Integrand size = 26, antiderivative size = 126 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {3 e^3 x}{c}-\frac {(d+e x)^3}{a+b x+c x^2}-\frac {3 e \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {3 e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2} \] Output:
3*e^3*x/c-(e*x+d)^3/(c*x^2+b*x+a)-3*e*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))* arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(1/2)+3/2*e^2*(-b*e +2*c*d)*ln(c*x^2+b*x+a)/c^2
Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.28 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {4 c e^3 x-\frac {2 \left (b e^3 (a+b x)+c^2 d^2 (d+3 e x)-c e^2 (3 a d+3 b d x+a e x)\right )}{a+x (b+c x)}+\frac {6 e \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-3 e^2 (-2 c d+b e) \log (a+x (b+c x))}{2 c^2} \] Input:
Integrate[((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^2,x]
Output:
(4*c*e^3*x - (2*(b*e^3*(a + b*x) + c^2*d^2*(d + 3*e*x) - c*e^2*(3*a*d + 3* b*d*x + a*e*x)))/(a + x*(b + c*x)) + (6*e*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b* d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*e ^2*(-2*c*d + b*e)*Log[a + x*(b + c*x)])/(2*c^2)
Time = 0.57 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1222 |
\(\displaystyle 3 e \int \frac {(d+e x)^2}{c x^2+b x+a}dx-\frac {(d+e x)^3}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle 3 e \int \left (\frac {e^2}{c}+\frac {c d^2-a e^2+e (2 c d-b e) x}{c \left (c x^2+b x+a\right )}\right )dx-\frac {(d+e x)^3}{a+b x+c x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 e \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {e^2 x}{c}\right )-\frac {(d+e x)^3}{a+b x+c x^2}\) |
Input:
Int[((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^2,x]
Output:
-((d + e*x)^3/(a + b*x + c*x^2)) + 3*e*((e^2*x)/c - ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + (e*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*c^2))
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 = 1.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {2 e^{3} x}{c}-\frac {\frac {-\frac {\left (a c \,e^{2}-b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) e x}{c}+\frac {a \,e^{3} b -3 a d \,e^{2} c +c^{2} d^{3}}{c}}{c \,x^{2}+b x +a}+3 e \left (\frac {\left (b \,e^{2}-2 d e c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a \,e^{2}-c \,d^{2}-\frac {\left (b \,e^{2}-2 d e c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{c}\) | \(187\) |
risch | \(\text {Expression too large to display}\) | \(2059\) |
Input:
int((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
2*e^3/c*x-1/c*((-(a*c*e^2-b^2*e^2+3*b*c*d*e-3*c^2*d^2)*e/c*x+(a*b*e^3-3*a* c*d*e^2+c^2*d^3)/c)/(c*x^2+b*x+a)+3*e*(1/2*(b*e^2-2*c*d*e)/c*ln(c*x^2+b*x+ a)+2*(a*e^2-c*d^2-1/2*(b*e^2-2*c*d*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. 515 vs. \(2 (120) = 240\).
Time = 0.09 (sec) , antiderivative size = 1050, normalized size of antiderivative = 8.33 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[1/2*(4*(b^2*c^2 - 4*a*c^3)*e^3*x^3 + 4*(b^3*c - 4*a*b*c^2)*e^3*x^2 - 2*(b ^2*c^2 - 4*a*c^3)*d^3 + 6*(a*b^2*c - 4*a^2*c^2)*d*e^2 - 2*(a*b^3 - 4*a^2*b *c)*e^3 - 3*(2*a*c^2*d^2*e - 2*a*b*c*d*e^2 + (a*b^2 - 2*a^2*c)*e^3 + (2*c^ 3*d^2*e - 2*b*c^2*d*e^2 + (b^2*c - 2*a*c^2)*e^3)*x^2 + (2*b*c^2*d^2*e - 2* b^2*c*d*e^2 + (b^3 - 2*a*b*c)*e^3)*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*(3*(b^2*c^2 - 4*a*c^3)*d^2*e - 3*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 7*a *b^2*c + 12*a^2*c^2)*e^3)*x + 3*(2*(a*b^2*c - 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^2 - (b^3*c - 4*a*b*c^2)*e^3)*x ^2 + (2*(b^3*c - 4*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c)*e^3)*x)*log(c*x^2 + b*x + a))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4* a*b*c^3)*x), 1/2*(4*(b^2*c^2 - 4*a*c^3)*e^3*x^3 + 4*(b^3*c - 4*a*b*c^2)*e^ 3*x^2 - 2*(b^2*c^2 - 4*a*c^3)*d^3 + 6*(a*b^2*c - 4*a^2*c^2)*d*e^2 - 2*(a*b ^3 - 4*a^2*b*c)*e^3 - 6*(2*a*c^2*d^2*e - 2*a*b*c*d*e^2 + (a*b^2 - 2*a^2*c) *e^3 + (2*c^3*d^2*e - 2*b*c^2*d*e^2 + (b^2*c - 2*a*c^2)*e^3)*x^2 + (2*b*c^ 2*d^2*e - 2*b^2*c*d*e^2 + (b^3 - 2*a*b*c)*e^3)*x)*sqrt(-b^2 + 4*a*c)*arcta n(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(3*(b^2*c^2 - 4*a*c^3 )*d^2*e - 3*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 7*a*b^2*c + 12*a^2*c^2)*e^3 )*x + 3*(2*(a*b^2*c - 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3 + (2*(b^2 *c^2 - 4*a*c^3)*d*e^2 - (b^3*c - 4*a*b*c^2)*e^3)*x^2 + (2*(b^3*c - 4*a*...
Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (121) = 242\).
Time = 4.70 (sec) , antiderivative size = 733, normalized size of antiderivative = 5.82 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a b e^{3} - 4 a c^{2} \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 12 a c d e^{2} + b^{2} c \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) - 3 b c d^{2} e}{6 a c e^{3} - 3 b^{2} e^{3} + 6 b c d e^{2} - 6 c^{2} d^{2} e} \right )} + \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a b e^{3} - 4 a c^{2} \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 12 a c d e^{2} + b^{2} c \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) - 3 b c d^{2} e}{6 a c e^{3} - 3 b^{2} e^{3} + 6 b c d e^{2} - 6 c^{2} d^{2} e} \right )} + \frac {- a b e^{3} + 3 a c d e^{2} - c^{2} d^{3} + x \left (a c e^{3} - b^{2} e^{3} + 3 b c d e^{2} - 3 c^{2} d^{2} e\right )}{a c^{2} + b c^{2} x + c^{3} x^{2}} + \frac {2 e^{3} x}{c} \] Input:
integrate((2*c*x+b)*(e*x+d)**3/(c*x**2+b*x+a)**2,x)
Output:
(-3*e**2*(b*e - 2*c*d)/(2*c**2) - 3*e*sqrt(-4*a*c + b**2)*(2*a*c*e**2 - b* *2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**2*(4*a*c - b**2)))*log(x + (-3*a* b*e**3 - 4*a*c**2*(-3*e**2*(b*e - 2*c*d)/(2*c**2) - 3*e*sqrt(-4*a*c + b**2 )*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**2*(4*a*c - b**2 ))) + 12*a*c*d*e**2 + b**2*c*(-3*e**2*(b*e - 2*c*d)/(2*c**2) - 3*e*sqrt(-4 *a*c + b**2)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**2*(4 *a*c - b**2))) - 3*b*c*d**2*e)/(6*a*c*e**3 - 3*b**2*e**3 + 6*b*c*d*e**2 - 6*c**2*d**2*e)) + (-3*e**2*(b*e - 2*c*d)/(2*c**2) + 3*e*sqrt(-4*a*c + b**2 )*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**2*(4*a*c - b**2 )))*log(x + (-3*a*b*e**3 - 4*a*c**2*(-3*e**2*(b*e - 2*c*d)/(2*c**2) + 3*e* sqrt(-4*a*c + b**2)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2* c**2*(4*a*c - b**2))) + 12*a*c*d*e**2 + b**2*c*(-3*e**2*(b*e - 2*c*d)/(2*c **2) + 3*e*sqrt(-4*a*c + b**2)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c** 2*d**2)/(2*c**2*(4*a*c - b**2))) - 3*b*c*d**2*e)/(6*a*c*e**3 - 3*b**2*e**3 + 6*b*c*d*e**2 - 6*c**2*d**2*e)) + (-a*b*e**3 + 3*a*c*d*e**2 - c**2*d**3 + x*(a*c*e**3 - b**2*e**3 + 3*b*c*d*e**2 - 3*c**2*d**2*e))/(a*c**2 + b*c** 2*x + c**3*x**2) + 2*e**3*x/c
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^2,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 = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.44 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 \, e^{3} x}{c} + \frac {3 \, {\left (2 \, c d e^{2} - b e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {3 \, {\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3} - 2 \, a c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} - \frac {c^{2} d^{3} - 3 \, a c d e^{2} + a b e^{3} + {\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{2}} \] Input:
integrate((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
2*e^3*x/c + 3/2*(2*c*d*e^2 - b*e^3)*log(c*x^2 + b*x + a)/c^2 + 3*(2*c^2*d^ 2*e - 2*b*c*d*e^2 + b^2*e^3 - 2*a*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4* a*c))/(sqrt(-b^2 + 4*a*c)*c^2) - (c^2*d^3 - 3*a*c*d*e^2 + a*b*e^3 + (3*c^2 *d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3)*x)/((c*x^2 + b*x + a)*c^2)
Time = 11.58 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.79 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2\,e^3\,x}{c}-\frac {\frac {c^2\,d^3-3\,a\,c\,d\,e^2+a\,b\,e^3}{c}+\frac {x\,\left (b^2\,e^3-3\,b\,c\,d\,e^2+3\,c^2\,d^2\,e-a\,c\,e^3\right )}{c}}{c^2\,x^2+b\,c\,x+a\,c}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (3\,b^3\,e^3-6\,d\,b^2\,c\,e^2-12\,a\,b\,c\,e^3+24\,a\,d\,c^2\,e^2\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}+\frac {3\,e\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2-2\,a\,c\,e^2\right )}{c^2\,\sqrt {4\,a\,c-b^2}} \] Input:
int(((b + 2*c*x)*(d + e*x)^3)/(a + b*x + c*x^2)^2,x)
Output:
(2*e^3*x)/c - ((c^2*d^3 + a*b*e^3 - 3*a*c*d*e^2)/c + (x*(b^2*e^3 + 3*c^2*d ^2*e - a*c*e^3 - 3*b*c*d*e^2))/c)/(a*c + c^2*x^2 + b*c*x) + (log(a + b*x + c*x^2)*(3*b^3*e^3 - 12*a*b*c*e^3 + 24*a*c^2*d*e^2 - 6*b^2*c*d*e^2))/(2*(4 *a*c^3 - b^2*c^2)) + (3*e*atan((b + 2*c*x)/(4*a*c - b^2)^(1/2))*(b^2*e^2 + 2*c^2*d^2 - 2*a*c*e^2 - 2*b*c*d*e))/(c^2*(4*a*c - b^2)^(1/2))
Time = 0.22 (sec) , antiderivative size = 1048, normalized size of antiderivative = 8.32 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((2*c*x+b)*(e*x+d)^3/(c*x^2+b*x+a)^2,x)
Output:
( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c*e* *3 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*e**3 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c*d*e **2 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c* e**3*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c* *2*d**2*e - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b *c**2*e**3*x**2 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) )*b**4*e**3*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) *b**3*c*d*e**2*x + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 ))*b**3*c*e**3*x**2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**2*d**2*e*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*b**2*c**2*d*e**2*x**2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x )/sqrt(4*a*c - b**2))*b*c**3*d**2*e*x**2 - 12*log(a + b*x + c*x**2)*a**2*b **2*c*e**3 + 24*log(a + b*x + c*x**2)*a**2*b*c**2*d*e**2 + 3*log(a + b*x + c*x**2)*a*b**4*e**3 - 6*log(a + b*x + c*x**2)*a*b**3*c*d*e**2 - 12*log(a + b*x + c*x**2)*a*b**3*c*e**3*x + 24*log(a + b*x + c*x**2)*a*b**2*c**2*d*e **2*x - 12*log(a + b*x + c*x**2)*a*b**2*c**2*e**3*x**2 + 24*log(a + b*x + c*x**2)*a*b*c**3*d*e**2*x**2 + 3*log(a + b*x + c*x**2)*b**5*e**3*x - 6*log (a + b*x + c*x**2)*b**4*c*d*e**2*x + 3*log(a + b*x + c*x**2)*b**4*c*e**3*x **2 - 6*log(a + b*x + c*x**2)*b**3*c**2*d*e**2*x**2 - 24*a**3*c**2*e**3...