\(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\) [2304]

Optimal result

Integrand size = 22, antiderivative size = 441 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1/4*(12*b*c*d-7*b^2*e+4*a*c*e+12*c*
(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-3/4*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))*c^(1/2)*(16*c^2*d^2+b*e^2*(3*b-2*(-4*a*c+b^2)^(1/2))-4*c*e*(4*b*d-a*e-d*(-4*a*c
+b^2)^(1/2)))/(-4*a*c+b^2)^(5/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+3/4*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))*c^(1/2)*(16*c^2*d^2+b*e^2*(3*b+2*(-4*a*c+b^2)^(1/2))-4*c*e
*(4*b*d-a*e+d*(-4*a*c+b^2)^(1/2)))/(-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 = 1.22 (sec) , antiderivative size = 441, 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, 836, 840, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {3 \sqrt {c} \left (-4 c e \left (-d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt {b^2-4 a c}\right )+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {3 \sqrt {c} \left (-4 c e \left (d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt {b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[In]

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

[Out]

-1/2*(Sqrt[d + e*x]*(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]*(12*
b*c*d - 7*b^2*e + 4*a*c*e + 12*c*(2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(16*c^2*
d^2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*
Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt
[b^2 - 4*a*c])*e]) + (3*Sqrt[c]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d + Sqrt[b^2 - 4*
a*c]*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[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 836

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

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 {\sqrt {d+e x} (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 {\frac {1}{2} \left (12 c d^2-7 b d e+2 a e^2\right )+\frac {5}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {\sqrt {d+e x} (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-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-12 b c d e+b^2 e^2+4 a c e^2\right )+3 c e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {\sqrt {d+e x} (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-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-3 c d e (2 c d-b e) \left (c d^2-b d e+a e^2\right )+\frac {3}{4} e \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-12 b c d e+b^2 e^2+4 a c e^2\right )+3 c e (2 c d-b e) \left (c d^2-b d e+a e^2\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 )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {\sqrt {d+e x} (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-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-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 )}{4 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (3 c \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-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 )}{4 \left (b^2-4 a c\right )^{5/2}} \\ & = -\frac {\sqrt {d+e x} (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-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\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 )}{2 \sqrt {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 {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\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 )}{2 \sqrt {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 = 15.80 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {(d+e x)^{5/2} \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{(a+x (b+c x))^2}-\frac {(d+e x)^{5/2} \left (-b^4 e^3+b^3 c e^2 (12 d-e x)+4 b c^2 \left (3 c d^2 (d-3 e x)-a e^2 (3 d+2 e x)\right )+4 c^2 \left (3 a^2 e^3+6 c^2 d^3 x+a c d e (5 d+4 e x)\right )+b^2 c e \left (-5 a e^2+c d (-23 d+14 e x)\right )\right )}{2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}+\frac {e \sqrt {d+e x} \left (-24 c^3 d^3 (2 d+e x)+b^2 e^3 (4 b d-3 a e+b e x)+4 c^2 d e (3 b d (7 d+3 e x)-a e (13 d+4 e x))+c e^2 \left (-12 a^2 e^2+4 a b e (11 d+2 e x)-b^2 d (41 d+14 e x)\right )\right )}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )}-\frac {3 \sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (32 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt {b^2-4 a c} d-6 a b e-2 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}+\frac {3 \sqrt {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (32 c^3 d^3+b^2 \left (-b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-6 b d+\sqrt {b^2-4 a c} d+3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt {b^2-4 a c} d-6 a b e+2 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]

[In]

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

[Out]

(((d + e*x)^(5/2)*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x))^2 - ((d + e*x)^(5/2)*(-(b^4*
e^3) + b^3*c*e^2*(12*d - e*x) + 4*b*c^2*(3*c*d^2*(d - 3*e*x) - a*e^2*(3*d + 2*e*x)) + 4*c^2*(3*a^2*e^3 + 6*c^2
*d^3*x + a*c*d*e*(5*d + 4*e*x)) + b^2*c*e*(-5*a*e^2 + c*d*(-23*d + 14*e*x))))/(2*(b^2 - 4*a*c)*(-(c*d^2) + e*(
b*d - a*e))*(a + x*(b + c*x))) + (e*Sqrt[d + e*x]*(-24*c^3*d^3*(2*d + e*x) + b^2*e^3*(4*b*d - 3*a*e + b*e*x) +
4*c^2*d*e*(3*b*d*(7*d + 3*e*x) - a*e*(13*d + 4*e*x)) + c*e^2*(-12*a^2*e^2 + 4*a*b*e*(11*d + 2*e*x) - b^2*d*(4
1*d + 14*e*x))))/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))) - (3*Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*
(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b
^2*d + 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/S
qrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(3/2)) + (3*Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4
*a*c])*e]*(32*c^3*d^3 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 3*a*e) +
2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt
[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c)*(c*d^2
+ e*(-(b*d) + a*e)))

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.20

method	result	size
pseudoelliptic	\(-\frac {3 \left (\sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e c \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\left (-\frac {b e}{2}+c d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+4 c^{2} d^{2}+\left (a \,e^{2}-4 b d e \right ) c +\frac {3 b^{2} e^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\left (\left (\left (\frac {b e}{2}-c d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+4 c^{2} d^{2}+\left (a \,e^{2}-4 b d e \right ) c +\frac {3 b^{2} e^{2}}{4}\right ) \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} e c \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\left (-2 c^{3} d \,x^{3}+\left (\left (b \,x^{3}-\frac {1}{3} x^{2} a \right ) e -\frac {10 x d \left (\frac {9 b x}{10}+a \right )}{3}\right ) c^{2}+\left (\left (\frac {19}{12} b^{2} x^{2}+a^{2}+\frac {4}{3} a b x \right ) e -\frac {5 \left (\frac {2 b x}{5}+a \right ) b d}{3}\right ) c +\frac {\left (\left (\frac {5 b x}{3}+a \right ) e +\frac {2 b d}{3}\right ) b^{2}}{4}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right )}{16 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}\)	\(530\)
derivativedivides	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (4 c \,a^{2} e^{4}+b^{2} a \,e^{4}-12 a b c d \,e^{3}+12 a \,c^{2} d^{2} e^{2}-b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 b \,c^{2} d^{3} e +8 c^{3} d^{4}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )\)	\(703\)
default	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (4 c \,a^{2} e^{4}+b^{2} a \,e^{4}-12 a b c d \,e^{3}+12 a \,c^{2} d^{2} e^{2}-b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 b \,c^{2} d^{3} e +8 c^{3} d^{4}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )\)	\(703\)

[In]

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

[Out]

-3/16/(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(2^(1/2)*(c*x^2+b*x+a)^2*e*c*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(
1/2)*((-1/2*b*e+c*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+4*c^2*d^2+(a*e^2-4*b*d*e)*c+3/4*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))+(((1/2*b*e-c*d)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)
+4*c^2*d^2+(a*e^2-4*b*d*e)*c+3/4*b^2*e^2)*2^(1/2)*(c*x^2+b*x+a)^2*e*c*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))+(-2*c^3*d*x^3+((b*x^3-1/3*x^2*a)*e-10/3*x*d*(9/10*b*x+a))*c^2+((19/
12*b^2*x^2+a^2+4/3*a*b*x)*e-5/3*(2/5*b*x+a)*b*d)*c+1/4*((5/3*b*x+a)*e+2/3*b*d)*b^2)*(-4*(a*c-1/4*b^2)*e^2)^(1/
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)/((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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11128 vs. \(2 (381) = 762\).

Time = 0.73 (sec) , antiderivative size = 11128, normalized size of antiderivative = 25.23 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate((e*x + d)^(3/2)/(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. 2116 vs. \(2 (381) = 762\).

Time = 1.29 (sec) , antiderivative size = 2116, normalized size of antiderivative = 4.80 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/16*(2*(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e
^2) + (8*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2*e - 8*(b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c)*d*e^2 + (b^4 -
16*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*e - 8*a*b^2*c*e +
16*a^2*c^2*e) - (32*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3*e - 48*(b^7*c^2 - 12*a*b^5*c^3
+ 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e^2 + 2*(11*b^8*c - 128*a*b^6*c^2 + 480*a^2*b^4*c^3 - 512*a^3*b^2*c^4 - 2
56*a^4*c^5)*d*e^3 - (3*b^9 - 32*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^4*b*c^4)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b
^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e +
8*a*b^3*c*e - 16*a^2*b*c^2*e + sqrt((2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2
*b*c^2*e)^2 - 4*(b^4*c*d^2 - 8*a*b^2*c^2*d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3*c*d*e - 16*a^2*b*c^2*d*e + a
*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e^2)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^
2*c^3)))/(((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^7 - 12*a*b^5*c + 48
*a^2*b^3*c^2 - 64*a^3*b*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*sqrt
(b^2 - 4*a*c)*e^2)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*abs(c)) - 3/16*(2*(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2
*e)^2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e^2) - (8*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4
*a*c)*d^2*e - 8*(b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c)*d*e^2 + (b^4 - 16*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-
4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e) - (32*(b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3*e - 48*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e^2
+ 2*(11*b^8*c - 128*a*b^6*c^2 + 480*a^2*b^4*c^3 - 512*a^3*b^2*c^4 - 256*a^4*c^5)*d*e^3 - (3*b^9 - 32*a*b^7*c +
96*a^2*b^5*c^2 - 256*a^4*b*c^4)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqr
t(e*x + d)/sqrt(-(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e - sqrt((2*b
^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)^2 - 4*(b^4*c*d^2 - 8*a*b^2*c^2*
d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3*c*d*e - 16*a^2*b*c^2*d*e + a*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e
^2)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(((b^6*c - 12*a*b^4*c^2 + 48*a^2
*b^2*c^3 - 64*a^3*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*sqrt(b^2 - 4
*a*c)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^4*e - 8*a*b^2*c*
e + 16*a^2*c^2*e)*abs(c)) + 1/4*(24*(e*x + d)^(7/2)*c^3*d*e - 72*(e*x + d)^(5/2)*c^3*d^2*e + 72*(e*x + d)^(3/2
)*c^3*d^3*e - 24*sqrt(e*x + d)*c^3*d^4*e - 12*(e*x + d)^(7/2)*b*c^2*e^2 + 72*(e*x + d)^(5/2)*b*c^2*d*e^2 - 108
*(e*x + d)^(3/2)*b*c^2*d^2*e^2 + 48*sqrt(e*x + d)*b*c^2*d^3*e^2 - 19*(e*x + d)^(5/2)*b^2*c*e^3 + 4*(e*x + d)^(
5/2)*a*c^2*e^3 + 46*(e*x + d)^(3/2)*b^2*c*d*e^3 + 32*(e*x + d)^(3/2)*a*c^2*d*e^3 - 27*sqrt(e*x + d)*b^2*c*d^2*
e^3 - 36*sqrt(e*x + d)*a*c^2*d^2*e^3 - 5*(e*x + d)^(3/2)*b^3*e^4 - 16*(e*x + d)^(3/2)*a*b*c*e^4 + 3*sqrt(e*x +
d)*b^3*d*e^4 + 36*sqrt(e*x + d)*a*b*c*d*e^4 - 3*sqrt(e*x + d)*a*b^2*e^5 - 12*sqrt(e*x + d)*a^2*c*e^5)/((b^4 -
8*a*b^2*c + 16*a^2*c^2)*((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 = 32.39 (sec) , antiderivative size = 13764, normalized size of antiderivative = 31.21 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log((27*c^3*e^3*(b*e - 2*c*d)*(5*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 192*a*c^3*d^2*e^2 + 336*b^2*c^2*d^2*
e^2 + 40*a*b^2*c*e^4 - 512*b*c^3*d^3*e - 80*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3))/(16*(4*a*c - b^2)^6) - (3*2^(1/2
)*((3*2^(1/2)*((3*c^2*e^3*(b^2*e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(4*a*c - b^2) - (3*2^(1/2)*c^2*e^2*(4
*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 -
512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81
920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^
3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b
^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 4480
0*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2
- 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e
+ 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 + 20480
0*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^
4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 + 204800*a^6*
b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/
2) + 524288*a^5*c^10*d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4
+ 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^1
1*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 6553
60*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400
*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1
024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*
e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040
*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3
*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*
b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/16 + (9*c^3*e^2*(
d + e*x)^(1/2)*(13*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 64*a*c^3*d^2*e^2 + 368*b^2*c^2*d^2*e^2 + 8*a*b^2*c
*e^4 - 512*b*c^3*d^3*e - 112*b^3*c*d*e^3 - 64*a*b*c^2*d*e^3))/(4*(4*a*c - b^2)^4))*(-(b^15*e^5 + e^5*(-(4*a*c
- b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840
*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5
- 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*
c^6*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20*a*b^13*c*e^5 - 50*b^14*c*
d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c
^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2*c^8*d^3*e^2 + 819200*a^
5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*
d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 819200*a^3*b^5*c^7*d^4
*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4
- 983040*a^6*b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/16)
*(-(9*(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5
- 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c
^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a
^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3
+ 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6
*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*
a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*b^12*c^2*d*e^4 - 1310720*a^5*
b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3
*d*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8*d^4*e + 166400*a^4*b^6*c^5*d
*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4))/(128*(a*b^20*e^2 + b^2
0*c*d^2 + 1048576*a^10*c^11*d^2 + 1048576*a^11*c^10*e^2 - b^21*d*e - 40*a*b^18*c^2*d^2 - 40*a^2*b^18*c*e^2 + 7
20*a^2*b^16*c^3*d^2 - 7680*a^3*b^14*c^4*d^2 + 53760*a^4*b^12*c^5*d^2 - 258048*a^5*b^10*c^6*d^2 + 860160*a^6*b^
8*c^7*d^2 - 1966080*a^7*b^6*c^8*d^2 + 2949120*a^8*b^4*c^9*d^2 - 2621440*a^9*b^2*c^10*d^2 + 720*a^3*b^16*c^2*e^
2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 196608
0*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2 - 1048576*a^10*b*c^10*d*e - 720*a^2*b^1
7*c^2*d*e + 7680*a^3*b^15*c^3*d*e - 53760*a^4*b^13*c^4*d*e + 258048*a^5*b^11*c^5*d*e - 860160*a^6*b^9*c^6*d*e
+ 1966080*a^7*b^7*c^7*d*e - 2949120*a^8*b^5*c^8*d*e + 2621440*a^9*b^3*c^9*d*e + 40*a*b^19*c*d*e)))^(1/2) - log
((27*c^3*e^3*(b*e - 2*c*d)*(5*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 192*a*c^3*d^2*e^2 + 336*b^2*c^2*d^2*e^2
+ 40*a*b^2*c*e^4 - 512*b*c^3*d^3*e - 80*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3))/(16*(4*a*c - b^2)^6) - (((3*c^2*e^3
*(b^2*e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(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)*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^5*c^10*d^5 - 36*b^10*c^5*d^5 + 7
20*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*e - 5760*a^2*b^6*c^7*d^5 + 23040
*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^
4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (315*b^12*c^3*d^3*e^2)/4 + (225*b^
13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^
4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^
6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9*c^5*d^4*e + (675*a*b^12*c^2*d*e
^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*e^3 + 14400*a^2*b^7*c^6*d^4*e -
(45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 + 115200*a^4*b^3*c^8*d^4*e + 1170
0*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 14400*a^6*b^2*c^7*d*e^4)/((4*a*c - b
^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^
5*c^10*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*
e - 5760*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3
*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (3
15*b^12*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b
^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b
^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9
*c^5*d^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*
e^3 + 14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 +
115200*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 1440
0*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2) - (9*c^3*e^2*(d + e*x)^(1/2)*(13*b^4*e^
4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 64*a*c^3*d^2*e^2 + 368*b^2*c^2*d^2*e^2 + 8*a*b^2*c*e^4 - 512*b*c^3*d^3*e -
112*b^3*c*d*e^3 - 64*a*b*c^2*d*e^3))/(4*(4*a*c - b^2)^4))*(-((9*b^15*e^5)/128 + (9*e^5*(-(4*a*c - b^2)^15)^(1/
2))/128 + 36864*a^5*c^10*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*b*c^7*e^5 + 11520*a^7*c^8*d*e^4
+ 90*b^11*c^4*d^4*e - 5760*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4*b^2*c^9*d^5 - (315*a^2*b^11*c^2
*e^5)/8 + (585*a^3*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 + 4320*a^6*b^3*c^6*e^5 + 46080*a^
6*c^9*d^3*e^2 - (315*b^12*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^13*c*e^5)/32 - (225*b^14*c*d*e^4
)/64 - 11700*a^2*b^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2
*e^3 - 72000*a^4*b^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^8*d^3*e^2 + 57600*a^5*b^3*c^7*d^
2*e^3 - 1800*a*b^9*c^5*d^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4*e + 1530*a*b^10*c^4*d^3*e^2 - 4
95*a*b^11*c^3*d^2*e^3 + 14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 - 57600*a^3*b^5*c^7*d^4*e - 2025*a
^3*b^8*c^4*d*e^4 + 115200*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*a^5*b^4*c^6*d*e^4 - 69120*a^6*b*
c^8*d^2*e^3 + 14400*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-((9*b^15*e^5)/128
+ (9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 + 36864*a^5*c^10*d^5 - 36*b^10*c^5*d^5 + 720*a*b^8*c^6*d^5 - 5760*a^7*
b*c^7*e^5 + 11520*a^7*c^8*d*e^4 + 90*b^11*c^4*d^4*e - 5760*a^2*b^6*c^7*d^5 + 23040*a^3*b^4*c^8*d^5 - 46080*a^4
*b^2*c^9*d^5 - (315*a^2*b^11*c^2*e^5)/8 + (585*a^3*b^9*c^3*e^5)/2 - 810*a^4*b^7*c^4*e^5 - 72*a^5*b^5*c^5*e^5 +
4320*a^6*b^3*c^6*e^5 + 46080*a^6*c^9*d^3*e^2 - (315*b^12*c^3*d^3*e^2)/4 + (225*b^13*c^2*d^2*e^3)/8 + (45*a*b^
13*c*e^5)/32 - (225*b^14*c*d*e^4)/64 - 11700*a^2*b^8*c^5*d^3*e^2 + 3150*a^2*b^9*c^4*d^2*e^3 + 43200*a^3*b^6*c^
6*d^3*e^2 - 7200*a^3*b^7*c^5*d^2*e^3 - 72000*a^4*b^4*c^7*d^3*e^2 - 7200*a^4*b^5*c^6*d^2*e^3 + 23040*a^5*b^2*c^
8*d^3*e^2 + 57600*a^5*b^3*c^7*d^2*e^3 - 1800*a*b^9*c^5*d^4*e + (675*a*b^12*c^2*d*e^4)/16 - 92160*a^5*b*c^9*d^4
*e + 1530*a*b^10*c^4*d^3*e^2 - 495*a*b^11*c^3*d^2*e^3 + 14400*a^2*b^7*c^6*d^4*e - (45*a^2*b^10*c^3*d*e^4)/4 -
57600*a^3*b^5*c^7*d^4*e - 2025*a^3*b^8*c^4*d*e^4 + 115200*a^4*b^3*c^8*d^4*e + 11700*a^4*b^6*c^5*d*e^4 - 25200*
a^5*b^4*c^6*d*e^4 - 69120*a^6*b*c^8*d^2*e^3 + 14400*a^6*b^2*c^7*d*e^4)/(a*b^20*e^2 + b^20*c*d^2 + 1048576*a^10
*c^11*d^2 + 1048576*a^11*c^10*e^2 - b^21*d*e - 40*a*b^18*c^2*d^2 - 40*a^2*b^18*c*e^2 + 720*a^2*b^16*c^3*d^2 -
7680*a^3*b^14*c^4*d^2 + 53760*a^4*b^12*c^5*d^2 - 258048*a^5*b^10*c^6*d^2 + 860160*a^6*b^8*c^7*d^2 - 1966080*a^
7*b^6*c^8*d^2 + 2949120*a^8*b^4*c^9*d^2 - 2621440*a^9*b^2*c^10*d^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*
e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 29
49120*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2 - 1048576*a^10*b*c^10*d*e - 720*a^2*b^17*c^2*d*e + 7680*a^3*b
^15*c^3*d*e - 53760*a^4*b^13*c^4*d*e + 258048*a^5*b^11*c^5*d*e - 860160*a^6*b^9*c^6*d*e + 1966080*a^7*b^7*c^7*
d*e - 2949120*a^8*b^5*c^8*d*e + 2621440*a^9*b^3*c^9*d*e + 40*a*b^19*c*d*e))^(1/2) - (((d + e*x)^(3/2)*(5*b^3*e
^4 - 72*c^3*d^3*e + 108*b*c^2*d^2*e^2 + 16*a*b*c*e^4 - 32*a*c^2*d*e^3 - 46*b^2*c*d*e^3))/(4*(b^4 + 16*a^2*c^2
- 8*a*b^2*c)) + (3*(d + e*x)^(1/2)*(a*b^2*e^5 + 4*a^2*c*e^5 - b^3*d*e^4 + 8*c^3*d^4*e + 12*a*c^2*d^2*e^3 - 16*
b*c^2*d^3*e^2 + 9*b^2*c*d^2*e^3 - 12*a*b*c*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (e*(d + e*x)^(5/2)*(72
*c^3*d^2 - 4*a*c^2*e^2 + 19*b^2*c*e^2 - 72*b*c^2*d*e))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (3*c*e*(2*c^2*d -
b*c*e)*(d + e*x)^(7/2))/(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(
(27*c^3*e^3*(b*e - 2*c*d)*(5*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 192*a*c^3*d^2*e^2 + 336*b^2*c^2*d^2*e^2
+ 40*a*b^2*c*e^4 - 512*b*c^3*d^3*e - 80*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3))/(16*(4*a*c - b^2)^6) - (((3*c^2*e^3*
(b^2*e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(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)*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c^10*d^5 + 36*b^10*c^5*d^5 - 720
*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e + 5760*a^2*b^6*c^7*d^5 - 23040*a
^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^9*c^3*e^5)/2 + 810*a^4*b^7*c^4*
e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*b^12*c^3*d^3*e^2)/4 - (225*b^13
*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*c^5*d^3*e^2 - 3150*a^2*b^9*c^4*
d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*c^7*d^3*e^2 + 7200*a^4*b^5*c^6*
d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^5*d^4*e - (675*a*b^12*c^2*d*e^4
)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3 - 14400*a^2*b^7*c^6*d^4*e + (4
5*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 115200*a^4*b^3*c^8*d^4*e - 11700*
a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2
)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c
^10*d^5 + 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e +
5760*a^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^
9*c^3*e^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*
b^12*c^3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*
c^5*d^3*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*
c^7*d^3*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^
5*d^4*e - (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3
- 14400*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 11
5200*a^4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a
^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2) - (9*c^3*e^2*(d + e*x)^(1/2)*(13*b^4*e^4 +
256*c^4*d^4 + 16*a^2*c^2*e^4 + 64*a*c^3*d^2*e^2 + 368*b^2*c^2*d^2*e^2 + 8*a*b^2*c*e^4 - 512*b*c^3*d^3*e - 112
*b^3*c*d*e^3 - 64*a*b*c^2*d*e^3))/(4*(4*a*c - b^2)^4))*(((9*e^5*(-(4*a*c - b^2)^15)^(1/2))/128 - (9*b^15*e^5)/
128 - 36864*a^5*c^10*d^5 + 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7*e^5 - 11520*a^7*c^8*d*e^4 - 90
*b^11*c^4*d^4*e + 5760*a^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*c^9*d^5 + (315*a^2*b^11*c^2*e^5
)/8 - (585*a^3*b^9*c^3*e^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320*a^6*b^3*c^6*e^5 - 46080*a^6*c^
9*d^3*e^2 + (315*b^12*c^3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*e^5)/32 + (225*b^14*c*d*e^4)/64
+ 11700*a^2*b^8*c^5*d^3*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3*e^2 + 7200*a^3*b^7*c^5*d^2*e^3
+ 72000*a^4*b^4*c^7*d^3*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3*e^2 - 57600*a^5*b^3*c^7*d^2*e^
3 + 1800*a*b^9*c^5*d^4*e - (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e - 1530*a*b^10*c^4*d^3*e^2 + 495*a
*b^11*c^3*d^2*e^3 - 14400*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600*a^3*b^5*c^7*d^4*e + 2025*a^3*b
^8*c^4*d*e^4 - 115200*a^4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b^4*c^6*d*e^4 + 69120*a^6*b*c^8*
d^2*e^3 - 14400*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(((9*e^5*(-(4*a*c - b^2)
^15)^(1/2))/128 - (9*b^15*e^5)/128 - 36864*a^5*c^10*d^5 + 36*b^10*c^5*d^5 - 720*a*b^8*c^6*d^5 + 5760*a^7*b*c^7
*e^5 - 11520*a^7*c^8*d*e^4 - 90*b^11*c^4*d^4*e + 5760*a^2*b^6*c^7*d^5 - 23040*a^3*b^4*c^8*d^5 + 46080*a^4*b^2*
c^9*d^5 + (315*a^2*b^11*c^2*e^5)/8 - (585*a^3*b^9*c^3*e^5)/2 + 810*a^4*b^7*c^4*e^5 + 72*a^5*b^5*c^5*e^5 - 4320
*a^6*b^3*c^6*e^5 - 46080*a^6*c^9*d^3*e^2 + (315*b^12*c^3*d^3*e^2)/4 - (225*b^13*c^2*d^2*e^3)/8 - (45*a*b^13*c*
e^5)/32 + (225*b^14*c*d*e^4)/64 + 11700*a^2*b^8*c^5*d^3*e^2 - 3150*a^2*b^9*c^4*d^2*e^3 - 43200*a^3*b^6*c^6*d^3
*e^2 + 7200*a^3*b^7*c^5*d^2*e^3 + 72000*a^4*b^4*c^7*d^3*e^2 + 7200*a^4*b^5*c^6*d^2*e^3 - 23040*a^5*b^2*c^8*d^3
*e^2 - 57600*a^5*b^3*c^7*d^2*e^3 + 1800*a*b^9*c^5*d^4*e - (675*a*b^12*c^2*d*e^4)/16 + 92160*a^5*b*c^9*d^4*e -
1530*a*b^10*c^4*d^3*e^2 + 495*a*b^11*c^3*d^2*e^3 - 14400*a^2*b^7*c^6*d^4*e + (45*a^2*b^10*c^3*d*e^4)/4 + 57600
*a^3*b^5*c^7*d^4*e + 2025*a^3*b^8*c^4*d*e^4 - 115200*a^4*b^3*c^8*d^4*e - 11700*a^4*b^6*c^5*d*e^4 + 25200*a^5*b
^4*c^6*d*e^4 + 69120*a^6*b*c^8*d^2*e^3 - 14400*a^6*b^2*c^7*d*e^4)/(a*b^20*e^2 + b^20*c*d^2 + 1048576*a^10*c^11
*d^2 + 1048576*a^11*c^10*e^2 - b^21*d*e - 40*a*b^18*c^2*d^2 - 40*a^2*b^18*c*e^2 + 720*a^2*b^16*c^3*d^2 - 7680*
a^3*b^14*c^4*d^2 + 53760*a^4*b^12*c^5*d^2 - 258048*a^5*b^10*c^6*d^2 + 860160*a^6*b^8*c^7*d^2 - 1966080*a^7*b^6
*c^8*d^2 + 2949120*a^8*b^4*c^9*d^2 - 2621440*a^9*b^2*c^10*d^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 +
53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120
*a^9*b^4*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2 - 1048576*a^10*b*c^10*d*e - 720*a^2*b^17*c^2*d*e + 7680*a^3*b^15*c
^3*d*e - 53760*a^4*b^13*c^4*d*e + 258048*a^5*b^11*c^5*d*e - 860160*a^6*b^9*c^6*d*e + 1966080*a^7*b^7*c^7*d*e -
2949120*a^8*b^5*c^8*d*e + 2621440*a^9*b^3*c^9*d*e + 40*a*b^19*c*d*e))^(1/2) + log((27*c^3*e^3*(b*e - 2*c*d)*(
5*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 192*a*c^3*d^2*e^2 + 336*b^2*c^2*d^2*e^2 + 40*a*b^2*c*e^4 - 512*b*c^
3*d^3*e - 80*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3))/(16*(4*a*c - b^2)^6) - (3*2^(1/2)*((3*2^(1/2)*((3*c^2*e^3*(b^2*
e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(4*a*c - b^2) - (3*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d +
e*x)^(1/2)*((e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5*c^10*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c
^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^4*d^4*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3
*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 4160*a^3*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 +
1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*e^2 + 1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d
^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^3*e^2 - 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a
^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*c^7*d^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 3
27680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b^9*c^5*d^4*e - 600*a*b^12*c^2*d*e^4 + 131072
0*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2*e^3 - 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^
10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4 - 1638400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6
*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 - 204800*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10
*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/2)*((e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5*c^10*d^5 + 512*b^
10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^4*d^4*e + 81920*a^
2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 4160*a^3*b^9*c^3*e^5
+ 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*e^2 + 1120*b^12*c^
3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^3*e^2 - 44800*a^2*
b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*c^7*d^3*e^2 + 1024
00*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b^9*c^5*d^4*e - 600
*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2*e^3 - 204800*a^2*
b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4 - 1638400*a^4*b^3*
c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 - 204800*a^6*b^2*c^
7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/16 + (9*c^3*e^2*(d + e*x)^(1/2)*(13*b^4*e^4 + 256*
c^4*d^4 + 16*a^2*c^2*e^4 + 64*a*c^3*d^2*e^2 + 368*b^2*c^2*d^2*e^2 + 8*a*b^2*c*e^4 - 512*b*c^3*d^3*e - 112*b^3*
c*d*e^3 - 64*a*b*c^2*d*e^3))/(4*(4*a*c - b^2)^4))*((e^5*(-(4*a*c - b^2)^15)^(1/2) - b^15*e^5 - 524288*a^5*c^10
*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7*c^8*d*e^4 - 1280*b^11*c^4*d^4
*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 560*a^2*b^11*c^2*e^5 - 4160*a^3
*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*e^5 - 655360*a^6*c^9*d^3*e^2 +
1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^4 + 166400*a^2*b^8*c^5*d^3*e^2
- 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d^2*e^3 + 1024000*a^4*b^4*c^7*d
^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^3*c^7*d^2*e^3 + 25600*a*b^9*c^
5*d^4*e - 600*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*e^2 + 7040*a*b^11*c^3*d^2*e^3
- 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e + 28800*a^3*b^8*c^4*d*e^4 - 163
8400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 983040*a^6*b*c^8*d^2*e^3 - 2048
00*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/16)*((9*(e^5*(-(4*a*c - b^2)^15)^(1/2
) - b^15*e^5 - 524288*a^5*c^10*d^5 + 512*b^10*c^5*d^5 - 10240*a*b^8*c^6*d^5 + 81920*a^7*b*c^7*e^5 - 163840*a^7
*c^8*d*e^4 - 1280*b^11*c^4*d^4*e + 81920*a^2*b^6*c^7*d^5 - 327680*a^3*b^4*c^8*d^5 + 655360*a^4*b^2*c^9*d^5 + 5
60*a^2*b^11*c^2*e^5 - 4160*a^3*b^9*c^3*e^5 + 11520*a^4*b^7*c^4*e^5 + 1024*a^5*b^5*c^5*e^5 - 61440*a^6*b^3*c^6*
e^5 - 655360*a^6*c^9*d^3*e^2 + 1120*b^12*c^3*d^3*e^2 - 400*b^13*c^2*d^2*e^3 - 20*a*b^13*c*e^5 + 50*b^14*c*d*e^
4 + 166400*a^2*b^8*c^5*d^3*e^2 - 44800*a^2*b^9*c^4*d^2*e^3 - 614400*a^3*b^6*c^6*d^3*e^2 + 102400*a^3*b^7*c^5*d
^2*e^3 + 1024000*a^4*b^4*c^7*d^3*e^2 + 102400*a^4*b^5*c^6*d^2*e^3 - 327680*a^5*b^2*c^8*d^3*e^2 - 819200*a^5*b^
3*c^7*d^2*e^3 + 25600*a*b^9*c^5*d^4*e - 600*a*b^12*c^2*d*e^4 + 1310720*a^5*b*c^9*d^4*e - 21760*a*b^10*c^4*d^3*
e^2 + 7040*a*b^11*c^3*d^2*e^3 - 204800*a^2*b^7*c^6*d^4*e + 160*a^2*b^10*c^3*d*e^4 + 819200*a^3*b^5*c^7*d^4*e +
28800*a^3*b^8*c^4*d*e^4 - 1638400*a^4*b^3*c^8*d^4*e - 166400*a^4*b^6*c^5*d*e^4 + 358400*a^5*b^4*c^6*d*e^4 + 9
83040*a^6*b*c^8*d^2*e^3 - 204800*a^6*b^2*c^7*d*e^4))/(128*(a*b^20*e^2 + b^20*c*d^2 + 1048576*a^10*c^11*d^2 + 1
048576*a^11*c^10*e^2 - b^21*d*e - 40*a*b^18*c^2*d^2 - 40*a^2*b^18*c*e^2 + 720*a^2*b^16*c^3*d^2 - 7680*a^3*b^14
*c^4*d^2 + 53760*a^4*b^12*c^5*d^2 - 258048*a^5*b^10*c^6*d^2 + 860160*a^6*b^8*c^7*d^2 - 1966080*a^7*b^6*c^8*d^2
+ 2949120*a^8*b^4*c^9*d^2 - 2621440*a^9*b^2*c^10*d^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a
^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4
*c^8*e^2 - 2621440*a^10*b^2*c^9*e^2 - 1048576*a^10*b*c^10*d*e - 720*a^2*b^17*c^2*d*e + 7680*a^3*b^15*c^3*d*e -
53760*a^4*b^13*c^4*d*e + 258048*a^5*b^11*c^5*d*e - 860160*a^6*b^9*c^6*d*e + 1966080*a^7*b^7*c^7*d*e - 2949120
*a^8*b^5*c^8*d*e + 2621440*a^9*b^3*c^9*d*e + 40*a*b^19*c*d*e)))^(1/2)