Integrand size = 22, antiderivative size = 310 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
-2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-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))*2^(1/2)*c^(1/2)*(2*c*d-e*(b+(-4 *a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a *c+b^2)^(1/2)))^(1/2)+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))*2^(1/2)*c^(1/2)*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))) /(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^( 1/2)
Time = 0.88 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )} \]
(-2*e)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[- 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e))) + (Sqrt[2]*Sqrt[c]*(2*c *d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sq rt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e)))
Time = 0.55 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1147, 1197, 27, 1480, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1147 |
| \(\displaystyle \frac {\int \frac {c d-b e-c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{a e^2-b d e+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {2 \int \frac {e (2 c d-b e-c (d+e x))}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \int \frac {2 c d-b e-c (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 e \left (\frac {c \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}-\frac {1}{2} c \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{a e^2-b d e+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 e \left (\frac {\sqrt {c} \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a e^2-b d e+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\) |
(-2*e)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (2*e*(-((Sqrt[c]*(2*c*d - (b + 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]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c* d - (b - Sqrt[b^2 - 4*a*c])*e])) + (Sqrt[c]*(1 + (2*c*d - b*e)/(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]])/(Sqrt[2]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(c *d^2 - b*d*e + a*e^2)
3.23.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol ] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Sim p[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x , x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[m, - 1]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[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] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
Time = 0.38 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(2 e \left (\frac {4 c \left (\frac {\left (b e -2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\right )\) | \(297\) |
| default | \(2 e \left (\frac {4 c \left (\frac {\left (b e -2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\right )\) | \(297\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {\sqrt {2}\, c \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{2}+\left (\frac {c \sqrt {2}\, \sqrt {e x +d}\, \left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{2}+\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right ) e}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {e x +d}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a \,e^{2}-b d e +c \,d^{2}\right )}\) | \(365\) |
2*e*(4/(a*e^2-b*d*e+c*d^2)*c*(1/8*(b*e-2*c*d-(-e^2*(4*a*c-b^2))^(1/2))/(-e ^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/ 2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c) ^(1/2))-1/8*(-b*e+2*c*d-(-e^2*(4*a*c-b^2))^(1/2))/(-e^2*(4*a*c-b^2))^(1/2) *2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d) ^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-1/(a*e^2- b*d*e+c*d^2)/(e*x+d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 11293 vs. \(2 (264) = 528\).
Time = 0.73 (sec) , antiderivative size = 11293, normalized size of antiderivative = 36.43 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (264) = 528\).
Time = 0.35 (sec) , antiderivative size = 1488, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
-2*e/((c*d^2 - b*d*e + a*e^2)*sqrt(e*x + d)) - 1/4*((c*d^2*e - b*d*e^2 + a *e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e - 2*(2*sqrt(b^2 - 4*a*c)*c^2*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^2 - sqrt (b^2 - 4*a*c)*a*b*e^4 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^3)*sqrt(-4*c^2 *d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) + (4* c^4*d^6*e - 12*b*c^3*d^5*e^2 + a^2*b^2*e^7 + (13*b^2*c^2 + 8*a*c^3)*d^4*e^ 3 - 2*(3*b^3*c + 8*a*b*c^2)*d^3*e^4 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e ^5 - 2*(a*b^3 + 2*a^2*b*c)*d*e^6)*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*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 + sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2* d*e^2 + 2*a*c*d*e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/( c^2*d^2 - b*c*d*e + a*c*e^2)))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^5*e - 3*sqrt(b^2 - 4*a*c)*a^2*b*d*e^5 + sqrt(b^2 - 4*a*c)*a ^3*e^6 + 3*(b^2*c + a*c^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^3 + 6*a*b*c)*sqr t(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs( c*d^2*e - b*d*e^2 + a*e^3)*abs(c)) + 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sq rt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e + 2*(2*sqrt (b^2 - 4*a*c)*c^2*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^2 - sqrt(b^2 - 4*a *c)*a*b*e^4 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^3)*sqrt(-4*c^2*d + 2*...
Time = 14.60 (sec) , antiderivative size = 23975, normalized size of antiderivative = 77.34 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
atan((((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^ 4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c ^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a* b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3 *c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16* a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^ 6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^ 2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5* c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/ 2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-( 4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^ 2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4* a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-...