 \, a b c^{10} d^{3} x^{10} - 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} - 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} - 50488 \, a b d^{3} +{\left (33075 \, b^{2} c^{10} d^{3} x^{10} - 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} - 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} - 50488 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

-1/13867189875*(10418625*(121*a^2 - 2*b^2)*c^11*d^3*x^11 - 471625*(9801*a^2 - 182*b^2)*c^9*d^3*x^9 + 12375*(48
0249*a^2 - 9410*b^2)*c^7*d^3*x^7 - 2079*(1334025*a^2 - 12622*b^2)*c^5*d^3*x^5 + 58313640*b^2*c^3*d^3*x^3 + 349
881840*b^2*c*d^3*x + 12006225*(105*b^2*c^11*d^3*x^11 - 385*b^2*c^9*d^3*x^9 + 495*b^2*c^7*d^3*x^7 - 231*b^2*c^5
*d^3*x^5)*arcsin(c*x)^2 + 24012450*(105*a*b*c^11*d^3*x^11 - 385*a*b*c^9*d^3*x^9 + 495*a*b*c^7*d^3*x^7 - 231*a*
b*c^5*d^3*x^5)*arcsin(c*x) + 6930*(33075*a*b*c^10*d^3*x^10 - 111475*a*b*c^8*d^3*x^8 + 117625*a*b*c^6*d^3*x^6 -
18933*a*b*c^4*d^3*x^4 - 25244*a*b*c^2*d^3*x^2 - 50488*a*b*d^3 + (33075*b^2*c^10*d^3*x^10 - 111475*b^2*c^8*d^3
*x^8 + 117625*b^2*c^6*d^3*x^6 - 18933*b^2*c^4*d^3*x^4 - 25244*b^2*c^2*d^3*x^2 - 50488*b^2*d^3)*arcsin(c*x))*sq
rt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 123.868, size = 702, normalized size = 1.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**3*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((-a**2*c**6*d**3*x**11/11 + a**2*c**4*d**3*x**9/3 - 3*a**2*c**2*d**3*x**7/7 + a**2*d**3*x**5/5 - 2*a
*b*c**6*d**3*x**11*asin(c*x)/11 - 2*a*b*c**5*d**3*x**10*sqrt(-c**2*x**2 + 1)/121 + 2*a*b*c**4*d**3*x**9*asin(c
*x)/3 + 182*a*b*c**3*d**3*x**8*sqrt(-c**2*x**2 + 1)/3267 - 6*a*b*c**2*d**3*x**7*asin(c*x)/7 - 9410*a*b*c*d**3*
x**6*sqrt(-c**2*x**2 + 1)/160083 + 2*a*b*d**3*x**5*asin(c*x)/5 + 12622*a*b*d**3*x**4*sqrt(-c**2*x**2 + 1)/(133
4025*c) + 50488*a*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(4002075*c**3) + 100976*a*b*d**3*sqrt(-c**2*x**2 + 1)/(4002
075*c**5) - b**2*c**6*d**3*x**11*asin(c*x)**2/11 + 2*b**2*c**6*d**3*x**11/1331 - 2*b**2*c**5*d**3*x**10*sqrt(-
c**2*x**2 + 1)*asin(c*x)/121 + b**2*c**4*d**3*x**9*asin(c*x)**2/3 - 182*b**2*c**4*d**3*x**9/29403 + 182*b**2*c
**3*d**3*x**8*sqrt(-c**2*x**2 + 1)*asin(c*x)/3267 - 3*b**2*c**2*d**3*x**7*asin(c*x)**2/7 + 9410*b**2*c**2*d**3
*x**7/1120581 - 9410*b**2*c*d**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/160083 + b**2*d**3*x**5*asin(c*x)**2/5 -
12622*b**2*d**3*x**5/6670125 + 12622*b**2*d**3*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(1334025*c) - 50488*b**2*d*
*3*x**3/(12006225*c**2) + 50488*b**2*d**3*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(4002075*c**3) - 100976*b**2*d**
3*x/(4002075*c**4) + 100976*b**2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(4002075*c**5), Ne(c, 0)), (a**2*d**3*x**
5/5, True))

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Giac [B]  time = 1.44143, size = 1168, normalized size = 2.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

-1/11*a^2*c^6*d^3*x^11 + 1/3*a^2*c^4*d^3*x^9 - 3/7*a^2*c^2*d^3*x^7 + 1/5*a^2*d^3*x^5 - 1/11*(c^2*x^2 - 1)^5*b^
2*d^3*x*arcsin(c*x)^2/c^4 - 2/11*(c^2*x^2 - 1)^5*a*b*d^3*x*arcsin(c*x)/c^4 - 4/33*(c^2*x^2 - 1)^4*b^2*d^3*x*ar
csin(c*x)^2/c^4 + 2/1331*(c^2*x^2 - 1)^5*b^2*d^3*x/c^4 - 8/33*(c^2*x^2 - 1)^4*a*b*d^3*x*arcsin(c*x)/c^4 - 1/23
1*(c^2*x^2 - 1)^3*b^2*d^3*x*arcsin(c*x)^2/c^4 - 2/121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c
^5 + 428/323433*(c^2*x^2 - 1)^4*b^2*d^3*x/c^4 - 2/231*(c^2*x^2 - 1)^3*a*b*d^3*x*arcsin(c*x)/c^4 + 2/385*(c^2*x
^2 - 1)^2*b^2*d^3*x*arcsin(c*x)^2/c^4 - 2/121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 - 8/297*(c^2*x^2
- 1)^4*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 - 148174/110937519*(c^2*x^2 - 1)^3*b^2*d^3*x/c^4 + 4/385*(c^
2*x^2 - 1)^2*a*b*d^3*x*arcsin(c*x)/c^4 - 8/1155*(c^2*x^2 - 1)*b^2*d^3*x*arcsin(c*x)^2/c^4 - 8/297*(c^2*x^2 - 1
)^4*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 - 2/1617*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 + 54877
04/4622396625*(c^2*x^2 - 1)^2*b^2*d^3*x/c^4 - 16/1155*(c^2*x^2 - 1)*a*b*d^3*x*arcsin(c*x)/c^4 + 16/1155*b^2*d^
3*x*arcsin(c*x)^2/c^4 - 2/1617*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 + 4/1925*(c^2*x^2 - 1)^2*sqrt(-c
^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 - 606416/13867189875*(c^2*x^2 - 1)*b^2*d^3*x/c^4 + 32/1155*a*b*d^3*x*arcsi
n(c*x)/c^4 + 4/1925*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 + 16/3465*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*arcs
in(c*x)/c^5 - 382986368/13867189875*b^2*d^3*x/c^4 + 16/3465*(-c^2*x^2 + 1)^(3/2)*a*b*d^3/c^5 + 32/1155*sqrt(-c
^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 + 32/1155*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5