Integrand size = 26, antiderivative size = 172 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {4 e^3 (3 c d-b e) x}{c^2}+\frac {2 e^4 x^2}{c}-\frac {(d+e x)^4}{a+b x+c x^2}-\frac {4 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{c^3} \]
4*e^3*(-b*e+3*c*d)*x/c^2+2*e^4*x^2/c-(e*x+d)^4/(c*x^2+b*x+a)+2*e^2*(3*c^2* d^2+b^2*e^2-c*e*(a*e+3*b*d))*ln(c*x^2+b*x+a)/c^3-4*e*(-b*e+2*c*d)*(c^2*d^2 +b^2*e^2-c*e*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a* c+b^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.40 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {c e^3 (8 c d-3 b e) x+c^2 e^4 x^2+\frac {b^2 e^4 (a+b x)-c^3 d^3 (d+4 e x)+2 c^2 d e^2 (3 a d+3 b d x+2 a e x)-c e^3 \left (a^2 e+4 b^2 d x+2 a b (2 d+e x)\right )}{a+x (b+c x)}+\frac {4 e (-2 c d+b e) \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log (a+x (b+c x))}{c^3} \]
(c*e^3*(8*c*d - 3*b*e)*x + c^2*e^4*x^2 + (b^2*e^4*(a + b*x) - c^3*d^3*(d + 4*e*x) + 2*c^2*d*e^2*(3*a*d + 3*b*d*x + 2*a*e*x) - c*e^3*(a^2*e + 4*b^2*d *x + 2*a*b*(2*d + e*x)))/(a + x*(b + c*x)) + (4*e*(-2*c*d + b*e)*(-(c^2*d^ 2) - b^2*e^2 + c*e*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/ Sqrt[-b^2 + 4*a*c] + 2*e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + x*(b + c*x)])/c^3
Time = 0.42 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1222, 1143, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle 4 e \int \frac {(d+e x)^3}{c x^2+b x+a}dx-\frac {(d+e x)^4}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle 4 e \int \left (\frac {x e^3}{c}+\frac {(3 c d-b e) e^2}{c^2}+\frac {c^2 d^3-3 a c e^2 d+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{c^2 \left (c x^2+b x+a\right )}\right )dx-\frac {(d+e x)^4}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 e \left (-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {e^3 x^2}{2 c}\right )-\frac {(d+e x)^4}{a+b x+c x^2}\) |
-((d + e*x)^4/(a + b*x + c*x^2)) + 4*e*((e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x ^2)/(2*c) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*ArcTanh [(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^3))
3.16.32.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
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]
Time = 0.92 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(-\frac {e^{3} \left (-c e \,x^{2}+3 b e x -8 c d x \right )}{c^{2}}+\frac {\frac {-\frac {\left (2 c \,e^{3} b a -4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+4 b^{2} c d \,e^{2}-6 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) e x}{c}-\frac {c \,e^{4} a^{2}-a \,b^{2} e^{4}+4 a b c d \,e^{3}-6 a \,c^{2} d^{2} e^{2}+c^{3} d^{4}}{c}}{c \,x^{2}+b x +a}+4 e \left (\frac {\left (-c \,e^{3} a +b^{2} e^{3}-3 b c d \,e^{2}+3 c^{2} d^{2} e \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a b \,e^{3}-3 a c d \,e^{2}+c^{2} d^{3}-\frac {\left (-c \,e^{3} a +b^{2} e^{3}-3 b c d \,e^{2}+3 c^{2} d^{2} e \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{c^{2}}\) | \(306\) |
| risch | \(\text {Expression too large to display}\) | \(4244\) |
-e^3/c^2*(-c*e*x^2+3*b*e*x-8*c*d*x)+1/c^2*((-(2*a*b*c*e^3-4*a*c^2*d*e^2-b^ 3*e^3+4*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*d^3)*e/c*x-(a^2*c*e^4-a*b^2*e^4+4* a*b*c*d*e^3-6*a*c^2*d^2*e^2+c^3*d^4)/c)/(c*x^2+b*x+a)+4*e*(1/2*(-a*c*e^3+b ^2*e^3-3*b*c*d*e^2+3*c^2*d^2*e)/c*ln(c*x^2+b*x+a)+2*(a*b*e^3-3*a*c*d*e^2+c ^2*d^3-1/2*(-a*c*e^3+b^2*e^3-3*b*c*d*e^2+3*c^2*d^2*e)*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. 840 vs. \(2 (168) = 336\).
Time = 0.31 (sec) , antiderivative size = 1701, normalized size of antiderivative = 9.89 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[((b^2*c^3 - 4*a*c^4)*e^4*x^4 - (b^2*c^3 - 4*a*c^4)*d^4 + 6*(a*b^2*c^2 - 4 *a^2*c^3)*d^2*e^2 - 4*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^4 + 2*(4*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^3) *e^4)*x^3 + (8*(b^3*c^2 - 4*a*b*c^3)*d*e^3 - (3*b^4*c - 13*a*b^2*c^2 + 4*a ^2*c^3)*e^4)*x^2 + 2*(2*a*c^3*d^3*e - 3*a*b*c^2*d^2*e^2 + 3*(a*b^2*c - 2*a ^2*c^2)*d*e^3 - (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*(b^2*c^2 - 2*a*c^3)*d*e^3 - (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (2*b*c^3*d^3 *e - 3*b^2*c^2*d^2*e^2 + 3*(b^3*c - 2*a*b*c^2)*d*e^3 - (b^4 - 3*a*b^2*c)*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)) - (4*(b^2*c^3 - 4*a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 4*(b^4*c - 7*a*b^2*c^2 + 12*a^2*c^3)*d* e^3 - (b^5 - 9*a*b^3*c + 20*a^2*b*c^2)*e^4)*x + 2*(3*(a*b^2*c^2 - 4*a^2*c^ 3)*d^2*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^ 3*c^2)*e^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^2 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^ 3 + (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e^4)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)* d^2*e^2 - 3*(b^4*c - 4*a*b^2*c^2)*d*e^3 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)* e^4)*x)*log(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5) *x^2 + (b^3*c^3 - 4*a*b*c^4)*x), ((b^2*c^3 - 4*a*c^4)*e^4*x^4 - (b^2*c^3 - 4*a*c^4)*d^4 + 6*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2 - 4*(a*b^3*c - 4*a^2*b*c ^2)*d*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^4 + 2*(4*(b^2*c^3 - 4*a...
Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (165) = 330\).
Time = 18.03 (sec) , antiderivative size = 1071, normalized size of antiderivative = 6.23 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=x \left (- \frac {3 b e^{4}}{c^{2}} + \frac {8 d e^{3}}{c}\right ) + \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {8 a^{2} c e^{4} - 4 a b^{2} e^{4} + 12 a b c d e^{3} + 4 a c^{3} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) - 24 a c^{2} d^{2} e^{2} - b^{2} c^{2} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) + 4 b c^{2} d^{3} e}{12 a b c e^{4} - 24 a c^{2} d e^{3} - 4 b^{3} e^{4} + 12 b^{2} c d e^{3} - 12 b c^{2} d^{2} e^{2} + 8 c^{3} d^{3} e} \right )} + \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {8 a^{2} c e^{4} - 4 a b^{2} e^{4} + 12 a b c d e^{3} + 4 a c^{3} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) - 24 a c^{2} d^{2} e^{2} - b^{2} c^{2} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \cdot \left (4 a c - b^{2}\right )}\right ) + 4 b c^{2} d^{3} e}{12 a b c e^{4} - 24 a c^{2} d e^{3} - 4 b^{3} e^{4} + 12 b^{2} c d e^{3} - 12 b c^{2} d^{2} e^{2} + 8 c^{3} d^{3} e} \right )} + \frac {- a^{2} c e^{4} + a b^{2} e^{4} - 4 a b c d e^{3} + 6 a c^{2} d^{2} e^{2} - c^{3} d^{4} + x \left (- 2 a b c e^{4} + 4 a c^{2} d e^{3} + b^{3} e^{4} - 4 b^{2} c d e^{3} + 6 b c^{2} d^{2} e^{2} - 4 c^{3} d^{3} e\right )}{a c^{3} + b c^{3} x + c^{4} x^{2}} + \frac {e^{4} x^{2}}{c} \]
x*(-3*b*e**4/c**2 + 8*d*e**3/c) + (-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d *e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2)))*log(x + (8*a**2 *c*e**4 - 4*a*b**2*e**4 + 12*a*b*c*d*e**3 + 4*a*c**3*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2 ))) - 24*a*c**2*d**2*e**2 - b**2*c**2*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b *c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c* e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) + 4*b*c**2* d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 4*b**3*e**4 + 12*b**2*c*d*e**3 - 12*b*c**2*d**2*e**2 + 8*c**3*d**3*e)) + (-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3 *a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2)))*log(x + (8*a**2*c*e**4 - 4*a*b**2*e**4 + 12*a*b*c*d*e**3 + 4*a*c**3*(-2*e**2*(a* c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b** 2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a *c - b**2))) - 24*a*c**2*d**2*e**2 - b**2*c**2*(-2*e**2*(a*c*e**2 - b**2*e **2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d )*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) + 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 4*b**3*e**4 + 12*b...
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.77 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} + \frac {4 \, {\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} + \frac {c^{3} e^{4} x^{2} + 8 \, c^{3} d e^{3} x - 3 \, b c^{2} e^{4} x}{c^{4}} - \frac {c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2} + 4 \, a b c d e^{3} - a b^{2} e^{4} + a^{2} c e^{4} + {\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{3}} \]
2*(3*c^2*d^2*e^2 - 3*b*c*d*e^3 + b^2*e^4 - a*c*e^4)*log(c*x^2 + b*x + a)/c ^3 + 4*(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)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4 *a*c)*c^3) + (c^3*e^4*x^2 + 8*c^3*d*e^3*x - 3*b*c^2*e^4*x)/c^4 - (c^3*d^4 - 6*a*c^2*d^2*e^2 + 4*a*b*c*d*e^3 - a*b^2*e^4 + a^2*c*e^4 + (4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*b^2*c*d*e^3 - 4*a*c^2*d*e^3 - b^3*e^4 + 2*a*b*c*e^4)* x)/((c*x^2 + b*x + a)*c^3)
Time = 0.34 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.08 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx=x\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{c^2}-\frac {4\,b\,e^4}{c^2}\right )-\frac {\frac {a^2\,c\,e^4-a\,b^2\,e^4+4\,a\,b\,c\,d\,e^3-6\,a\,c^2\,d^2\,e^2+c^3\,d^4}{c}-\frac {x\,\left (b^3\,e^4-4\,b^2\,c\,d\,e^3+6\,b\,c^2\,d^2\,e^2-2\,a\,b\,c\,e^4-4\,c^3\,d^3\,e+4\,a\,c^2\,d\,e^3\right )}{c}}{c^3\,x^2+b\,c^2\,x+a\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (16\,a^2\,c^2\,e^4-20\,a\,b^2\,c\,e^4+48\,a\,b\,c^2\,d\,e^3-48\,a\,c^3\,d^2\,e^2+4\,b^4\,e^4-12\,b^3\,c\,d\,e^3+12\,b^2\,c^2\,d^2\,e^2\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}+\frac {e^4\,x^2}{c}-\frac {4\,e\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-b\,c\,d\,e+c^2\,d^2-3\,a\,c\,e^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \]
x*((b*e^4 + 8*c*d*e^3)/c^2 - (4*b*e^4)/c^2) - ((c^3*d^4 - a*b^2*e^4 + a^2* c*e^4 - 6*a*c^2*d^2*e^2 + 4*a*b*c*d*e^3)/c - (x*(b^3*e^4 - 4*c^3*d^3*e + 6 *b*c^2*d^2*e^2 - 2*a*b*c*e^4 + 4*a*c^2*d*e^3 - 4*b^2*c*d*e^3))/c)/(a*c^2 + c^3*x^2 + b*c^2*x) - (log(a + b*x + c*x^2)*(4*b^4*e^4 + 16*a^2*c^2*e^4 - 48*a*c^3*d^2*e^2 + 12*b^2*c^2*d^2*e^2 - 20*a*b^2*c*e^4 - 12*b^3*c*d*e^3 + 48*a*b*c^2*d*e^3))/(2*(4*a*c^4 - b^2*c^3)) + (e^4*x^2)/c - (4*e*atan((b + 2*c*x)/(4*a*c - b^2)^(1/2))*(b*e - 2*c*d)*(b^2*e^2 + c^2*d^2 - 3*a*c*e^2 - b*c*d*e))/(c^3*(4*a*c - b^2)^(1/2))