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

Optimal. Leaf size=338 \[ \frac{2 b c^3 d e x^5 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{d e \left (1-c^2 x^2\right )^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{2 b^2 d e \left (1-c^2 x^2\right )^2 \sqrt{c d x+d} \sqrt{e-c e x}}{125 c^2}+\frac{8 b^2 d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{225 c^2}+\frac{16 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}{75 c^2} \]

[Out]

(16*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(75*c^2) + (8*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^
2))/(225*c^2) + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2)/(125*c^2) + (2*b*d*e*x*Sqrt[d + c*
d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Arc
Sin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2
)/(5*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.506942, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4739, 4677, 194, 4645, 12, 1247, 698} \[ \frac{2 b c^3 d e x^5 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{d e \left (1-c^2 x^2\right )^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{2 b^2 d e \left (1-c^2 x^2\right )^2 \sqrt{c d x+d} \sqrt{e-c e x}}{125 c^2}+\frac{8 b^2 d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{225 c^2}+\frac{16 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}{75 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(75*c^2) + (8*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^
2))/(225*c^2) + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2)/(125*c^2) + (2*b*d*e*x*Sqrt[d + c*
d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Arc
Sin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2
)/(5*c^2)

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

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

Rubi steps

\begin{align*} \int x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac{\left (2 b d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{\left (2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx}{5 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{\left (2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{\left (b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{15-10 c^2 x+3 c^4 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac{\left (b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-c^2 x}}+4 \sqrt{1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}\\ &=\frac{16 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}{75 c^2}+\frac{8 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )}{225 c^2}+\frac{2 b^2 d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2}{125 c^2}+\frac{2 b d e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{d e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}\\ \end{align*}

Mathematica [A]  time = 0.815709, size = 207, normalized size = 0.61 \[ -\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left (225 a^2 \left (c^2 x^2-1\right )^3+30 a b c x \sqrt{1-c^2 x^2} \left (3 c^4 x^4-10 c^2 x^2+15\right )+30 b \sin ^{-1}(c x) \left (15 a \left (c^2 x^2-1\right )^3+b c x \sqrt{1-c^2 x^2} \left (3 c^4 x^4-10 c^2 x^2+15\right )\right )+2 b^2 \left (-9 c^6 x^6+47 c^4 x^4-187 c^2 x^2+149\right )+225 b^2 \left (c^2 x^2-1\right )^3 \sin ^{-1}(c x)^2\right )}{1125 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(225*a^2*(-1 + c^2*x^2)^3 + 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^
2 + 3*c^4*x^4) + 2*b^2*(149 - 187*c^2*x^2 + 47*c^4*x^4 - 9*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt
[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcSin[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcSin[c*x]^2))/(1125*c^2*
(-1 + c^2*x^2))

________________________________________________________________________________________

Maple [F]  time = 0.35, size = 0, normalized size = 0. \begin{align*} \int x \left ( cdx+d \right ) ^{{\frac{3}{2}}} \left ( -cex+e \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.53316, size = 691, normalized size = 2.04 \begin{align*} -\frac{{\left (9 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} d e x^{6} -{\left (675 \, a^{2} - 94 \, b^{2}\right )} c^{4} d e x^{4} +{\left (675 \, a^{2} - 374 \, b^{2}\right )} c^{2} d e x^{2} -{\left (225 \, a^{2} - 298 \, b^{2}\right )} d e + 225 \,{\left (b^{2} c^{6} d e x^{6} - 3 \, b^{2} c^{4} d e x^{4} + 3 \, b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 450 \,{\left (a b c^{6} d e x^{6} - 3 \, a b c^{4} d e x^{4} + 3 \, a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right ) + 30 \,{\left (3 \, a b c^{5} d e x^{5} - 10 \, a b c^{3} d e x^{3} + 15 \, a b c d e x +{\left (3 \, b^{2} c^{5} d e x^{5} - 10 \, b^{2} c^{3} d e x^{3} + 15 \, b^{2} c d e x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{1125 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1125*(9*(25*a^2 - 2*b^2)*c^6*d*e*x^6 - (675*a^2 - 94*b^2)*c^4*d*e*x^4 + (675*a^2 - 374*b^2)*c^2*d*e*x^2 - (
225*a^2 - 298*b^2)*d*e + 225*(b^2*c^6*d*e*x^6 - 3*b^2*c^4*d*e*x^4 + 3*b^2*c^2*d*e*x^2 - b^2*d*e)*arcsin(c*x)^2
+ 450*(a*b*c^6*d*e*x^6 - 3*a*b*c^4*d*e*x^4 + 3*a*b*c^2*d*e*x^2 - a*b*d*e)*arcsin(c*x) + 30*(3*a*b*c^5*d*e*x^5
- 10*a*b*c^3*d*e*x^3 + 15*a*b*c*d*e*x + (3*b^2*c^5*d*e*x^5 - 10*b^2*c^3*d*e*x^3 + 15*b^2*c*d*e*x)*arcsin(c*x)
)*sqrt(-c^2*x^2 + 1))*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^4*x^2 - c^2)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.1531, size = 1897, normalized size = 5.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108000*(7200*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d)*(3*(c*d*x + d)*((c*d*x + d)/(c
^3*d^3) - 4/(c^3*d^2)) + 17/(c^3*d)) - 10/c^3)*a^2*c^2*abs(d)*e/d^2 + 960*(15*(c*d*x + d)^(3/2)*sqrt(-(c*d*x +
d)*d*e + 2*d^2*e)*((c*d*x + d)*(3*(c*d*x + d)*((c*d*x + d)/(c^3*d^3) - 4/(c^3*d^2)) + 17/(c^3*d)) - 10/c^3)*a
rcsin(c*x) + (9*(c*d*x + d)^5 - 45*(c*d*x + d)^4*d + 85*(c*d*x + d)^3*d^2 - 75*(c*d*x + d)^2*d^3)*e^(1/2)/(c^3
*d^(3/2)*abs(d)))*a*b*c^2*abs(d)*e/d^2 + 8*(900*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d
)*(3*(c*d*x + d)*((c*d*x + d)/(c^3*d^3) - 4/(c^3*d^2)) + 17/(c^3*d)) - 10/c^3)*arcsin(c*x)^2 - (1560*pi*d^3 -
(1080*(c^2*x^2 - 1)^2*c*d^4*x*arcsin(-c*x) + 12960*(c^2*x^2 - 1)*c*d^4*x*arcsin(-c*x) + 1350*(-c^2*x^2 + 1)^(3
/2)*c*d^4*x + 5400*(c^2*x^2 - 1)^2*d^4*arcsin(-c*x) + 17280*c*d^4*x*arcsin(-c*x) - 216*(c^2*x^2 - 1)^2*sqrt(-c
^2*x^2 + 1)*d^4 - 8775*sqrt(-c^2*x^2 + 1)*c*d^4*x + 21600*(c^2*x^2 - 1)*d^4*arcsin(-c*x) + 4320*(-c^2*x^2 + 1)
^(3/2)*d^4 + 8775*d^4*arcsin(-c*x) - 17280*sqrt(-c^2*x^2 + 1)*d^4 - 4500*(4*c*d*x*arcsin(-c*x) - sqrt(-c^2*x^2
+ 1)*c*d*x + 2*(c^2*x^2 - 1)*d*arcsin(-c*x) + d*arcsin(-c*x) - 4*sqrt(-c^2*x^2 + 1)*d)*d^3 + 1700*(6*(c^2*x^2
- 1)*c*d^2*x*arcsin(-c*x) + 24*c*d^2*x*arcsin(-c*x) - 9*sqrt(-c^2*x^2 + 1)*c*d^2*x + 18*(c^2*x^2 - 1)*d^2*arc
sin(-c*x) + 2*(-c^2*x^2 + 1)^(3/2)*d^2 + 9*d^2*arcsin(-c*x) - 24*sqrt(-c^2*x^2 + 1)*d^2)*d^2 - 225*(96*(c^2*x^
2 - 1)*c*d^3*x*arcsin(-c*x) + 6*(-c^2*x^2 + 1)^(3/2)*c*d^3*x + 24*(c^2*x^2 - 1)^2*d^3*arcsin(-c*x) + 192*c*d^3
*x*arcsin(-c*x) - 87*sqrt(-c^2*x^2 + 1)*c*d^3*x + 192*(c^2*x^2 - 1)*d^3*arcsin(-c*x) + 32*(-c^2*x^2 + 1)^(3/2)
*d^3 + 87*d^3*arcsin(-c*x) - 192*sqrt(-c^2*x^2 + 1)*d^3)*d)/d)*sqrt(d)*e^(1/2)/(c^3*abs(d)))*b^2*c^2*abs(d)*e/
d^2 + 225*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*a^2*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5)*abs(d
)*e/(c*d^3) + 150*(3*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5
)*arcsin(c*x) - ((c*d*x + d)^3 - 3*(c*d*x + d)^2*d)*e^(-11/2)/(d^(9/2)*abs(d)))*a*b*abs(d)*e/(c*d^3) + 25*(9*(
c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5)*arcsin(c*x)^2 + sqrt
(d)*(6*pi*e^(-6)/d^2 - (6*(c^2*x^2 - 1)*c*d^2*x*arcsin(-c*x) + 24*c*d^2*x*arcsin(-c*x) - 9*sqrt(-c^2*x^2 + 1)*
c*d^2*x + 18*(c^2*x^2 - 1)*d^2*arcsin(-c*x) + 2*(-c^2*x^2 + 1)^(3/2)*d^2 + 9*d^2*arcsin(-c*x) - 24*sqrt(-c^2*x
^2 + 1)*d^2 - 9*(4*c*d*x*arcsin(-c*x) - sqrt(-c^2*x^2 + 1)*c*d*x + 2*(c^2*x^2 - 1)*d*arcsin(-c*x) + d*arcsin(-
c*x) - 4*sqrt(-c^2*x^2 + 1)*d)*d)*e^(-6)/d^4)*e^(1/2)/abs(d))*b^2*abs(d)*e/(c*d^3))/c