\(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\) [2302]

Optimal result

Integrand size = 22, antiderivative size = 751 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/2*(e*x+d)^(5/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1/4*(12*b*c*d*(3*a*e^2+c*d^2)-4*a*c
*e*(5*a*e^2+7*c*d^2)-b^2*(a*e^3+11*c*d^2*e)+(-b*e+2*c*d)*(12*c^2*d^2+b^2*e^2-4*c*e*(-2*a*e+3*b*d))*x)*(e*x+d)^
(1/2)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))
)^(1/2))*(96*c^4*d^4-b^3*e^4*(b-(-4*a*c+b^2)^(1/2))-8*c^3*d^2*e*(24*b*d-19*a*e-3*d*(-4*a*c+b^2)^(1/2))-2*b*c*e
^3*(5*b^2*d-9*a*b*e-5*b*d*(-4*a*c+b^2)^(1/2)+8*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(53*b^2*d^2+4*a*e*(5*a*e+4*d*
(-4*a*c+b^2)^(1/2))-2*b*d*(38*a*e+9*d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(5/2)*2^(1/2)/(2*c*d-e*(b-(-4
*a*c+b^2)^(1/2)))^(1/2)+1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(96*
c^4*d^4-b^3*e^4*(b+(-4*a*c+b^2)^(1/2))-8*c^3*d^2*e*(24*b*d-19*a*e+3*d*(-4*a*c+b^2)^(1/2))-2*b*c*e^3*(5*b^2*d-9
*a*b*e+5*b*d*(-4*a*c+b^2)^(1/2)-8*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(53*b^2*d^2-4*a*e*(-5*a*e+4*d*(-4*a*c+b^2)
^(1/2))+2*b*d*(-38*a*e+9*d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(5/2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(
1/2)))^(1/2)

Rubi [A] (verified)

Time = 10.32 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 832, 840, 1180, 214} \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\left (-8 c^3 d^2 e \left (-3 d \sqrt {b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (-2 b d \left (9 d \sqrt {b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt {b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (-5 b d \sqrt {b^2-4 a c}+8 a e \sqrt {b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt {b^2-4 a c}\right )+96 c^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (-8 c^3 d^2 e \left (3 d \sqrt {b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (2 b d \left (9 d \sqrt {b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt {b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (5 b d \sqrt {b^2-4 a c}-8 a e \sqrt {b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt {b^2-4 a c}+b\right )+96 c^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (Sqrt[d + e*x]*(1
2*b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2
+ b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(b - Sqrt
[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2
- 4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*
e) - 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d
^4 - b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^
2*d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 2*b*d*(9*Sqrt[b^2
- 4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d - 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), 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] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (12 c d^2-11 b d e+10 a e^2\right )+\frac {1}{2} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{4} \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 c \left (b^2-4 a c\right )^2} \\ & = -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} d e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+\frac {1}{4} e \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c \left (b^2-4 a c\right )^2} \\ & = -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 c \left (b^2-4 a c\right )^{5/2}}+\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 c \left (b^2-4 a c\right )^{5/2}} \\ & = -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.35 (sec) , antiderivative size = 720, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {-4 \sqrt {c} \sqrt {b^2-4 a c} \sqrt {d+e x} \left (b^4 e^3 x^2+b^2 \left (a^2 e^3+c^2 d x \left (-8 d^2+55 d e x-10 e^2 x^2\right )+a c e \left (7 d^2-58 d e x+5 e^2 x^2\right )\right )+4 c \left (5 a^3 e^3-6 c^3 d^3 x^3-a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )-4 b c \left (a^2 e^2 (9 d-7 e x)+9 c^2 d^2 x^2 (d-e x)+a c \left (5 d^3-14 d^2 e x+11 d e^2 x^2-4 e^3 x^3\right )\right )+b^3 \left (2 a e^3 x+c \left (2 d^3+13 d^2 e x-16 d e^2 x^2-e^3 x^3\right )\right )\right )-\sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) (a+x (b+c x))^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )+\sqrt {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-18 b d+3 \sqrt {b^2-4 a c} d+13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) (a+x (b+c x))^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{16 c^{3/2} \left (b^2-4 a c\right )^{5/2} (a+x (b+c x))^2} \]

[In]

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

(-4*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*(b^4*e^3*x^2 + b^2*(a^2*e^3 + c^2*d*x*(-8*d^2 + 55*d*e*x - 10*e^2*
x^2) + a*c*e*(7*d^2 - 58*d*e*x + 5*e^2*x^2)) + 4*c*(5*a^3*e^3 - 6*c^3*d^3*x^3 - a*c^2*d*x*(10*d^2 + d*e*x + 8*
e^2*x^2) + a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)) - 4*b*c*(a^2*e^2*(9*d - 7*e*x) + 9*c^2*d^2*x^2*(d - e*x) +
a*c*(5*d^3 - 14*d^2*e*x + 11*d*e^2*x^2 - 4*e^3*x^3)) + b^3*(2*a*e^3*x + c*(2*d^3 + 13*d^2*e*x - 16*d*e^2*x^2 -
e^3*x^3))) - Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2
*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*
a*Sqrt[b^2 - 4*a*c]*e))*(a + x*(b + c*x))^2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^
2 - 4*a*c]*e]] + Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c
^2*d*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt
[b^2 - 4*a*c]*d + 13*a*e)))*(a + x*(b + c*x))^2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e]])/(16*c^(3/2)*(b^2 - 4*a*c)^(5/2)*(a + x*(b + c*x))^2)

Maple [A] (verified)

Time = 3.08 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(-\frac {5 \left (\sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (-\frac {2 \left (\frac {3 c^{2} d^{2}}{4}+e \left (a e -\frac {3 b d}{4}\right ) c -\frac {b^{2} e^{2}}{16}\right ) \left (b e -2 c d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}}{5}+\left (3 c^{2} d^{2}+\left (a \,e^{2}-3 b d e \right ) c +\frac {b^{2} e^{2}}{2}\right ) \left (\frac {4 c^{2} d^{2}}{5}+e \left (a e -\frac {4 b d}{5}\right ) c -\frac {b^{2} e^{2}}{20}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e \left (\frac {2 \left (\frac {3 c^{2} d^{2}}{4}+e \left (a e -\frac {3 b d}{4}\right ) c -\frac {b^{2} e^{2}}{16}\right ) \left (b e -2 c d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}}{5}+\left (3 c^{2} d^{2}+\left (a \,e^{2}-3 b d e \right ) c +\frac {b^{2} e^{2}}{2}\right ) \left (\frac {4 c^{2} d^{2}}{5}+e \left (a e -\frac {4 b d}{5}\right ) c -\frac {b^{2} e^{2}}{20}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \left (-\frac {6 c^{4} d^{3} x^{3}}{5}-2 x \left (\frac {4 a \,e^{2} x^{2}}{5}+\frac {d x \left (-9 b x +a \right ) e}{10}+d^{2} \left (\frac {9 b x}{10}+a \right )\right ) d \,c^{3}+\left (\frac {9 \left (\frac {4 b x}{9}+a \right ) x^{2} a \,e^{3}}{5}+\frac {4 x d \left (-\frac {5}{8} b^{2} x^{2}-\frac {11}{4} a b x +a^{2}\right ) e^{2}}{5}+\frac {11 \left (\frac {5}{4} b^{2} x^{2}+\frac {14}{11} a b x +a^{2}\right ) d^{2} e}{5}-\left (\frac {2 b x}{5}+a \right ) b \,d^{3}\right ) c^{2}+\left (\left (-\frac {1}{20} b^{3} x^{3}+a^{3}+\frac {1}{4} a \,b^{2} x^{2}+\frac {7}{5} a^{2} b x \right ) e^{3}-\frac {9 \left (\frac {4}{9} b^{2} x^{2}+\frac {29}{18} a b x +a^{2}\right ) b d \,e^{2}}{5}+\frac {7 \left (\frac {13 b x}{7}+a \right ) b^{2} d^{2} e}{20}+\frac {b^{3} d^{3}}{10}\right ) c +\frac {b^{2} e^{3} \left (b x +a \right )^{2}}{20}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\right )\right )}{16 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2} c}\)	\(785\)
derivativedivides	\(\text {Expression too large to display}\)	\(1277\)
default	\(\text {Expression too large to display}\)	\(1277\)

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-5/16/(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(2^(1/2)*(c*x^2+b*x+a)^2*e*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/
2)*(-2/5*(3/4*c^2*d^2+e*(a*e-3/4*b*d)*c-1/16*b^2*e^2)*(b*e-2*c*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+(3*c^2*d^2+(a*e
^2-3*b*d*e)*c+1/2*b^2*e^2)*(4/5*c^2*d^2+e*(a*e-4/5*b*d)*c-1/20*b^2*e^2))*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*
e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)*(2^(1/2)*(
c*x^2+b*x+a)^2*e*(2/5*(3/4*c^2*d^2+e*(a*e-3/4*b*d)*c-1/16*b^2*e^2)*(b*e-2*c*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+(3
*c^2*d^2+(a*e^2-3*b*d*e)*c+1/2*b^2*e^2)*(4/5*c^2*d^2+e*(a*e-4/5*b*d)*c-1/20*b^2*e^2))*arctan(c*(e*x+d)^(1/2)*2
^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(-6/5*c^4*d^3*x^3-2*x*
(4/5*a*e^2*x^2+1/10*d*x*(-9*b*x+a)*e+d^2*(9/10*b*x+a))*d*c^3+(9/5*(4/9*b*x+a)*x^2*a*e^3+4/5*x*d*(-5/8*b^2*x^2-
11/4*a*b*x+a^2)*e^2+11/5*(5/4*b^2*x^2+14/11*a*b*x+a^2)*d^2*e-(2/5*b*x+a)*b*d^3)*c^2+((-1/20*b^3*x^3+a^3+1/4*a*
b^2*x^2+7/5*a^2*b*x)*e^3-9/5*(4/9*b^2*x^2+29/18*a*b*x+a^2)*b*d*e^2+7/20*(13/7*b*x+a)*b^2*d^2*e+1/10*b^3*d^3)*c
+1/20*b^2*e^3*(b*x+a)^2)*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/((-b*e+2*c*d+(-4*(
a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)/(a*c-1/4*b^2)^2/(c*x^2+b*
x+a)^2/c

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8780 vs. \(2 (675) = 1350\).

Time = 2.18 (sec) , antiderivative size = 8780, normalized size of antiderivative = 11.69 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^3, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3037 vs. \(2 (675) = 1350\).

Time = 2.00 (sec) , antiderivative size = 3037, normalized size of antiderivative = 4.04 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(192*(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*d^5*e - 480*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*
b^3*c^8 - 64*a^3*b*c^9)*d^4*e^2 + 4*(101*b^8*c^5 - 1136*a*b^6*c^6 + 3936*a^2*b^4*c^7 - 2816*a^3*b^2*c^8 - 4864
*a^4*c^9)*d^3*e^3 - 6*(21*b^9*c^4 - 176*a*b^7*c^5 + 96*a^2*b^5*c^6 + 2304*a^3*b^3*c^7 - 4864*a^4*b*c^8)*d^2*e^
4 + 4*(2*b^10*c^3 + 23*a*b^8*c^4 - 448*a^2*b^6*c^5 + 1888*a^3*b^4*c^6 - 2048*a^4*b^2*c^7 - 1280*a^5*c^8)*d*e^5
+ (b^11*c^2 - 30*a*b^9*c^3 + 224*a^2*b^7*c^4 - 448*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2560*a^5*b*c^7)*e^6 - (24*
c^3*d^3*e - 36*b*c^2*d^2*e^2 + 2*(5*b^2*c + 16*a*c^2)*d*e^3 + (b^3 - 16*a*b*c)*e^4)*(b^4*c*e - 8*a*b^2*c^2*e +
16*a^2*c^3*e)^2 - 2*(24*(b^2*c^5 - 4*a*c^6)*sqrt(b^2 - 4*a*c)*d^4*e - 48*(b^3*c^4 - 4*a*b*c^5)*sqrt(b^2 - 4*a
*c)*d^3*e^2 + (25*b^4*c^3 - 56*a*b^2*c^4 - 176*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (b^5*c^2 + 40*a*b^3*c^3 -
176*a^2*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (a*b^4*c^2 + 16*a^2*b^2*c^3 - 80*a^3*c^4)*sqrt(b^2 - 4*a*c)*e^5)*abs(
b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c^2*d - 16*a*b^2*c^3*d
+ 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e + sqrt((2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d
- b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e)^2 - 4*(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e
+ 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)*(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4)))/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)))/(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt(b^2 - 4*a*c)*d - (b^8*c - 16*a*b^6*c^2 + 96*a^2*b^
4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5 + (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*
c))*e)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e)*abs(c)) - 1/4*(192*(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9
- 64*a^3*c^10)*d^5*e - 480*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*d^4*e^2 + 4*(101*b^8*c^5
- 1136*a*b^6*c^6 + 3936*a^2*b^4*c^7 - 2816*a^3*b^2*c^8 - 4864*a^4*c^9)*d^3*e^3 - 6*(21*b^9*c^4 - 176*a*b^7*c^5
+ 96*a^2*b^5*c^6 + 2304*a^3*b^3*c^7 - 4864*a^4*b*c^8)*d^2*e^4 + 4*(2*b^10*c^3 + 23*a*b^8*c^4 - 448*a^2*b^6*c^
5 + 1888*a^3*b^4*c^6 - 2048*a^4*b^2*c^7 - 1280*a^5*c^8)*d*e^5 + (b^11*c^2 - 30*a*b^9*c^3 + 224*a^2*b^7*c^4 - 4
48*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2560*a^5*b*c^7)*e^6 - (24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 2*(5*b^2*c + 16*a*
c^2)*d*e^3 + (b^3 - 16*a*b*c)*e^4)*(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e)^2 + 2*(24*(b^2*c^5 - 4*a*c^6)*sqrt
(b^2 - 4*a*c)*d^4*e - 48*(b^3*c^4 - 4*a*b*c^5)*sqrt(b^2 - 4*a*c)*d^3*e^2 + (25*b^4*c^3 - 56*a*b^2*c^4 - 176*a^
2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (b^5*c^2 + 40*a*b^3*c^3 - 176*a^2*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (a*b^4*c
^2 + 16*a^2*b^2*c^3 - 80*a^3*c^4)*sqrt(b^2 - 4*a*c)*e^5)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e))*arctan(2
*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2
*b*c^3*e - sqrt((2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e)^2 - 4
*(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^
2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)))/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2
*c^4)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3
*c^5)*sqrt(b^2 - 4*a*c)*d + (b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5 - (b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*c))*e)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e
)*abs(c)) + 1/4*(24*(e*x + d)^(7/2)*c^4*d^3*e - 72*(e*x + d)^(5/2)*c^4*d^4*e + 72*(e*x + d)^(3/2)*c^4*d^5*e -
24*sqrt(e*x + d)*c^4*d^6*e - 36*(e*x + d)^(7/2)*b*c^3*d^2*e^2 + 144*(e*x + d)^(5/2)*b*c^3*d^3*e^2 - 180*(e*x +
d)^(3/2)*b*c^3*d^4*e^2 + 72*sqrt(e*x + d)*b*c^3*d^5*e^2 + 10*(e*x + d)^(7/2)*b^2*c^2*d*e^3 + 32*(e*x + d)^(7/
2)*a*c^3*d*e^3 - 85*(e*x + d)^(5/2)*b^2*c^2*d^2*e^3 - 92*(e*x + d)^(5/2)*a*c^3*d^2*e^3 + 148*(e*x + d)^(3/2)*b
^2*c^2*d^3*e^3 + 128*(e*x + d)^(3/2)*a*c^3*d^3*e^3 - 73*sqrt(e*x + d)*b^2*c^2*d^4*e^3 - 68*sqrt(e*x + d)*a*c^3
*d^4*e^3 + (e*x + d)^(7/2)*b^3*c*e^4 - 16*(e*x + d)^(7/2)*a*b*c^2*e^4 + 13*(e*x + d)^(5/2)*b^3*c*d*e^4 + 92*(e
*x + d)^(5/2)*a*b*c^2*d*e^4 - 42*(e*x + d)^(3/2)*b^3*c*d^2*e^4 - 192*(e*x + d)^(3/2)*a*b*c^2*d^2*e^4 + 26*sqrt
(e*x + d)*b^3*c*d^3*e^4 + 136*sqrt(e*x + d)*a*b*c^2*d^3*e^4 - (e*x + d)^(5/2)*b^4*e^5 - 5*(e*x + d)^(5/2)*a*b^
2*c*e^5 - 36*(e*x + d)^(5/2)*a^2*c^2*e^5 + 2*(e*x + d)^(3/2)*b^4*d*e^5 + 68*(e*x + d)^(3/2)*a*b^2*c*d*e^5 + 56
*(e*x + d)^(3/2)*a^2*c^2*d*e^5 - sqrt(e*x + d)*b^4*d^2*e^5 - 70*sqrt(e*x + d)*a*b^2*c*d^2*e^5 - 64*sqrt(e*x +
d)*a^2*c^2*d^2*e^5 - 2*(e*x + d)^(3/2)*a*b^3*e^6 - 28*(e*x + d)^(3/2)*a^2*b*c*e^6 + 2*sqrt(e*x + d)*a*b^3*d*e^
6 + 64*sqrt(e*x + d)*a^2*b*c*d*e^6 - sqrt(e*x + d)*a^2*b^2*e^7 - 20*sqrt(e*x + d)*a^3*c*e^7)/((b^4*c - 8*a*b^2
*c^2 + 16*a^2*c^3)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e + a*e^2)^2)

Mupad [B] (verification not implemented)

Time = 35.39 (sec) , antiderivative size = 20000, normalized size of antiderivative = 26.63 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x)

[Out]

log((e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)^2*(35*b^6*e^6 + 27648*c^6*d^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*
d^4*e^2 + 9456*a^2*b^2*c^2*e^6 + 57024*a^2*c^4*d^2*e^4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b
^4*c^2*d^2*e^4 - 1176*a*b^4*c*e^6 - 82944*b*c^5*d^5*e + 756*b^5*c*d*e^5 - 152064*a*b*c^4*d^3*e^3 - 9504*a*b^3*
c^2*d*e^5 - 57024*a^2*b*c^3*d*e^5 + 85536*a*b^2*c^3*d^2*e^4))/(64*c*(4*a*c - b^2)^6) - (2^(1/2)*((2^(1/2)*((c*
e^3*(24*c^3*d^4 + a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2*d^2*e^2 + 25*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e
- 44*a*b*c*d*e^3))/(4*a*c - b^2) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 +
4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a
^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^
7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*
a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*
c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 -
21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c
*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 +
144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7
*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a
^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2
+ 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^
2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*
c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^
3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c
^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b
*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*
c - b^2)^10))^(1/2))/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(
1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2
*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^
3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 137
62560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^
14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7
*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^
5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3
*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18
063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*
d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896
*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e -
630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 215
60*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 1032192
0*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472
896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 +
2580480*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))/16 + ((d + e*x)^(1/2)*(b^8*e^10 + 800*a^4*c^4*e^10
+ 4608*c^8*d^8*e^2 + 13440*a*c^7*d^6*e^4 - 18432*b*c^7*d^7*e^3 + 314*a^2*b^4*c^2*e^10 + 208*a^3*b^2*c^3*e^10 +
12320*a^2*c^6*d^4*e^6 + 4032*a^3*c^5*d^2*e^8 + 28896*b^2*c^6*d^6*e^4 - 22176*b^3*c^5*d^5*e^5 + 8330*b^4*c^4*d
^4*e^6 - 1204*b^5*c^3*d^3*e^7 - 42*b^6*c^2*d^2*e^8 - 36*a*b^6*c*e^10 + 20*b^7*c*d*e^9 + 15456*a^2*b^2*c^4*d^2*
e^8 - 40320*a*b*c^6*d^5*e^5 - 196*a*b^5*c^2*d*e^9 - 4032*a^3*b*c^4*d*e^9 + 44240*a*b^2*c^5*d^4*e^6 - 21280*a*b
^3*c^4*d^3*e^7 + 4116*a*b^4*c^3*d^2*e^8 - 24640*a^2*b*c^5*d^3*e^7 - 3136*a^2*b^3*c^3*d*e^9))/(8*c*(4*a*c - b^2
)^4))*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8
*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 294
9120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*
b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5
*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 2
1*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^1
5)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^
2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 -
2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9
*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*
a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e
^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e
^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*
e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*
e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*
e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c
^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))/16)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*
(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c
^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^
7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*
a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*
c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^
15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3
064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d
^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3
*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 -
1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*
c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 3225
60*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b
^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^1
2*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4
*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 1892352
0*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(128*(1048576*a^10*c^13 + b^20*c^3 - 40*a*b^18*c^4 + 720*a^2*
b^16*c^5 - 7680*a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 258048*a^5*b^10*c^8 + 860160*a^6*b^8*c^9 - 1966080*a^7*b^6
*c^10 + 2949120*a^8*b^4*c^11 - 2621440*a^9*b^2*c^12)))^(1/2) - log((e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)^
2*(35*b^6*e^6 + 27648*c^6*d^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*d^4*e^2 + 9456*a^2*b^2*c^2*e^6 + 57024*a^2*c^4*
d^2*e^4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b^4*c^2*d^2*e^4 - 1176*a*b^4*c*e^6 - 82944*b*c^5
*d^5*e + 756*b^5*c*d*e^5 - 152064*a*b*c^4*d^3*e^3 - 9504*a*b^3*c^2*d*e^5 - 57024*a^2*b*c^3*d*e^5 + 85536*a*b^2
*c^3*d^2*e^4))/(64*c*(4*a*c - b^2)^6) - (((c*e^3*(24*c^3*d^4 + a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2
*d^2*e^2 + 25*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e - 44*a*b*c*d*e^3))/(4*a*c - b^2) + 8*c^2*e^2*(4*a*c - b^2)*(b*e -
2*c*d)*(d + e*x)^(1/2)*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 + (b^2*e^7*(-(4*a*c - b^2)^15
)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b
^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e
^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*
a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c
^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7
)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 + (21*b*c*d*e^6*(-(4*a*c - b^2)^15)
^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2
*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13
335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^
4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d
^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d
^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3
+ (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d
*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^
5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2
*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^1
0*c^7*d^7 + (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*
d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2
*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^
6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 +
(735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c -
b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/
128 + (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4
515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^
4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5
*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5
*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320
*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^1
0*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*
a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 1612
80*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a
^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2) - ((d + e
*x)^(1/2)*(b^8*e^10 + 800*a^4*c^4*e^10 + 4608*c^8*d^8*e^2 + 13440*a*c^7*d^6*e^4 - 18432*b*c^7*d^7*e^3 + 314*a^
2*b^4*c^2*e^10 + 208*a^3*b^2*c^3*e^10 + 12320*a^2*c^6*d^4*e^6 + 4032*a^3*c^5*d^2*e^8 + 28896*b^2*c^6*d^6*e^4 -
22176*b^3*c^5*d^5*e^5 + 8330*b^4*c^4*d^4*e^6 - 1204*b^5*c^3*d^3*e^7 - 42*b^6*c^2*d^2*e^8 - 36*a*b^6*c*e^10 +
20*b^7*c*d*e^9 + 15456*a^2*b^2*c^4*d^2*e^8 - 40320*a*b*c^6*d^5*e^5 - 196*a*b^5*c^2*d*e^9 - 4032*a^3*b*c^4*d*e^
9 + 44240*a*b^2*c^5*d^4*e^6 - 21280*a*b^3*c^4*d^3*e^7 + 4116*a*b^4*c^3*d^2*e^8 - 24640*a^2*b*c^5*d^3*e^7 - 313
6*a^2*b^3*c^3*d*e^9))/(8*c*(4*a*c - b^2)^4))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 + (b^2*e
^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c
^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32
- (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*
b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*
e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/1
28 - (55*a*b^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 + (21*b*c*d*e^
6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d
^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*
b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 973
00*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 -
193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e
^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 15
75*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e
+ (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^
6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 -
147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))*(-((b^17*e^7)/128 + 36864*
a^5*c^12*d^7 - 36*b^10*c^7*d^7 + (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8
*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^
2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*
e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b
^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 - (21*c
^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 - (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128
+ (21*b^16*c*d*e^6)/128 + (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2
*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 -
16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^
5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10
*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2
*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^
11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3
*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^1
0*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c
^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(1048576*a^10*c^13
+ b^20*c^3 - 40*a*b^18*c^4 + 720*a^2*b^16*c^5 - 7680*a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 258048*a^5*b^10*c^8
+ 860160*a^6*b^8*c^9 - 1966080*a^7*b^6*c^10 + 2949120*a^8*b^4*c^11 - 2621440*a^9*b^2*c^12))^(1/2) - (((d + e*x
)^(1/2)*(20*a^3*c*e^7 + 24*c^4*d^6*e + a^2*b^2*e^7 + b^4*d^2*e^5 + 68*a*c^3*d^4*e^3 - 72*b*c^3*d^5*e^2 - 26*b^
3*c*d^3*e^4 + 64*a^2*c^2*d^2*e^5 + 73*b^2*c^2*d^4*e^3 - 2*a*b^3*d*e^6 - 64*a^2*b*c*d*e^6 - 136*a*b*c^2*d^3*e^4
+ 70*a*b^2*c*d^2*e^5))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - ((d + e*x)^(3/2)*(b^4*d*e^5 - a*b^3*e^6 + 36*c^
4*d^5*e + 64*a*c^3*d^3*e^3 + 28*a^2*c^2*d*e^5 - 90*b*c^3*d^4*e^2 - 21*b^3*c*d^2*e^4 + 74*b^2*c^2*d^3*e^3 - 14*
a^2*b*c*e^6 + 34*a*b^2*c*d*e^5 - 96*a*b*c^2*d^2*e^4))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (e*(d + e*x)^(7/2
)*(b^3*e^3 + 24*c^3*d^3 - 16*a*b*c*e^3 + 32*a*c^2*d*e^2 - 36*b*c^2*d^2*e + 10*b^2*c*d*e^2))/(4*(b^4 + 16*a^2*c
^2 - 8*a*b^2*c)) + (e*(d + e*x)^(5/2)*(b^4*e^4 + 72*c^4*d^4 + 36*a^2*c^2*e^4 + 92*a*c^3*d^2*e^2 + 85*b^2*c^2*d
^2*e^2 + 5*a*b^2*c*e^4 - 144*b*c^3*d^3*e - 13*b^3*c*d*e^3 - 92*a*b*c^2*d*e^3))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^
2*c)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2
*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^
2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) - log((e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)^2*(35*
b^6*e^6 + 27648*c^6*d^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*d^4*e^2 + 9456*a^2*b^2*c^2*e^6 + 57024*a^2*c^4*d^2*e^
4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b^4*c^2*d^2*e^4 - 1176*a*b^4*c*e^6 - 82944*b*c^5*d^5*e
+ 756*b^5*c*d*e^5 - 152064*a*b*c^4*d^3*e^3 - 9504*a*b^3*c^2*d*e^5 - 57024*a^2*b*c^3*d*e^5 + 85536*a*b^2*c^3*d
^2*e^4))/(64*c*(4*a*c - b^2)^6) - (((c*e^3*(24*c^3*d^4 + a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2*d^2*e
^2 + 25*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e - 44*a*b*c*d*e^3))/(4*a*c - b^2) + 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d
)*(d + e*x)^(1/2)*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2
))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9
*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8
+ (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^
11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3
*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128
+ (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2)
)/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*
c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^
3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14
910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4
- 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e -
(315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (26
95*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/
16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^
6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 +
20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*
d^7 - (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6
+ 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*
c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7
+ 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b
^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^
15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 -
(21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^
2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3
- 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d
^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c
^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b
^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*
d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^
7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4
*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4
*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2) - ((d + e*x)^(1
/2)*(b^8*e^10 + 800*a^4*c^4*e^10 + 4608*c^8*d^8*e^2 + 13440*a*c^7*d^6*e^4 - 18432*b*c^7*d^7*e^3 + 314*a^2*b^4*
c^2*e^10 + 208*a^3*b^2*c^3*e^10 + 12320*a^2*c^6*d^4*e^6 + 4032*a^3*c^5*d^2*e^8 + 28896*b^2*c^6*d^6*e^4 - 22176
*b^3*c^5*d^5*e^5 + 8330*b^4*c^4*d^4*e^6 - 1204*b^5*c^3*d^3*e^7 - 42*b^6*c^2*d^2*e^8 - 36*a*b^6*c*e^10 + 20*b^7
*c*d*e^9 + 15456*a^2*b^2*c^4*d^2*e^8 - 40320*a*b*c^6*d^5*e^5 - 196*a*b^5*c^2*d*e^9 - 4032*a^3*b*c^4*d*e^9 + 44
240*a*b^2*c^5*d^4*e^6 - 21280*a*b^3*c^4*d^3*e^7 + 4116*a*b^4*c^3*d^2*e^8 - 24640*a^2*b*c^5*d^3*e^7 - 3136*a^2*
b^3*c^3*d*e^9))/(8*c*(4*a*c - b^2)^4))*(-((b^17*e^7)/128 + 36864*a^5*c^12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(-(
4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 + 26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6
*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635
*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 - 5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^
7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8
- (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (
55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-(4
*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4
)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^
6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2 - 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4
*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 19320
0*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2
520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b
^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105
*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e -
6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 14784
0*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))*(-((b^17*e^7)/128 + 36864*a^5*c^
12*d^7 - 36*b^10*c^7*d^7 - (b^2*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + 720*a*b^8*c^8*d^7 - 13440*a^8*b*c^8*e^7 +
26880*a^8*c^9*d*e^6 + 126*b^11*c^6*d^6*e - 5760*a^2*b^6*c^9*d^7 + 23040*a^3*b^4*c^10*d^7 - 46080*a^4*b^2*c^11
*d^7 + (285*a^2*b^13*c^2*e^7)/32 - (635*a^3*b^11*c^3*e^7)/8 + (545*a^4*b^9*c^4*e^7)/2 + 342*a^5*b^7*c^5*e^7 -
5320*a^6*b^5*c^6*e^7 + 14560*a^7*b^3*c^7*e^7 + 107520*a^6*c^11*d^5*e^2 + 98560*a^7*c^10*d^3*e^4 - (651*b^12*c^
5*d^5*e^2)/4 + (735*b^13*c^4*d^4*e^3)/8 - (1225*b^14*c^3*d^3*e^4)/64 - (21*b^15*c^2*d^2*e^5)/128 + (21*c^2*d^2
*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (55*a*b^15*c*e^7)/128 + (25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2))/128 + (21
*b^16*c*d*e^6)/128 - (21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2))/128 - 23940*a^2*b^8*c^7*d^5*e^2 + 9450*a^2*b^9*c
^6*d^4*e^3 + (4515*a^2*b^10*c^5*d^3*e^4)/4 - (8505*a^2*b^11*c^4*d^2*e^5)/8 + 87360*a^3*b^6*c^8*d^5*e^2 - 16800
*a^3*b^7*c^7*d^4*e^3 - 20125*a^3*b^8*c^6*d^3*e^4 + (13335*a^3*b^9*c^5*d^2*e^5)/2 - 141120*a^4*b^4*c^9*d^5*e^2
- 50400*a^4*b^5*c^8*d^4*e^3 + 97300*a^4*b^6*c^7*d^3*e^4 - 14910*a^4*b^7*c^6*d^2*e^5 + 32256*a^5*b^2*c^10*d^5*e
^2 + 241920*a^5*b^3*c^9*d^4*e^3 - 193200*a^5*b^4*c^8*d^3*e^4 - 16632*a^5*b^5*c^7*d^2*e^5 + 96320*a^6*b^2*c^9*d
^3*e^4 + 124320*a^6*b^3*c^8*d^2*e^5 - 2520*a*b^9*c^7*d^6*e - (315*a*b^14*c^2*d*e^6)/64 - 129024*a^5*b*c^11*d^6
*e + 3150*a*b^10*c^6*d^5*e^2 - 1575*a*b^11*c^5*d^4*e^3 + (2695*a*b^12*c^4*d^3*e^4)/16 + (1995*a*b^13*c^3*d^2*e
^5)/32 + 20160*a^2*b^7*c^8*d^6*e + (105*a^2*b^12*c^3*d*e^6)/16 - 80640*a^3*b^5*c^9*d^6*e + (2625*a^3*b^10*c^4*
d*e^6)/4 + 161280*a^4*b^3*c^10*d^6*e - 6615*a^4*b^8*c^5*d*e^6 + 27132*a^5*b^6*c^6*d*e^6 - 268800*a^6*b*c^10*d^
4*e^3 - 48720*a^6*b^4*c^7*d*e^6 - 147840*a^7*b*c^9*d^2*e^5 + 20160*a^7*b^2*c^8*d*e^6)/(1048576*a^10*c^13 + b^2
0*c^3 - 40*a*b^18*c^4 + 720*a^2*b^16*c^5 - 7680*a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 258048*a^5*b^10*c^8 + 8601
60*a^6*b^8*c^9 - 1966080*a^7*b^6*c^10 + 2949120*a^8*b^4*c^11 - 2621440*a^9*b^2*c^12))^(1/2) + log((e^3*(b*e -
2*c*d)*(a*e^2 + c*d^2 - b*d*e)^2*(35*b^6*e^6 + 27648*c^6*d^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*d^4*e^2 + 9456*a
^2*b^2*c^2*e^6 + 57024*a^2*c^4*d^2*e^4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b^4*c^2*d^2*e^4 -
1176*a*b^4*c*e^6 - 82944*b*c^5*d^5*e + 756*b^5*c*d*e^5 - 152064*a*b*c^4*d^3*e^3 - 9504*a*b^3*c^2*d*e^5 - 5702
4*a^2*b*c^3*d*e^5 + 85536*a*b^2*c^3*d^2*e^4))/(64*c*(4*a*c - b^2)^6) - (2^(1/2)*((2^(1/2)*((c*e^3*(24*c^3*d^4
+ a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2*d^2*e^2 + 25*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e - 44*a*b*c*d*e^3
))/(4*a*c - b^2) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 4718592*a^5*c^12
*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3
440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*
b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7
- 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 2
0832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*e^5*
(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 - 21*b*c*
d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*
c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 25760
00*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e
^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*
b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 +
15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403
200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580
480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643
840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 -
6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1
/2))/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b
^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2
949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^
4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d
^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 -
21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7*(-(4*a*c - b^2)
^15)^(1/2) + 21*b^16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*
a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2
- 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c
^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 412876
8*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2
*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d
*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^
3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^
6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*
d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2
*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))/16 + ((d + e*x)^(1/2)*(b^8*e^10 + 800*a^4*c^4*e^10 + 4608*c^8*d^8*e
^2 + 13440*a*c^7*d^6*e^4 - 18432*b*c^7*d^7*e^3 + 314*a^2*b^4*c^2*e^10 + 208*a^3*b^2*c^3*e^10 + 12320*a^2*c^6*d
^4*e^6 + 4032*a^3*c^5*d^2*e^8 + 28896*b^2*c^6*d^6*e^4 - 22176*b^3*c^5*d^5*e^5 + 8330*b^4*c^4*d^4*e^6 - 1204*b^
5*c^3*d^3*e^7 - 42*b^6*c^2*d^2*e^8 - 36*a*b^6*c*e^10 + 20*b^7*c*d*e^9 + 15456*a^2*b^2*c^4*d^2*e^8 - 40320*a*b*
c^6*d^5*e^5 - 196*a*b^5*c^2*d*e^9 - 4032*a^3*b*c^4*d*e^9 + 44240*a*b^2*c^5*d^4*e^6 - 21280*a*b^3*c^4*d^3*e^7 +
4116*a*b^4*c^3*d^2*e^8 - 24640*a^2*b*c^5*d^3*e^7 - 3136*a^2*b^3*c^3*d*e^9))/(8*c*(4*a*c - b^2)^4))*(-(b^17*e^
7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 17203
20*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^1
0*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43
776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*
a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e
^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^
16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^
3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7
*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 64512
00*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5
*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^
6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^
5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^1
3*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^
10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a
^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(c^3*(
4*a*c - b^2)^10))^(1/2))/16)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 - b^2*e^7*(-(4*a*c - b^2)^
15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 73728
0*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^
11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7
+ 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 24
50*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 + 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 + 25*a*
c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 - 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c
^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 1118208
0*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5
- 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7
*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 21
28896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6
*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3
+ 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10
321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 +
3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*
e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(128*(1048576*a^10*c^13 + b^20*c^3 - 40*a*b^18*c^4 + 720*a^2*b^16*c^5 - 7680*
a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 258048*a^5*b^10*c^8 + 860160*a^6*b^8*c^9 - 1966080*a^7*b^6*c^10 + 2949120*
a^8*b^4*c^11 - 2621440*a^9*b^2*c^12)))^(1/2)