Integrand size = 26, antiderivative size = 303 \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\frac {2 c d-b e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\sqrt {b^2-4 a c} e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (c d^2-b d e+a e^2\right )^3}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]
1/2*(-b*e+2*c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2+(2*c^2*d^2+b^2*e^2-2*c*e*(a *e+b*d))/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*( 3*a*e+b*d))*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3+1/2*(-b*e+2*c*d)*(c^2*d^2+b^2* e^2-c*e*(3*a*e+b*d))*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^3+e*(3*c^2*d^2+b^ 2*e^2-c*e*(a*e+3*b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^ (1/2)/(a*e^2-b*d*e+c*d^2)^3
Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.88 \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\frac {\frac {(2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {2 \left (c d^2+e (-b d+a e)\right ) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{d+e x}+2 \sqrt {-b^2+4 a c} e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )-2 (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^3} \]
(((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (2*(c*d^2 + e* (-(b*d) + a*e))*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))/(d + e*x) + 2*S qrt[-b^2 + 4*a*c]*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] - 2*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Log[d + e*x] + (2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3* a*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)
Time = 0.73 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1200, 2009}
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.
| \(\displaystyle \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {e (2 c d-b e) \left (c e (3 a e+b d)-b^2 e^2-c^2 d^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {e \left (2 c e (a e+b d)-b^2 e^2-2 c^2 d^2\right )}{(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {c x (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b^2 c e \left (3 c d^2-4 a e^2\right )+b c^2 d \left (c d^2-9 a e^2\right )+2 a c^2 e \left (3 c d^2-a e^2\right )+b^4 \left (-e^3\right )+3 b^3 c d e^2}{\left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}+\frac {e (b e-2 c d)}{(d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {(2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 c d-b e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\) |
(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + (2*c^2*d^2 + b^2*e ^2 - 2*c*e*(b*d + a*e))/((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*ArcTanh[(b + 2*c*x)/S qrt[b^2 - 4*a*c]])/(c*d^2 - b*d*e + a*e^2)^3 - ((2*c*d - b*e)*(c^2*d^2 + b ^2*e^2 - c*e*(b*d + 3*a*e))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 + ((2* c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Log[a + b*x + c*x^2])/( 2*(c*d^2 - b*d*e + a*e^2)^3)
3.16.31.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 0.49 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(-\frac {2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {\left (3 c \,e^{3} b a -6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}-\frac {b e -2 c d}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{2}}+\frac {\frac {\left (3 a b \,c^{2} e^{3}-6 a \,c^{3} d \,e^{2}-b^{3} c \,e^{3}+3 b^{2} c^{2} d \,e^{2}-3 b \,c^{3} d^{2} e +2 c^{4} d^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-9 a b \,c^{2} d \,e^{2}+6 a \,c^{3} d^{2} e -b^{4} e^{3}+3 b^{3} c d \,e^{2}-3 b^{2} c^{2} d^{2} e +b \,d^{3} c^{3}-\frac {\left (3 a b \,c^{2} e^{3}-6 a \,c^{3} d \,e^{2}-b^{3} c \,e^{3}+3 b^{2} c^{2} d \,e^{2}-3 b \,c^{3} d^{2} e +2 c^{4} d^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\) | \(452\) |
| risch | \(\text {Expression too large to display}\) | \(2509\) |
-(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-(3* a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/(a* e^2-b*d*e+c*d^2)^3*ln(e*x+d)-1/2*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2 +1/(a*e^2-b*d*e+c*d^2)^3*(1/2*(3*a*b*c^2*e^3-6*a*c^3*d*e^2-b^3*c*e^3+3*b^2 *c^2*d*e^2-3*b*c^3*d^2*e+2*c^4*d^3)/c*ln(c*x^2+b*x+a)+2*(-2*a^2*c^2*e^3+4* a*b^2*c*e^3-9*a*b*c^2*d*e^2+6*a*c^3*d^2*e-b^4*e^3+3*b^3*c*d*e^2-3*b^2*c^2* d^2*e+b*d^3*c^3-1/2*(3*a*b*c^2*e^3-6*a*c^3*d*e^2-b^3*c*e^3+3*b^2*c^2*d*e^2 -3*b*c^3*d^2*e+2*c^4*d^3)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b ^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (295) = 590\).
Time = 8.86 (sec) , antiderivative size = 1961, normalized size of antiderivative = 6.47 \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
[1/2*(6*c^3*d^5 - 13*b*c^2*d^4*e - a^2*b*e^5 + 2*(5*b^2*c + 2*a*c^2)*d^3*e ^2 - 3*(b^3 + 2*a*b*c)*d^2*e^3 + 2*(2*a*b^2 - a^2*c)*d*e^4 - (3*c^2*d^4*e - 3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (b^ 2 - a*c)*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + (b^2 - a*c)*d*e^4)* x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4 *a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 2*(2*c^3*d^4*e - 4*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4 + (a*b^2 - 2*a^2*c)*e^5)*x + (2*c^3*d^5 - 3*b *c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^ 3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)* e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*log(c*x^2 + b*x + a) - 2*(2*c^3*d^5 - 3*b*c^2* d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3 *e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5)* x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^ 3 - 3*a*b*c)*d*e^4)*x)*log(e*x + d))/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^ 3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e ^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a* b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e ^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + ...
Timed out. \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (295) = 590\).
Time = 0.27 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.48 \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} - \frac {{\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} - \frac {{\left (3 \, b^{2} c^{2} d^{2} e - 12 \, a c^{3} d^{2} e - 3 \, b^{3} c d e^{2} + 12 \, a b c^{2} d e^{2} + b^{4} e^{3} - 5 \, a b^{2} c e^{3} + 4 \, a^{2} c^{2} e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 10 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - 3 \, b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 4 \, a b^{2} d e^{4} - 2 \, a^{2} c d e^{4} - a^{2} b e^{5} + 2 \, {\left (2 \, c^{3} d^{4} e - 4 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4} + a b^{2} e^{5} - 2 \, a^{2} c e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (e x + d\right )}^{2}} \]
1/2*(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*log(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4* e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (2*c^3*d^3*e - 3*b*c^2*d^2*e^ 2 + 3*b^2*c*d*e^3 - 6*a*c^2*d*e^3 - b^3*e^4 + 3*a*b*c*e^4)*log(abs(e*x + d ))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3* d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d* e^6 + a^3*e^7) - (3*b^2*c^2*d^2*e - 12*a*c^3*d^2*e - 3*b^3*c*d*e^2 + 12*a* b*c^2*d*e^2 + b^4*e^3 - 5*a*b^2*c*e^3 + 4*a^2*c^2*e^3)*arctan((2*c*x + b)/ sqrt(-b^2 + 4*a*c))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2* d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^ 4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) + 1/2*(6*c^3*d^5 - 13*b*c ^2*d^4*e + 10*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - 3*b^3*d^2*e^3 - 6*a*b*c*d^ 2*e^3 + 4*a*b^2*d*e^4 - 2*a^2*c*d*e^4 - a^2*b*e^5 + 2*(2*c^3*d^4*e - 4*b*c ^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4 + a*b^2*e^5 - 2*a^2*c*e^5)*x)/((c *d^2 - b*d*e + a*e^2)^3*(e*x + d)^2)
Time = 21.72 (sec) , antiderivative size = 2608, normalized size of antiderivative = 8.61 \[ \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
(log(2*a*e^5*(b^2 - 4*a*c)^(5/2) + 32*a*b^5*e^5 - 192*a*c^5*d^5 + 32*b^6*e ^5*x + 48*b^2*c^4*d^5 + 18*b^3*e^5*x*(b^2 - 4*a*c)^(3/2) + 3*b^5*e^5*x*(b^ 2 - 4*a*c)^(1/2) - 96*c^5*d^5*x*(b^2 - 4*a*c)^(1/2) - 208*a^2*b^3*c*e^5 + 320*a^3*b*c^2*e^5 - 704*a^3*c^3*d*e^4 - 48*b^3*c^3*d^4*e - 16*b^5*c*d^2*e^ 3 - 64*a^3*c^3*e^5*x + 1152*a^2*c^4*d^3*e^2 + 48*b^4*c^2*d^3*e^2 + 33*b*d* e^4*(b^2 - 4*a*c)^(5/2) + 11*b*e^5*x*(b^2 - 4*a*c)^(5/2) + 24*a*b^2*e^5*(b ^2 - 4*a*c)^(3/2) + 6*a*b^4*e^5*(b^2 - 4*a*c)^(1/2) - 48*b*c^4*d^5*(b^2 - 4*a*c)^(1/2) - 18*b^3*d*e^4*(b^2 - 4*a*c)^(3/2) - 15*b^5*d*e^4*(b^2 - 4*a* c)^(1/2) - 44*c*d^2*e^3*(b^2 - 4*a*c)^(5/2) - 72*c^3*d^4*e*(b^2 - 4*a*c)^( 3/2) - 22*c*d*e^4*x*(b^2 - 4*a*c)^(5/2) + 192*a*b*c^4*d^4*e - 128*a*b^4*c* d*e^4 - 120*b^3*c^2*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 224*a*b^4*c*e^5*x - 576* a*c^5*d^4*e*x - 160*b^5*c*d*e^4*x + 144*b^2*c^4*d^4*e*x + 72*b*c^2*d^3*e^2 *(b^2 - 4*a*c)^(3/2) + 120*b^2*c^3*d^4*e*(b^2 - 4*a*c)^(1/2) + 60*b^4*c*d^ 2*e^3*(b^2 - 4*a*c)^(1/2) - 144*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(3/2) - 480*a* b^2*c^3*d^3*e^2 + 320*a*b^3*c^2*d^2*e^3 - 1024*a^2*b*c^3*d^2*e^3 + 688*a^2 *b^2*c^2*d*e^4 + 400*a^2*b^2*c^2*e^5*x + 1408*a^2*c^4*d^2*e^3*x - 288*b^3* c^3*d^3*e^2*x + 304*b^4*c^2*d^2*e^3*x + 216*b*c^2*d^2*e^3*x*(b^2 - 4*a*c)^ (3/2) - 1568*a*b^2*c^3*d^2*e^3*x - 240*b^2*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(1/ 2) + 120*b^3*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(1/2) + 240*b*c^4*d^4*e*x*(b^2 - 4*a*c)^(1/2) - 108*b^2*c*d*e^4*x*(b^2 - 4*a*c)^(3/2) - 30*b^4*c*d*e^4*x...