Integrand size = 25, antiderivative size = 198 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\frac {e (c e f+2 c d g-b e g) x}{c^2}+\frac {e^2 g x^2}{2 c}-\frac {\left (2 c^3 d^2 f-b^3 e^2 g+b c e (b e f+2 b d g+3 a e g)-c^2 (b d (2 e f+d g)+2 a e (e f+2 d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2 e^2 g+c^2 d (2 e f+d g)-c e (b e f+2 b d g+a e g)\right ) \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:
e*(-b*e*g+2*c*d*g+c*e*f)*x/c^2+1/2*e^2*g*x^2/c-(2*c^3*d^2*f-b^3*e^2*g+b*c* e*(3*a*e*g+2*b*d*g+b*e*f)-c^2*(b*d*(d*g+2*e*f)+2*a*e*(2*d*g+e*f)))*arctanh ((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(1/2)+1/2*(b^2*e^2*g+c^2*d *(d*g+2*e*f)-c*e*(a*e*g+2*b*d*g+b*e*f))*ln(c*x^2+b*x+a)/c^3
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\frac {2 c e (c e f+2 c d g-b e g) x+c^2 e^2 g x^2-\frac {2 \left (-2 c^3 d^2 f+b^3 e^2 g-b c e (b e f+2 b d g+3 a e g)+c^2 (b d (2 e f+d g)+2 a e (e f+2 d g))\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (b^2 e^2 g+c^2 d (2 e f+d g)-c e (b e f+2 b d g+a e g)\right ) \log (a+x (b+c x))}{2 c^3} \] Input:
Integrate[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2),x]
Output:
(2*c*e*(c*e*f + 2*c*d*g - b*e*g)*x + c^2*e^2*g*x^2 - (2*(-2*c^3*d^2*f + b^ 3*e^2*g - b*c*e*(b*e*f + 2*b*d*g + 3*a*e*g) + c^2*(b*d*(2*e*f + d*g) + 2*a *e*(e*f + 2*d*g)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a *c] + (b^2*e^2*g + c^2*d*(2*e*f + d*g) - c*e*(b*e*f + 2*b*d*g + a*e*g))*Lo g[a + x*(b + c*x)])/(2*c^3)
Time = 0.78 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {x \left (-c e (a e g+2 b d g+b e f)+b^2 e^2 g+c^2 d (d g+2 e f)\right )+a b e^2 g-a c e (2 d g+e f)+c^2 d^2 f}{c^2 \left (a+b x+c x^2\right )}+\frac {e (-b e g+2 c d g+c e f)}{c^2}+\frac {e^2 g x}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 a e (2 d g+e f)+b d (d g+2 e f))+b c e (3 a e g+2 b d g+b e f)+b^3 \left (-e^2\right ) g+2 c^3 d^2 f\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right ) \left (-c e (a e g+2 b d g+b e f)+b^2 e^2 g+c^2 d (d g+2 e f)\right )}{2 c^3}+\frac {e x (-b e g+2 c d g+c e f)}{c^2}+\frac {e^2 g x^2}{2 c}\) |
Input:
Int[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2),x]
Output:
(e*(c*e*f + 2*c*d*g - b*e*g)*x)/c^2 + (e^2*g*x^2)/(2*c) - ((2*c^3*d^2*f - b^3*e^2*g + b*c*e*(b*e*f + 2*b*d*g + 3*a*e*g) - c^2*(b*d*(2*e*f + d*g) + 2 *a*e*(e*f + 2*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + ((b^2*e^2*g + c^2*d*(2*e*f + d*g) - c*e*(b*e*f + 2*b*d*g + a* e*g))*Log[a + b*x + c*x^2])/(2*c^3)
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 = 1.77 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {e \left (-\frac {1}{2} c e g \,x^{2}+b e g x -2 c d g x -c e f x \right )}{c^{2}}+\frac {\frac {\left (-a c \,e^{2} g +b^{2} e^{2} g -2 b c d e g -b c \,e^{2} f +c^{2} d^{2} g +2 c^{2} d e f \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a b \,e^{2} g -2 a c d e g -a c \,e^{2} f +c^{2} d^{2} f -\frac {\left (-a c \,e^{2} g +b^{2} e^{2} g -2 b c d e g -b c \,e^{2} f +c^{2} d^{2} g +2 c^{2} d e f \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2}}\) | \(222\) |
| risch | \(\text {Expression too large to display}\) | \(9713\) |
Input:
int((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-e/c^2*(-1/2*c*e*g*x^2+b*e*g*x-2*c*d*g*x-c*e*f*x)+1/c^2*(1/2*(-a*c*e^2*g+b ^2*e^2*g-2*b*c*d*e*g-b*c*e^2*f+c^2*d^2*g+2*c^2*d*e*f)/c*ln(c*x^2+b*x+a)+2* (a*b*e^2*g-2*a*c*d*e*g-a*c*e^2*f+c^2*d^2*f-1/2*(-a*c*e^2*g+b^2*e^2*g-2*b*c *d*e*g-b*c*e^2*f+c^2*d^2*g+2*c^2*d*e*f)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c *x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.14 (sec) , antiderivative size = 696, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/2*((b^2*c^2 - 4*a*c^3)*e^2*g*x^2 + sqrt(b^2 - 4*a*c)*((2*c^3*d^2 - 2*b* c^2*d*e + (b^2*c - 2*a*c^2)*e^2)*f - (b*c^2*d^2 - 2*(b^2*c - 2*a*c^2)*d*e + (b^3 - 3*a*b*c)*e^2)*g)*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*((b^2*c^2 - 4*a*c^3)*e^2*f + (2*(b^2*c^2 - 4*a*c^3)*d*e - (b^3*c - 4*a*b*c^2)*e^2)*g)*x + ((2*(b^2*c^ 2 - 4*a*c^3)*d*e - (b^3*c - 4*a*b*c^2)*e^2)*f + ((b^2*c^2 - 4*a*c^3)*d^2 - 2*(b^3*c - 4*a*b*c^2)*d*e + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^2)*g)*log(c*x ^2 + b*x + a))/(b^2*c^3 - 4*a*c^4), 1/2*((b^2*c^2 - 4*a*c^3)*e^2*g*x^2 - 2 *sqrt(-b^2 + 4*a*c)*((2*c^3*d^2 - 2*b*c^2*d*e + (b^2*c - 2*a*c^2)*e^2)*f - (b*c^2*d^2 - 2*(b^2*c - 2*a*c^2)*d*e + (b^3 - 3*a*b*c)*e^2)*g)*arctan(-sq rt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*((b^2*c^2 - 4*a*c^3)*e^2*f + (2*(b^2*c^2 - 4*a*c^3)*d*e - (b^3*c - 4*a*b*c^2)*e^2)*g)*x + ((2*(b^2*c ^2 - 4*a*c^3)*d*e - (b^3*c - 4*a*b*c^2)*e^2)*f + ((b^2*c^2 - 4*a*c^3)*d^2 - 2*(b^3*c - 4*a*b*c^2)*d*e + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^2)*g)*log(c* x^2 + b*x + a))/(b^2*c^3 - 4*a*c^4)]
Leaf count of result is larger than twice the leaf count of optimal. 1532 vs. \(2 (204) = 408\).
Time = 5.93 (sec) , antiderivative size = 1532, normalized size of antiderivative = 7.74 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**2*(g*x+f)/(c*x**2+b*x+a),x)
Output:
x*(-b*e**2*g/c**2 + 2*d*e*g/c + e**2*f/c) + (-sqrt(-4*a*c + b**2)*(3*a*b*c *e**2*g - 4*a*c**2*d*e*g - 2*a*c**2*e**2*f - b**3*e**2*g + 2*b**2*c*d*e*g + b**2*c*e**2*f - b*c**2*d**2*g - 2*b*c**2*d*e*f + 2*c**3*d**2*f)/(2*c**3* (4*a*c - b**2)) - (a*c*e**2*g - b**2*e**2*g + 2*b*c*d*e*g + b*c*e**2*f - c **2*d**2*g - 2*c**2*d*e*f)/(2*c**3))*log(x + (2*a**2*c*e**2*g - a*b**2*e** 2*g + 2*a*b*c*d*e*g + a*b*c*e**2*f + 4*a*c**3*(-sqrt(-4*a*c + b**2)*(3*a*b *c*e**2*g - 4*a*c**2*d*e*g - 2*a*c**2*e**2*f - b**3*e**2*g + 2*b**2*c*d*e* g + b**2*c*e**2*f - b*c**2*d**2*g - 2*b*c**2*d*e*f + 2*c**3*d**2*f)/(2*c** 3*(4*a*c - b**2)) - (a*c*e**2*g - b**2*e**2*g + 2*b*c*d*e*g + b*c*e**2*f - c**2*d**2*g - 2*c**2*d*e*f)/(2*c**3)) - 2*a*c**2*d**2*g - 4*a*c**2*d*e*f - b**2*c**2*(-sqrt(-4*a*c + b**2)*(3*a*b*c*e**2*g - 4*a*c**2*d*e*g - 2*a*c **2*e**2*f - b**3*e**2*g + 2*b**2*c*d*e*g + b**2*c*e**2*f - b*c**2*d**2*g - 2*b*c**2*d*e*f + 2*c**3*d**2*f)/(2*c**3*(4*a*c - b**2)) - (a*c*e**2*g - b**2*e**2*g + 2*b*c*d*e*g + b*c*e**2*f - c**2*d**2*g - 2*c**2*d*e*f)/(2*c* *3)) + b*c**2*d**2*f)/(3*a*b*c*e**2*g - 4*a*c**2*d*e*g - 2*a*c**2*e**2*f - b**3*e**2*g + 2*b**2*c*d*e*g + b**2*c*e**2*f - b*c**2*d**2*g - 2*b*c**2*d *e*f + 2*c**3*d**2*f)) + (sqrt(-4*a*c + b**2)*(3*a*b*c*e**2*g - 4*a*c**2*d *e*g - 2*a*c**2*e**2*f - b**3*e**2*g + 2*b**2*c*d*e*g + b**2*c*e**2*f - b* c**2*d**2*g - 2*b*c**2*d*e*f + 2*c**3*d**2*f)/(2*c**3*(4*a*c - b**2)) - (a *c*e**2*g - b**2*e**2*g + 2*b*c*d*e*g + b*c*e**2*f - c**2*d**2*g - 2*c*...
Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(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 = 0.20 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\frac {c e^{2} g x^{2} + 2 \, c e^{2} f x + 4 \, c d e g x - 2 \, b e^{2} g x}{2 \, c^{2}} + \frac {{\left (2 \, c^{2} d e f - b c e^{2} f + c^{2} d^{2} g - 2 \, b c d e g + b^{2} e^{2} g - a c e^{2} g\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (2 \, c^{3} d^{2} f - 2 \, b c^{2} d e f + b^{2} c e^{2} f - 2 \, a c^{2} e^{2} f - b c^{2} d^{2} g + 2 \, b^{2} c d e g - 4 \, a c^{2} d e g - b^{3} e^{2} g + 3 \, a b c e^{2} g\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*(c*e^2*g*x^2 + 2*c*e^2*f*x + 4*c*d*e*g*x - 2*b*e^2*g*x)/c^2 + 1/2*(2*c ^2*d*e*f - b*c*e^2*f + c^2*d^2*g - 2*b*c*d*e*g + b^2*e^2*g - a*c*e^2*g)*lo g(c*x^2 + b*x + a)/c^3 + (2*c^3*d^2*f - 2*b*c^2*d*e*f + b^2*c*e^2*f - 2*a* c^2*e^2*f - b*c^2*d^2*g + 2*b^2*c*d*e*g - 4*a*c^2*d*e*g - b^3*e^2*g + 3*a* b*c*e^2*g)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)
Time = 12.25 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=x\,\left (\frac {f\,e^2+2\,d\,g\,e}{c}-\frac {b\,e^2\,g}{c^2}\right )-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,g\,a^2\,c^2\,e^2-5\,g\,a\,b^2\,c\,e^2+8\,g\,a\,b\,c^2\,d\,e+4\,f\,a\,b\,c^2\,e^2-4\,g\,a\,c^3\,d^2-8\,f\,a\,c^3\,d\,e+g\,b^4\,e^2-2\,g\,b^3\,c\,d\,e-f\,b^3\,c\,e^2+g\,b^2\,c^2\,d^2+2\,f\,b^2\,c^2\,d\,e\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}+\frac {e^2\,g\,x^2}{2\,c}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (g\,b^3\,e^2-2\,g\,b^2\,c\,d\,e-f\,b^2\,c\,e^2+g\,b\,c^2\,d^2+2\,f\,b\,c^2\,d\,e-3\,a\,g\,b\,c\,e^2-2\,f\,c^3\,d^2+4\,a\,g\,c^2\,d\,e+2\,a\,f\,c^2\,e^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \] Input:
int(((f + g*x)*(d + e*x)^2)/(a + b*x + c*x^2),x)
Output:
x*((e^2*f + 2*d*e*g)/c - (b*e^2*g)/c^2) - (log(a + b*x + c*x^2)*(b^4*e^2*g + 4*a^2*c^2*e^2*g + b^2*c^2*d^2*g - 4*a*c^3*d^2*g - b^3*c*e^2*f + 4*a*b*c ^2*e^2*f - 5*a*b^2*c*e^2*g + 2*b^2*c^2*d*e*f - 8*a*c^3*d*e*f - 2*b^3*c*d*e *g + 8*a*b*c^2*d*e*g))/(2*(4*a*c^4 - b^2*c^3)) + (e^2*g*x^2)/(2*c) - (atan (b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(b^3*e^2*g - 2*c^3*d ^2*f + 2*a*c^2*e^2*f + b*c^2*d^2*g - b^2*c*e^2*f - 3*a*b*c*e^2*g + 4*a*c^2 *d*e*g + 2*b*c^2*d*e*f - 2*b^2*c*d*e*g))/(c^3*(4*a*c - b^2)^(1/2))
Time = 0.28 (sec) , antiderivative size = 721, normalized size of antiderivative = 3.64 \[ \int \frac {(d+e x)^2 (f+g x)}{a+b x+c x^2} \, dx=\frac {-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} e^{2} g +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) c^{3} d^{2} f -4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} c^{2} e^{2} g +4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,c^{3} d^{2} g +8 a \,c^{3} e^{2} f x +4 a \,c^{3} e^{2} g \,x^{2}+2 b^{3} c \,e^{2} g x -2 b^{2} c^{2} e^{2} f x -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} e^{2} g +\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3} c \,e^{2} f -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{2} c^{2} d^{2} g -b^{2} c^{2} e^{2} g \,x^{2}+6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c \,e^{2} g -8 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} d e g +4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c d e g -4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} d e f -8 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b \,c^{2} d e g -4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} e^{2} f +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,e^{2} f -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} d^{2} g +5 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c \,e^{2} g -4 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a b \,c^{2} e^{2} f +8 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,c^{3} d e f +2 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{3} c d e g -2 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{2} c^{2} d e f -8 a b \,c^{2} e^{2} g x +16 a \,c^{3} d e g x -4 b^{2} c^{2} d e g x}{2 c^{3} \left (4 a c -b^{2}\right )} \] Input:
int((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a),x)
Output:
(6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*e**2*g - 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*d*e*g - 4 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*e**2*f - 2 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e**2*g + 4*s qrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*d*e*g + 2*sq rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*e**2*f - 2*sq rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d**2*g - 4*sq rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d*e*f + 4*sqr t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**3*d**2*f - 4*log(a + b*x + c*x**2)*a**2*c**2*e**2*g + 5*log(a + b*x + c*x**2)*a*b**2*c*e**2* g - 8*log(a + b*x + c*x**2)*a*b*c**2*d*e*g - 4*log(a + b*x + c*x**2)*a*b*c **2*e**2*f + 4*log(a + b*x + c*x**2)*a*c**3*d**2*g + 8*log(a + b*x + c*x** 2)*a*c**3*d*e*f - log(a + b*x + c*x**2)*b**4*e**2*g + 2*log(a + b*x + c*x* *2)*b**3*c*d*e*g + log(a + b*x + c*x**2)*b**3*c*e**2*f - log(a + b*x + c*x **2)*b**2*c**2*d**2*g - 2*log(a + b*x + c*x**2)*b**2*c**2*d*e*f - 8*a*b*c* *2*e**2*g*x + 16*a*c**3*d*e*g*x + 8*a*c**3*e**2*f*x + 4*a*c**3*e**2*g*x**2 + 2*b**3*c*e**2*g*x - 4*b**2*c**2*d*e*g*x - 2*b**2*c**2*e**2*f*x - b**2*c **2*e**2*g*x**2)/(2*c**3*(4*a*c - b**2))