Integrand size = 27, antiderivative size = 644 \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=-\frac {b^3 e g-b^2 c (e f+d g)+2 a c^2 (e f+d g)+b c (c d f-3 a e g)+c \left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right ) \left (a+b x+c x^2\right )}+\frac {2 c \left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac {\left (b^2 e^2 g^2 (b e f+b d g-2 a e g)-2 c^3 d f \left (e^2 f^2+d e f g+d^2 g^2\right )+2 c e g \left (a^2 e^2 g^2+a b e g (e f+d g)-b^2 (e f+d g)^2\right )-c^2 \left (4 a d e^2 f g^2-b \left (e^3 f^3+5 d e^2 f^2 g+5 d^2 e f g^2+d^3 g^3\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {e^4 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2 (e f-d g)}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac {(c e f+c d g-b e g) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2 \left (c f^2-g (b f-a g)\right )^2} \]
(-b^3*e*g+b^2*c*(d*g+e*f)-2*a*c^2*(d*g+e*f)-b*c*(-3*a*e*g+c*d*f)-c*(2*c^2* d*f+b^2*e*g-c*(2*a*e*g+b*d*g+b*e*f))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/( c*f^2-g*(-a*g+b*f))/(c*x^2+b*x+a)+2*c*(2*c^2*d*f+b^2*e*g-c*(2*a*e*g+b*d*g+ b*e*f))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^2-b* d*e+c*d^2)/(c*f^2-g*(-a*g+b*f))+e^4*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^2/(-d*g+ e*f)-g^4*ln(g*x+f)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^2-1/2*(-b*e*g+c*d*g+c*e* f)*(c*(d^2*g^2+e^2*f^2)+e*g*(2*a*e*g-b*(d*g+e*f)))*ln(c*x^2+b*x+a)/(a*e^2- b*d*e+c*d^2)^2/(c*f^2-g*(-a*g+b*f))^2+(b^2*e^2*g^2*(-2*a*e*g+b*d*g+b*e*f)- 2*c^3*d*f*(d^2*g^2+d*e*f*g+e^2*f^2)+2*c*e*g*(a^2*e^2*g^2+a*b*e*g*(d*g+e*f) -b^2*(d*g+e*f)^2)-c^2*(4*a*d*e^2*f*g^2-b*(d^3*g^3+5*d^2*e*f*g^2+5*d*e^2*f^ 2*g+e^3*f^3)))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^2 /(c*f^2-g*(-a*g+b*f))^2/(-4*a*c+b^2)^(1/2)
Time = 1.43 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\frac {-b^3 e g+b^2 c (d g+e (f-g x))-2 c^2 (a d g+c d f x+a e (f-g x))+b c (3 a e g+c (-d f+e f x+d g x))}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (-c f^2+g (b f-a g)\right ) (a+x (b+c x))}+\frac {\left (4 c^5 d^3 f^3+b^4 e^2 g^2 (b e f+b d g-2 a e g)-2 b^2 c e g \left (-6 a^2 e^2 g^2+2 a b e g (e f+d g)+b^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right )+2 c^4 d f \left (-3 b d f (e f+d g)+2 a \left (3 e^2 f^2+d e f g+3 d^2 g^2\right )\right )+c^2 \left (-12 a^3 e^3 g^3-6 a^2 b e^2 g^2 (e f+d g)+12 a b^2 e g \left (e^2 f^2+d e f g+d^2 g^2\right )+b^3 \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )\right )-2 c^3 \left (-4 b^2 d^2 e f^2 g+2 a^2 e g \left (e^2 f^2-5 d e f g+d^2 g^2\right )+a b \left (3 e^3 f^3+11 d e^2 f^2 g+11 d^2 e f g^2+3 d^3 g^3\right )\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )^2 \left (c f^2+g (-b f+a g)\right )^2}+\frac {e^4 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^2 (e f-d g)}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (c f^2+g (-b f+a g)\right )^2}-\frac {(c e f+c d g-b e g) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2 \left (c f^2+g (-b f+a g)\right )^2} \]
(-(b^3*e*g) + b^2*c*(d*g + e*(f - g*x)) - 2*c^2*(a*d*g + c*d*f*x + a*e*(f - g*x)) + b*c*(3*a*e*g + c*(-(d*f) + e*f*x + d*g*x)))/((b^2 - 4*a*c)*(-(c* d^2) + e*(b*d - a*e))*(-(c*f^2) + g*(b*f - a*g))*(a + x*(b + c*x))) + ((4* c^5*d^3*f^3 + b^4*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*b^2*c*e*g*(-6*a^2* e^2*g^2 + 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) + 2*c ^4*d*f*(-3*b*d*f*(e*f + d*g) + 2*a*(3*e^2*f^2 + d*e*f*g + 3*d^2*g^2)) + c^ 2*(-12*a^3*e^3*g^3 - 6*a^2*b*e^2*g^2*(e*f + d*g) + 12*a*b^2*e*g*(e^2*f^2 + d*e*f*g + d^2*g^2) + b^3*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)) - 2*c^3*(-4*b^2*d^2*e*f^2*g + 2*a^2*e*g*(e^2*f^2 - 5*d*e*f*g + d^2*g^2) + a*b*(3*e^3*f^3 + 11*d*e^2*f^2*g + 11*d^2*e*f*g^2 + 3*d^3*g^3)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2) + (e^4*Log[d + e*x])/((c*d^2 + e*(-( b*d) + a*e))^2*(e*f - d*g)) - (g^4*Log[f + g*x])/((e*f - d*g)*(c*f^2 + g*( -(b*f) + a*g))^2) - ((c*e*f + c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g* (2*a*e*g - b*(e*f + d*g)))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a *e))^2*(c*f^2 + g*(-(b*f) + a*g))^2)
Time = 1.72 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1289, 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.
| \(\displaystyle \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (\frac {-c e g \left (a^2 e^2 g^2+2 a b e g (d g+e f)-\left (b^2 \left (2 d^2 g^2+3 d e f g+2 e^2 f^2\right )\right )\right )-b^2 e^2 g^2 (-2 a e g+b d g+b e f)+c^2 \left (2 a d e^2 f g^2-b (d g+e f)^3\right )-c x (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )+c^3 d f \left (d^2 g^2+d e f g+e^2 f^2\right )}{\left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2 \left (a g^2-b f g+c f^2\right )^2}+\frac {-c (a e g+b d g+b e f)+b^2 e g-c x (-b e g+c d g+c e f)+c^2 d f}{\left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right ) \left (a g^2-b f g+c f^2\right )}-\frac {e^5}{(d+e x) (d g-e f) \left (a e^2-b d e+c d^2\right )^2}-\frac {g^5}{(f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c e g \left (a^2 e^2 g^2+a b e g (d g+e f)-b^2 (d g+e f)^2\right )+b^2 e^2 g^2 (-2 a e g+b d g+b e f)-c^2 \left (4 a d e^2 f g^2-b \left (d^3 g^3+5 d^2 e f g^2+5 d e^2 f^2 g+e^3 f^3\right )\right )-2 c^3 d f \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {2 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac {c x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )+b c (c d f-3 a e g)+2 a c^2 (d g+e f)+b^3 e g-b^2 c (d g+e f)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac {\log \left (a+b x+c x^2\right ) (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {e^4 \log (d+e x)}{(e f-d g) \left (a e^2-b d e+c d^2\right )^2}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (a g^2-b f g+c f^2\right )^2}\) |
-((b^3*e*g - b^2*c*(e*f + d*g) + 2*a*c^2*(e*f + d*g) + b*c*(c*d*f - 3*a*e* g) + c*(2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/((b^2 - 4*a* c)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))*(a + b*x + c*x^2))) + ( 2*c*(2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*ArcTanh[(b + 2*c*x )/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))) + ((b^2*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*c^3*d*f*(e ^2*f^2 + d*e*f*g + d^2*g^2) + 2*c*e*g*(a^2*e^2*g^2 + a*b*e*g*(e*f + d*g) - b^2*(e*f + d*g)^2) - c^2*(4*a*d*e^2*f*g^2 - b*(e^3*f^3 + 5*d*e^2*f^2*g + 5*d^2*e*f*g^2 + d^3*g^3)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b ^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*(c*f^2 - g*(b*f - a*g))^2) + (e^4*Lo g[d + e*x])/((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)) - (g^4*Log[f + g*x])/( (e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2) - ((c*e*f + c*d*g - b*e*g)*(c*(e^2* f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*Log[a + b*x + c*x^2])/(2*( c*d^2 - b*d*e + a*e^2)^2*(c*f^2 - g*(b*f - a*g))^2)
3.9.18.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(2227\) vs. \(2(635)=1270\).
Time = 1.49 (sec) , antiderivative size = 2228, normalized size of antiderivative = 3.46
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2228\) |
| risch | \(\text {Expression too large to display}\) | \(29824\) |
-e^4/(d*g-e*f)/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)-1/(a*e^2-b*d*e+c*d^2)^2/(a* g^2-b*f*g+c*f^2)^2*((c*(2*a^3*c*e^3*g^3-a^2*b^2*e^3*g^3-a^2*b*c*d*e^2*g^3- a^2*b*c*e^3*f*g^2+2*a^2*c^2*d^2*e*g^3-2*a^2*c^2*d*e^2*f*g^2+2*a^2*c^2*e^3* f^2*g+a*b^3*d*e^2*g^3+a*b^3*e^3*f*g^2-2*a*b^2*c*d^2*e*g^3-2*a*b^2*c*e^3*f^ 2*g+a*b*c^2*d^3*g^3+a*b*c^2*d^2*e*f*g^2+a*b*c^2*d*e^2*f^2*g+a*b*c^2*e^3*f^ 3-2*a*c^3*d^3*f*g^2+2*a*c^3*d^2*e*f^2*g-2*a*c^3*d*e^2*f^3-b^4*d*e^2*f*g^2+ 2*b^3*c*d^2*e*f*g^2+2*b^3*c*d*e^2*f^2*g-b^2*c^2*d^3*f*g^2-5*b^2*c^2*d^2*e* f^2*g-b^2*c^2*d*e^2*f^3+3*b*c^3*d^3*f^2*g+3*b*c^3*d^2*e*f^3-2*c^4*d^3*f^3) /(4*a*c-b^2)*x+(3*a^3*b*c*e^3*g^3-2*a^3*c^2*d*e^2*g^3-2*a^3*c^2*e^3*f*g^2- a^2*b^3*e^3*g^3-2*a^2*b^2*c*d*e^2*g^3-2*a^2*b^2*c*e^3*f*g^2+5*a^2*b*c^2*d^ 2*e*g^3+3*a^2*b*c^2*d*e^2*f*g^2+5*a^2*b*c^2*e^3*f^2*g-2*a^2*c^3*d^3*g^3-2* a^2*c^3*d^2*e*f*g^2-2*a^2*c^3*d*e^2*f^2*g-2*a^2*c^3*e^3*f^3+a*b^4*d*e^2*g^ 3+a*b^4*e^3*f*g^2-2*a*b^3*c*d^2*e*g^3+a*b^3*c*d*e^2*f*g^2-2*a*b^3*c*e^3*f^ 2*g+a*b^2*c^2*d^3*g^3-3*a*b^2*c^2*d^2*e*f*g^2-3*a*b^2*c^2*d*e^2*f^2*g+a*b^ 2*c^2*e^3*f^3+a*b*c^3*d^3*f*g^2+7*a*b*c^3*d^2*e*f^2*g+a*b*c^3*d*e^2*f^3-2* a*c^4*d^3*f^2*g-2*a*c^4*d^2*e*f^3-b^5*d*e^2*f*g^2+2*b^4*c*d^2*e*f*g^2+2*b^ 4*c*d*e^2*f^2*g-b^3*c^2*d^3*f*g^2-4*b^3*c^2*d^2*e*f^2*g-b^3*c^2*d*e^2*f^3+ 2*b^2*c^3*d^3*f^2*g+2*b^2*c^3*d^2*e*f^3-b*c^4*d^3*f^3)/(4*a*c-b^2))/(c*x^2 +b*x+a)+1/(4*a*c-b^2)*(1/2*(-8*a^2*b*c^2*e^3*g^3+8*a^2*c^3*d*e^2*g^3+8*a^2 *c^3*e^3*f*g^2+2*a*b^3*c*e^3*g^3+2*a*b^2*c^2*d*e^2*g^3+2*a*b^2*c^2*e^3*...
Timed out. \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 3400 vs. \(2 (634) = 1268\).
Time = 0.30 (sec) , antiderivative size = 3400, normalized size of antiderivative = 5.28 \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
e^5*log(abs(e*x + d))/(c^2*d^4*e^2*f - 2*b*c*d^3*e^3*f + b^2*d^2*e^4*f + 2 *a*c*d^2*e^4*f - 2*a*b*d*e^5*f + a^2*e^6*f - c^2*d^5*e*g + 2*b*c*d^4*e^2*g - b^2*d^3*e^3*g - 2*a*c*d^3*e^3*g + 2*a*b*d^2*e^4*g - a^2*d*e^5*g) - g^5* log(abs(g*x + f))/(c^2*e*f^5*g - c^2*d*f^4*g^2 - 2*b*c*e*f^4*g^2 + 2*b*c*d *f^3*g^3 + b^2*e*f^3*g^3 + 2*a*c*e*f^3*g^3 - b^2*d*f^2*g^4 - 2*a*c*d*f^2*g ^4 - 2*a*b*e*f^2*g^4 + 2*a*b*d*f*g^5 + a^2*e*f*g^5 - a^2*d*g^6) - 1/2*(c^2 *e^3*f^3 + c^2*d*e^2*f^2*g - 2*b*c*e^3*f^2*g + c^2*d^2*e*f*g^2 - 2*b*c*d*e ^2*f*g^2 + b^2*e^3*f*g^2 + 2*a*c*e^3*f*g^2 + c^2*d^3*g^3 - 2*b*c*d^2*e*g^3 + b^2*d*e^2*g^3 + 2*a*c*d*e^2*g^3 - 2*a*b*e^3*g^3)*log(c*x^2 + b*x + a)/( c^4*d^4*f^4 - 2*b*c^3*d^3*e*f^4 + b^2*c^2*d^2*e^2*f^4 + 2*a*c^3*d^2*e^2*f^ 4 - 2*a*b*c^2*d*e^3*f^4 + a^2*c^2*e^4*f^4 - 2*b*c^3*d^4*f^3*g + 4*b^2*c^2* d^3*e*f^3*g - 2*b^3*c*d^2*e^2*f^3*g - 4*a*b*c^2*d^2*e^2*f^3*g + 4*a*b^2*c* d*e^3*f^3*g - 2*a^2*b*c*e^4*f^3*g + b^2*c^2*d^4*f^2*g^2 + 2*a*c^3*d^4*f^2* g^2 - 2*b^3*c*d^3*e*f^2*g^2 - 4*a*b*c^2*d^3*e*f^2*g^2 + b^4*d^2*e^2*f^2*g^ 2 + 4*a*b^2*c*d^2*e^2*f^2*g^2 + 4*a^2*c^2*d^2*e^2*f^2*g^2 - 2*a*b^3*d*e^3* f^2*g^2 - 4*a^2*b*c*d*e^3*f^2*g^2 + a^2*b^2*e^4*f^2*g^2 + 2*a^3*c*e^4*f^2* g^2 - 2*a*b*c^2*d^4*f*g^3 + 4*a*b^2*c*d^3*e*f*g^3 - 2*a*b^3*d^2*e^2*f*g^3 - 4*a^2*b*c*d^2*e^2*f*g^3 + 4*a^2*b^2*d*e^3*f*g^3 - 2*a^3*b*e^4*f*g^3 + a^ 2*c^2*d^4*g^4 - 2*a^2*b*c*d^3*e*g^4 + a^2*b^2*d^2*e^2*g^4 + 2*a^3*c*d^2*e^ 2*g^4 - 2*a^3*b*d*e^3*g^4 + a^4*e^4*g^4) - (4*c^5*d^3*f^3 - 6*b*c^4*d^2...
Time = 54.58 (sec) , antiderivative size = 130035, normalized size of antiderivative = 201.92 \[ \int \frac {1}{(d+e x) (f+g x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((b^3*e*g + 2*a*c^2*d*g + 2*a*c^2*e*f + b*c^2*d*f - b^2*c*d*g - b^2*c*e*f - 3*a*b*c*e*g)/(4*a*c^3*d^2*f^2 + 4*a^3*c*e^2*g^2 - a^2*b^2*e^2*g^2 + 4*a^ 2*c^2*d^2*g^2 + 4*a^2*c^2*e^2*f^2 - b^2*c^2*d^2*f^2 + a*b^3*d*e*g^2 + b^3* c*d*e*f^2 + a*b^3*e^2*f*g + b^3*c*d^2*f*g - a*b^2*c*d^2*g^2 - a*b^2*c*e^2* f^2 - b^4*d*e*f*g - 4*a*b*c^2*d*e*f^2 - 4*a^2*b*c*d*e*g^2 - 4*a*b*c^2*d^2* f*g - 4*a^2*b*c*e^2*f*g + 4*a*b^2*c*d*e*f*g) - (x*(2*a*c^2*e*g - 2*c^3*d*f + b*c^2*d*g + b*c^2*e*f - b^2*c*e*g))/(4*a*c^3*d^2*f^2 + 4*a^3*c*e^2*g^2 - a^2*b^2*e^2*g^2 + 4*a^2*c^2*d^2*g^2 + 4*a^2*c^2*e^2*f^2 - b^2*c^2*d^2*f^ 2 + a*b^3*d*e*g^2 + b^3*c*d*e*f^2 + a*b^3*e^2*f*g + b^3*c*d^2*f*g - a*b^2* c*d^2*g^2 - a*b^2*c*e^2*f^2 - b^4*d*e*f*g - 4*a*b*c^2*d*e*f^2 - 4*a^2*b*c* d*e*g^2 - 4*a*b*c^2*d^2*f*g - 4*a^2*b*c*e^2*f*g + 4*a*b^2*c*d*e*f*g))/(a + b*x + c*x^2) + symsum(log((12*a^2*c^5*e^6*g^6 - 3*b^2*c^5*d^2*e^4*g^6 - 3 *b^2*c^5*e^6*f^2*g^4 + 4*c^7*d^2*e^4*f^2*g^4 - 2*a*b^2*c^4*e^6*g^6 + 16*a* c^6*d^2*e^4*g^6 + 3*b^3*c^4*d*e^5*g^6 + 16*a*c^6*e^6*f^2*g^4 + 3*b^3*c^4*e ^6*f*g^5 - 4*b*c^6*d*e^5*f^2*g^4 - 4*b*c^6*d^2*e^4*f*g^5 - 16*a*b*c^5*d*e^ 5*g^6 - 16*a*b*c^5*e^6*f*g^5 + 16*a*c^6*d*e^5*f*g^5)/(16*a^2*c^6*d^4*f^4 + a^4*b^4*e^4*g^4 + 16*a^4*c^4*d^4*g^4 + 16*a^4*c^4*e^4*f^4 + b^4*c^4*d^4*f ^4 + 16*a^6*c^2*e^4*g^4 + a^2*b^4*c^2*d^4*g^4 + a^2*b^4*c^2*e^4*f^4 - 8*a^ 3*b^2*c^3*d^4*g^4 - 8*a^3*b^2*c^3*e^4*f^4 + a^2*b^6*d^2*e^2*g^4 + 32*a^3*c ^5*d^2*e^2*f^4 + 32*a^5*c^3*d^2*e^2*g^4 + b^6*c^2*d^2*e^2*f^4 + a^2*b^6...