Integrand size = 21, antiderivative size = 204 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=-\frac {d}{3 a x^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {b^2 d-a c d-a b e}{a^3 x}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)}{a^4}+\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x+c x^2\right )}{2 a^4} \] Output:
-1/3*d/a/x^3+1/2*(-a*e+b*d)/a^2/x^2-(-a*b*e-a*c*d+b^2*d)/a^3/x-(3*a^2*b*c* e+2*a^2*c^2*d-a*b^3*e-4*a*b^2*c*d+b^4*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1 /2))/a^4/(-4*a*c+b^2)^(1/2)-(a^2*c*e-a*b^2*e-2*a*b*c*d+b^3*d)*ln(x)/a^4+1/ 2*(a^2*c*e-a*b^2*e-2*a*b*c*d+b^3*d)*ln(c*x^2+b*x+a)/a^4
Time = 0.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {2 a^3 d}{x^3}+\frac {3 a^2 (b d-a e)}{x^2}+\frac {6 a \left (-b^2 d+a c d+a b e\right )}{x}+\frac {6 \left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-6 \left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)+3 \left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (a+x (b+c x))}{6 a^4} \] Input:
Integrate[(d + e*x)/(x^4*(a + b*x + c*x^2)),x]
Output:
((-2*a^3*d)/x^3 + (3*a^2*(b*d - a*e))/x^2 + (6*a*(-(b^2*d) + a*c*d + a*b*e ))/x + (6*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*ArcT an[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 6*(b^3*d - 2*a*b* c*d - a*b^2*e + a^2*c*e)*Log[x] + 3*(b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e )*Log[a + x*(b + c*x)])/(6*a^4)
Time = 0.49 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {-a b e-a c d+b^2 d}{a^3 x^2}+\frac {a e-b d}{a^2 x^3}+\frac {-a^2 c e+a b^2 e+2 a b c d+b^3 (-d)}{a^4 x}+\frac {c x \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right )+2 a^2 b c e+a^2 c^2 d-a b^3 e-3 a b^2 c d+b^4 d}{a^4 \left (a+b x+c x^2\right )}+\frac {d}{a x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a b e-a c d+b^2 d}{a^3 x}+\frac {b d-a e}{2 a^2 x^2}-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right )}{a^4 \sqrt {b^2-4 a c}}+\frac {\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {\log (x) \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right )}{a^4}-\frac {d}{3 a x^3}\) |
Input:
Int[(d + e*x)/(x^4*(a + b*x + c*x^2)),x]
Output:
-1/3*d/(a*x^3) + (b*d - a*e)/(2*a^2*x^2) - (b^2*d - a*c*d - a*b*e)/(a^3*x) - ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - ((b^3*d - 2*a*b*c* d - a*b^2*e + a^2*c*e)*Log[x])/a^4 + ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c *e)*Log[a + b*x + c*x^2])/(2*a^4)
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 = 1.33 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.19
| method | result | size |
| default | \(-\frac {d}{3 a \,x^{3}}-\frac {a e -b d}{2 a^{2} x^{2}}-\frac {-a b e -a c d +b^{2} d}{a^{3} x}+\frac {\left (-a^{2} c e +e a \,b^{2}+2 a b c d -b^{3} d \right ) \ln \left (x \right )}{a^{4}}+\frac {\frac {\left (a^{2} c^{2} e -a \,b^{2} c e -2 a b \,c^{2} d +b^{3} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (2 a^{2} b c e +a^{2} c^{2} d -a \,b^{3} e -3 c d a \,b^{2}+b^{4} d -\frac {\left (a^{2} c^{2} e -a \,b^{2} c e -2 a b \,c^{2} d +b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a^{4}}\) | \(243\) |
| risch | \(\frac {\frac {\left (a b e +a c d -b^{2} d \right ) x^{2}}{a^{3}}-\frac {\left (a e -b d \right ) x}{2 a^{2}}-\frac {d}{3 a}}{x^{3}}-\frac {\ln \left (x \right ) c e}{a^{2}}+\frac {\ln \left (x \right ) e \,b^{2}}{a^{3}}+\frac {2 b c d \ln \left (x \right )}{a^{3}}-\frac {\ln \left (x \right ) b^{3} d}{a^{4}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{5}-a^{4} b^{2}\right ) \textit {\_Z}^{2}+\left (-4 e \,c^{2} a^{3}+5 a^{2} b^{2} c e +8 a^{2} b \,c^{2} d -a \,b^{4} e -6 a \,b^{3} c d +b^{5} d \right ) \textit {\_Z} +e^{2} a \,c^{3}-c^{3} d e b +c^{4} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6 a^{7} c -2 a^{6} b^{2}\right ) \textit {\_R}^{2}+\left (-3 a^{5} c^{2} e +2 a^{4} b^{2} c e +5 a^{4} b \,c^{2} d -2 a^{3} b^{3} c d \right ) \textit {\_R} +a^{2} b^{2} c^{2} e^{2}+2 a^{2} b \,c^{3} d e +a^{2} c^{4} d^{2}-2 a \,b^{3} c^{2} d e -2 a \,b^{2} c^{3} d^{2}+b^{4} c^{2} d^{2}\right ) x -a^{7} b \,\textit {\_R}^{2}+\left (-2 a^{5} b c e -c^{2} d \,a^{5}+a^{4} b^{3} e +3 a^{4} b^{2} c d -a^{3} b^{4} d \right ) \textit {\_R} -a^{3} b \,c^{2} e^{2}-a^{3} c^{3} d e +a^{2} b^{3} c \,e^{2}+4 a^{2} b^{2} c^{2} d e +2 a^{2} b \,c^{3} d^{2}-2 a \,b^{4} c d e -3 a \,b^{3} c^{2} d^{2}+b^{5} c \,d^{2}\right )\right )\) | \(466\) |
Input:
int((e*x+d)/x^4/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/3*d/a/x^3-1/2*(a*e-b*d)/a^2/x^2-(-a*b*e-a*c*d+b^2*d)/a^3/x+1/a^4*(-a^2* c*e+a*b^2*e+2*a*b*c*d-b^3*d)*ln(x)+1/a^4*(1/2*(a^2*c^2*e-a*b^2*c*e-2*a*b*c ^2*d+b^3*c*d)/c*ln(c*x^2+b*x+a)+2*(2*a^2*b*c*e+a^2*c^2*d-a*b^3*e-3*c*d*a*b ^2+b^4*d-1/2*(a^2*c^2*e-a*b^2*c*e-2*a*b*c^2*d+b^3*c*d)*b/c)/(4*a*c-b^2)^(1 /2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.74 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.37 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)/x^4/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(b^2 - 4*a*c)*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3*a^ 2*b*c)*e)*x^3*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)) + 3*((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d - (a *b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e)*x^3*log(c*x^2 + b*x + a) - 6*((b^5 - 6* a*b^3*c + 8*a^2*b*c^2)*d - (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e)*x^3*log(x) - 6*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d - (a^2*b^3 - 4*a^3*b*c)*e)*x^2 - 2*(a^3*b^2 - 4*a^4*c)*d + 3*((a^2*b^3 - 4*a^3*b*c)*d - (a^3*b^2 - 4*a^4*c )*e)*x)/((a^4*b^2 - 4*a^5*c)*x^3), -1/6*(6*sqrt(-b^2 + 4*a*c)*((b^4 - 4*a* b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3*a^2*b*c)*e)*x^3*arctan(-sqrt(-b^2 + 4*a* c)*(2*c*x + b)/(b^2 - 4*a*c)) - 3*((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d - (a* b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e)*x^3*log(c*x^2 + b*x + a) + 6*((b^5 - 6*a *b^3*c + 8*a^2*b*c^2)*d - (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e)*x^3*log(x) + 6*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d - (a^2*b^3 - 4*a^3*b*c)*e)*x^2 + 2*(a^3*b^2 - 4*a^4*c)*d - 3*((a^2*b^3 - 4*a^3*b*c)*d - (a^3*b^2 - 4*a^4*c) *e)*x)/((a^4*b^2 - 4*a^5*c)*x^3)]
Timed out. \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)/x**4/(c*x**2+b*x+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/x^4/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
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
Time = 0.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=\frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{4}} - \frac {2 \, a^{3} d + 6 \, {\left (a b^{2} d - a^{2} c d - a^{2} b e\right )} x^{2} - 3 \, {\left (a^{2} b d - a^{3} e\right )} x}{6 \, a^{4} x^{3}} \] Input:
integrate((e*x+d)/x^4/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*(b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*log(c*x^2 + b*x + a)/a^4 - (b^ 3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*log(abs(x))/a^4 + (b^4*d - 4*a*b^2*c* d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4* a*c))/(sqrt(-b^2 + 4*a*c)*a^4) - 1/6*(2*a^3*d + 6*(a*b^2*d - a^2*c*d - a^2 *b*e)*x^2 - 3*(a^2*b*d - a^3*e)*x)/(a^4*x^3)
Time = 12.10 (sec) , antiderivative size = 1063, normalized size of antiderivative = 5.21 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
int((d + e*x)/(x^4*(a + b*x + c*x^2)),x)
Output:
(log(2*a^2*b^4*e + 6*a^4*c^2*e - 2*a*b^5*d - 2*b^6*d*x + 2*a*b^5*e*x + 2*a *b^4*d*(b^2 - 4*a*c)^(1/2) + 2*b^5*d*x*(b^2 - 4*a*c)^(1/2) + 11*a^2*b^3*c* d - 13*a^3*b*c^2*d - 9*a^3*b^2*c*e + 2*a^3*c^3*d*x - 2*a^2*b^3*e*(b^2 - 4* a*c)^(1/2) + a^3*c^2*d*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*e*x*(b^2 - 4*a*c)^(1/ 2) - 10*a^2*b^3*c*e*x + 9*a^3*b*c^2*e*x - 5*a^2*b^2*c*d*(b^2 - 4*a*c)^(1/2 ) - 3*a^3*c^2*e*x*(b^2 - 4*a*c)^(1/2) - 17*a^2*b^2*c^2*d*x + 12*a*b^4*c*d* x + 3*a^3*b*c*e*(b^2 - 4*a*c)^(1/2) - 8*a*b^3*c*d*x*(b^2 - 4*a*c)^(1/2) + 7*a^2*b*c^2*d*x*(b^2 - 4*a*c)^(1/2) + 6*a^2*b^2*c*e*x*(b^2 - 4*a*c)^(1/2)) *(b^4*d*(b^2 - 4*a*c)^(1/2) - b^5*d + 4*a^3*c^2*e + a*b^4*e + 6*a*b^3*c*d - a*b^3*e*(b^2 - 4*a*c)^(1/2) - 8*a^2*b*c^2*d - 5*a^2*b^2*c*e + 2*a^2*c^2* d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*c*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^5*c - a^4*b^2)) - (d/(3*a) + (x*(a*e - b*d))/(2* a^2) - (x^2*(a*b*e - b^2*d + a*c*d))/a^3)/x^3 - (log(2*a^2*b^4*e + 6*a^4*c ^2*e - 2*a*b^5*d - 2*b^6*d*x + 2*a*b^5*e*x - 2*a*b^4*d*(b^2 - 4*a*c)^(1/2) - 2*b^5*d*x*(b^2 - 4*a*c)^(1/2) + 11*a^2*b^3*c*d - 13*a^3*b*c^2*d - 9*a^3 *b^2*c*e + 2*a^3*c^3*d*x + 2*a^2*b^3*e*(b^2 - 4*a*c)^(1/2) - a^3*c^2*d*(b^ 2 - 4*a*c)^(1/2) + 2*a*b^4*e*x*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^3*c*e*x + 9* a^3*b*c^2*e*x + 5*a^2*b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^3*c^2*e*x*(b^2 - 4 *a*c)^(1/2) - 17*a^2*b^2*c^2*d*x + 12*a*b^4*c*d*x - 3*a^3*b*c*e*(b^2 - 4*a *c)^(1/2) + 8*a*b^3*c*d*x*(b^2 - 4*a*c)^(1/2) - 7*a^2*b*c^2*d*x*(b^2 - ...
Time = 0.23 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.74 \[ \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx=\frac {12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c^{2} d \,x^{3}-6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} e \,x^{3}-15 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c e \,x^{3}-24 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b \,c^{2} d \,x^{3}+18 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{3} c d \,x^{3}+30 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c e \,x^{3}+48 \,\mathrm {log}\left (x \right ) a^{2} b \,c^{2} d \,x^{3}-36 \,\mathrm {log}\left (x \right ) a \,b^{3} c d \,x^{3}-8 a^{4} c d +2 a^{3} b^{2} d +18 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b c e \,x^{3}-24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c d \,x^{3}+6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4} d \,x^{3}+12 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{3} c^{2} e \,x^{3}+3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{4} e \,x^{3}-24 \,\mathrm {log}\left (x \right ) a^{3} c^{2} e \,x^{3}-6 \,\mathrm {log}\left (x \right ) a \,b^{4} e \,x^{3}+12 a^{3} b c d x +24 a^{3} b c e \,x^{2}-30 a^{2} b^{2} c d \,x^{2}-3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{5} d \,x^{3}+6 \,\mathrm {log}\left (x \right ) b^{5} d \,x^{3}-12 a^{4} c e x +3 a^{3} b^{2} e x +24 a^{3} c^{2} d \,x^{2}-3 a^{2} b^{3} d x -6 a^{2} b^{3} e \,x^{2}+6 a \,b^{4} d \,x^{2}}{6 a^{4} x^{3} \left (4 a c -b^{2}\right )} \] Input:
int((e*x+d)/x^4/(c*x^2+b*x+a),x)
Output:
(18*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c*e*x** 3 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**2*d *x**3 - 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*e *x**3 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2* c*d*x**3 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4* d*x**3 + 12*log(a + b*x + c*x**2)*a**3*c**2*e*x**3 - 15*log(a + b*x + c*x* *2)*a**2*b**2*c*e*x**3 - 24*log(a + b*x + c*x**2)*a**2*b*c**2*d*x**3 + 3*l og(a + b*x + c*x**2)*a*b**4*e*x**3 + 18*log(a + b*x + c*x**2)*a*b**3*c*d*x **3 - 3*log(a + b*x + c*x**2)*b**5*d*x**3 - 24*log(x)*a**3*c**2*e*x**3 + 3 0*log(x)*a**2*b**2*c*e*x**3 + 48*log(x)*a**2*b*c**2*d*x**3 - 6*log(x)*a*b* *4*e*x**3 - 36*log(x)*a*b**3*c*d*x**3 + 6*log(x)*b**5*d*x**3 - 8*a**4*c*d - 12*a**4*c*e*x + 2*a**3*b**2*d + 3*a**3*b**2*e*x + 12*a**3*b*c*d*x + 24*a **3*b*c*e*x**2 + 24*a**3*c**2*d*x**2 - 3*a**2*b**3*d*x - 6*a**2*b**3*e*x** 2 - 30*a**2*b**2*c*d*x**2 + 6*a*b**4*d*x**2)/(6*a**4*x**3*(4*a*c - b**2))