Integrand size = 28, antiderivative size = 223 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{a+b x+c x^2}-\frac {\sqrt {2} \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-(e*x+d)^(1/2)/(c*x^2+b*x+a)-e*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)/(-4*a*c+b^2)^(1/2)/(2*c *d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+e*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)/(-4*a*c+b^2)^(1/2 )/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{a+x (b+c x)}-\frac {i \sqrt {2} \sqrt {c} e \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {i \sqrt {2} \sqrt {c} e \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}} \]
-(Sqrt[d + e*x]/(a + x*(b + c*x))) - (I*Sqrt[2]*Sqrt[c]*e*ArcTan[(Sqrt[2]* Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt [-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (I*Sqrt[2]*S qrt[c]*e*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt [-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e])
Time = 0.36 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1222, 1149, 1406, 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.
| \(\displaystyle \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {1}{2} e \int \frac {1}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx-\frac {\sqrt {d+e x}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1149 |
| \(\displaystyle e^2 \int \frac {1}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}-\frac {\sqrt {d+e x}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle e^2 \left (\frac {c \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{e \sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{e \sqrt {b^2-4 a c}}\right )-\frac {\sqrt {d+e x}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle e^2 \left (\frac {\sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )-\frac {\sqrt {d+e x}}{a+b x+c x^2}\) |
-(Sqrt[d + e*x]/(a + b*x + c*x^2)) + e^2*(-((Sqrt[2]*Sqrt[c]*ArcTanh[(Sqrt [2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt [b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) + (Sqrt[2]*Sqrt[ c]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4* a*c])*e]])/(Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))
3.17.20.3.1 Defintions of rubi rules used
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[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Sym bol] :> Simp[2*e Subst[Int[1/(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}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Time = 0.38 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {e x +d}}{c \,x^{2}+b x +a}-\frac {e^{2} c \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {e^{2} c \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\) | \(219\) |
| derivativedivides | \(2 e^{2} \left (-\frac {\sqrt {e x +d}}{2 \left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )}+2 c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\) | \(252\) |
| default | \(2 e^{2} \left (-\frac {\sqrt {e x +d}}{2 \left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )}+2 c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\) | \(252\) |
-(e*x+d)^(1/2)/(c*x^2+b*x+a)-e^2*c/(-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))-e^2*c/(-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))
Leaf count of result is larger than twice the leaf count of optimal. 2750 vs. \(2 (181) = 362\).
Time = 0.42 (sec) , antiderivative size = 2750, normalized size of antiderivative = 12.33 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/2*(sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/((b^2 *c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a ^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))* ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b ^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2* sqrt(e*x + d)*c*e^4 + sqrt(1/2)*((b^2 - 4*a*c)*e^4 - sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2) *d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*(2*(b^2 *c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a ^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3))*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e ^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2 *c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c )*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e ^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2) )) - sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/((b^2* c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^ 2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*( (b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^ 2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*s qrt(e*x + d)*c*e^4 - sqrt(1/2)*((b^2 - 4*a*c)*e^4 - sqrt(e^6/((b^2*c^2 ...
Timed out. \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (181) = 362\).
Time = 0.50 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.14 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {e x + d} e^{2}}{{\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e + a e^{2}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} e^{2} {\left | e \right |} - {\left (2 \, c d e^{2} - b e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c d^{2} - \sqrt {b^{2} - 4 \, a c} b d e + \sqrt {b^{2} - 4 \, a c} a e^{2}\right )} {\left | c \right |} {\left | e \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} e^{2} {\left | e \right |} + {\left (2 \, c d e^{2} - b e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c d^{2} - \sqrt {b^{2} - 4 \, a c} b d e + \sqrt {b^{2} - 4 \, a c} a e^{2}\right )} {\left | c \right |} {\left | e \right |}} \]
-sqrt(e*x + d)*e^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) + 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) *sqrt(b^2 - 4*a*c)*e^2*abs(e) - (2*c*d*e^2 - b*e^3)*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*d - b*e + sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((sqrt(b^ 2 - 4*a*c)*c*d^2 - sqrt(b^2 - 4*a*c)*b*d*e + sqrt(b^2 - 4*a*c)*a*e^2)*abs( c)*abs(e)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^ 2 - 4*a*c)*e^2*abs(e) + (2*c*d*e^2 - b*e^3)*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*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((sqrt(b^2 - 4*a* c)*c*d^2 - sqrt(b^2 - 4*a*c)*b*d*e + sqrt(b^2 - 4*a*c)*a*e^2)*abs(c)*abs(e ))
Time = 11.41 (sec) , antiderivative size = 4814, normalized size of antiderivative = 21.59 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
atan((((4*b^2*c^2*e^4 - 16*a*c^3*e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(- (4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*( a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^ 2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8 *b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))*(-(b^3 *e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2* c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5 *d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d* e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3 )^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4 *c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a ^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i + ((16*a*c^3*e ^4 - 4*b^2*c^2*e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^( 1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c* d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2* b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16* b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))*(-(b^3*e^3 + e^3*(-(4*a* c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4 *e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2 *d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*...