Integrand size = 42, antiderivative size = 143 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {(c e f+c d g-b e g)^2}{c^2 g^3 (2 c f-b g) (c f-b g-c g x)}+\frac {(e f-d g)^2 \log (f+g x)}{g^3 (2 c f-b g)^2}+\frac {(3 c e f-c d g-b e g) (c e f+c d g-b e g) \log (c f-b g-c g x)}{c^2 g^3 (2 c f-b g)^2} \] Output:
(-b*e*g+c*d*g+c*e*f)^2/c^2/g^3/(-b*g+2*c*f)/(-c*g*x-b*g+c*f)+(-d*g+e*f)^2* ln(g*x+f)/g^3/(-b*g+2*c*f)^2+(-b*e*g-c*d*g+3*c*e*f)*(-b*e*g+c*d*g+c*e*f)*l n(-c*g*x-b*g+c*f)/c^2/g^3/(-b*g+2*c*f)^2
Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {\frac {(c e f+c d g-b e g)^2}{c^2 (2 c f-b g) (-b g+c (f-g x))}+\frac {(e f-d g)^2 \log (f+g x)}{(-2 c f+b g)^2}+\frac {\left (-4 b c e^2 f g+b^2 e^2 g^2+c^2 \left (3 e^2 f^2+2 d e f g-d^2 g^2\right )\right ) \log (c f-b g-c g x)}{c^2 (-2 c f+b g)^2}}{g^3} \] Input:
Integrate[((d + e*x)^2*(f + g*x))/(c*f^2 - b*f*g - b*g^2*x - c*g^2*x^2)^2, x]
Output:
((c*e*f + c*d*g - b*e*g)^2/(c^2*(2*c*f - b*g)*(-(b*g) + c*(f - g*x))) + (( e*f - d*g)^2*Log[f + g*x])/(-2*c*f + b*g)^2 + ((-4*b*c*e^2*f*g + b^2*e^2*g ^2 + c^2*(3*e^2*f^2 + 2*d*e*f*g - d^2*g^2))*Log[c*f - b*g - c*g*x])/(c^2*( -2*c*f + b*g)^2))/g^3
Time = 0.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1207, 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.
| \(\displaystyle \int \frac {(d+e x)^2 (f+g x)}{\left (-b f g-b g^2 x+c f^2-c g^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1207 |
| \(\displaystyle c^2 g^4 \int \left (\frac {(e f-d g)^2}{c^2 g^6 (2 c f-b g)^2 (f+g x)}-\frac {(3 c e f-c d g-b e g) (c e f+c d g-b e g)}{c^3 g^6 (2 c f-b g)^2 (c f-b g-c g x)}+\frac {(c e f+c d g-b e g)^2}{c^3 g^6 (2 c f-b g) (c f-b g-c g x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^2 g^4 \left (\frac {(-b e g+c d g+c e f)^2}{c^4 g^7 (2 c f-b g) (-b g+c f-c g x)}+\frac {(-b e g-c d g+3 c e f) (-b e g+c d g+c e f) \log (-b g+c f-c g x)}{c^4 g^7 (2 c f-b g)^2}+\frac {(e f-d g)^2 \log (f+g x)}{c^2 g^7 (2 c f-b g)^2}\right )\) |
Input:
Int[((d + e*x)^2*(f + g*x))/(c*f^2 - b*f*g - b*g^2*x - c*g^2*x^2)^2,x]
Output:
c^2*g^4*((c*e*f + c*d*g - b*e*g)^2/(c^4*g^7*(2*c*f - b*g)*(c*f - b*g - c*g *x)) + ((e*f - d*g)^2*Log[f + g*x])/(c^2*g^7*(2*c*f - b*g)^2) + ((3*c*e*f - c*d*g - b*e*g)*(c*e*f + c*d*g - b*e*g)*Log[c*f - b*g - c*g*x])/(c^4*g^7* (2*c*f - b*g)^2))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1 /c^p Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[p, -1] && IntegersQ[m, n] && NiceSqrtQ[b^2 - 4* a*c]
Time = 1.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(\frac {\left (b^{2} e^{2} g^{2}-4 b c \,e^{2} f g -c^{2} d^{2} g^{2}+2 c^{2} d e f g +3 c^{2} e^{2} f^{2}\right ) \ln \left (x g c +b g -c f \right )}{g^{3} \left (b g -2 c f \right )^{2} c^{2}}-\frac {-b^{2} e^{2} g^{2}+2 b c d e \,g^{2}+2 b c \,e^{2} f g -c^{2} d^{2} g^{2}-2 c^{2} d e f g -c^{2} e^{2} f^{2}}{c^{2} g^{3} \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \ln \left (g x +f \right )}{g^{3} \left (b g -2 c f \right )^{2}}\) | \(217\) |
| norman | \(\frac {\frac {\left (b^{2} e^{2} g^{2}-2 b c d e \,g^{2}-2 b c \,e^{2} f g +c^{2} d^{2} g^{2}+2 c^{2} d e f g +c^{2} e^{2} f^{2}\right ) x}{c^{2} g^{2} \left (b g -2 c f \right )}+\frac {\left (b^{2} e^{2} g^{2}-2 b c d e \,g^{2}-2 b c \,e^{2} f g +c^{2} d^{2} g^{2}+2 c^{2} d e f g +c^{2} e^{2} f^{2}\right ) f}{c^{2} g^{3} \left (b g -2 c f \right )}}{\left (g x +f \right ) \left (x g c +b g -c f \right )}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \ln \left (g x +f \right )}{g^{3} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}+\frac {\left (b^{2} e^{2} g^{2}-4 b c \,e^{2} f g -c^{2} d^{2} g^{2}+2 c^{2} d e f g +3 c^{2} e^{2} f^{2}\right ) \ln \left (x g c +b g -c f \right )}{c^{2} g^{3} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}\) | \(327\) |
| risch | \(\frac {b^{2} e^{2}}{c^{2} g \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}-\frac {2 b d e}{c g \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}-\frac {2 b \,e^{2} f}{c \,g^{2} \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}+\frac {d^{2}}{g \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}+\frac {2 d e f}{g^{2} \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}+\frac {e^{2} f^{2}}{g^{3} \left (b g -2 c f \right ) \left (x g c +b g -c f \right )}+\frac {\ln \left (g x +f \right ) d^{2}}{g \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}-\frac {2 \ln \left (g x +f \right ) d e f}{g^{2} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}+\frac {\ln \left (g x +f \right ) e^{2} f^{2}}{g^{3} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}+\frac {\ln \left (-x g c -b g +c f \right ) b^{2} e^{2}}{c^{2} g \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}-\frac {4 \ln \left (-x g c -b g +c f \right ) b \,e^{2} f}{c \,g^{2} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}-\frac {\ln \left (-x g c -b g +c f \right ) d^{2}}{g \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}+\frac {2 \ln \left (-x g c -b g +c f \right ) d e f}{g^{2} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}+\frac {3 \ln \left (-x g c -b g +c f \right ) e^{2} f^{2}}{g^{3} \left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right )}\) | \(566\) |
| parallelrisch | \(\frac {-2 c^{3} e^{2} f^{3}-2 b^{2} d e \,g^{3} c -4 b^{2} e^{2} f \,g^{2} c +\ln \left (x g c +b g -c f \right ) b^{3} e^{2} g^{3}-3 \ln \left (x g c +b g -c f \right ) c^{3} e^{2} f^{3}-\ln \left (g x +f \right ) c^{3} e^{2} f^{3}-2 \ln \left (g x +f \right ) x \,c^{3} d e f \,g^{2}-4 \ln \left (x g c +b g -c f \right ) x b \,c^{2} e^{2} f \,g^{2}+2 \ln \left (x g c +b g -c f \right ) x \,c^{3} d e f \,g^{2}-2 \ln \left (g x +f \right ) b \,c^{2} d e f \,g^{2}+2 \ln \left (x g c +b g -c f \right ) b \,c^{2} d e f \,g^{2}+6 b \,c^{2} d e f \,g^{2}+5 b \,c^{2} e^{2} f^{2} g -4 c^{3} d e \,f^{2} g +b \,c^{2} d^{2} g^{3}-2 c^{3} d^{2} f \,g^{2}+\ln \left (g x +f \right ) x \,c^{3} d^{2} g^{3}-\ln \left (x g c +b g -c f \right ) x \,c^{3} d^{2} g^{3}+\ln \left (g x +f \right ) b \,c^{2} d^{2} g^{3}-\ln \left (g x +f \right ) c^{3} d^{2} f \,g^{2}-\ln \left (x g c +b g -c f \right ) b \,c^{2} d^{2} g^{3}+\ln \left (x g c +b g -c f \right ) c^{3} d^{2} f \,g^{2}+b^{3} e^{2} g^{3}+2 \ln \left (g x +f \right ) c^{3} d e \,f^{2} g -5 \ln \left (x g c +b g -c f \right ) b^{2} c \,e^{2} f \,g^{2}+7 \ln \left (x g c +b g -c f \right ) b \,c^{2} e^{2} f^{2} g -2 \ln \left (x g c +b g -c f \right ) c^{3} d e \,f^{2} g +\ln \left (g x +f \right ) x \,c^{3} e^{2} f^{2} g +\ln \left (x g c +b g -c f \right ) x \,b^{2} c \,e^{2} g^{3}+3 \ln \left (x g c +b g -c f \right ) x \,c^{3} e^{2} f^{2} g +\ln \left (g x +f \right ) b \,c^{2} e^{2} f^{2} g}{\left (b^{2} g^{2}-4 b c f g +4 c^{2} f^{2}\right ) \left (x g c +b g -c f \right ) c^{2} g^{3}}\) | \(634\) |
Input:
int((e*x+d)^2*(g*x+f)/(-c*g^2*x^2-b*g^2*x-b*f*g+c*f^2)^2,x,method=_RETURNV ERBOSE)
Output:
(b^2*e^2*g^2-4*b*c*e^2*f*g-c^2*d^2*g^2+2*c^2*d*e*f*g+3*c^2*e^2*f^2)/g^3/(b *g-2*c*f)^2/c^2*ln(c*g*x+b*g-c*f)-(-b^2*e^2*g^2+2*b*c*d*e*g^2+2*b*c*e^2*f* g-c^2*d^2*g^2-2*c^2*d*e*f*g-c^2*e^2*f^2)/c^2/g^3/(b*g-2*c*f)/(c*g*x+b*g-c* f)+(d^2*g^2-2*d*e*f*g+e^2*f^2)/g^3/(b*g-2*c*f)^2*ln(g*x+f)
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (143) = 286\).
Time = 0.10 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.13 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {2 \, c^{3} e^{2} f^{3} + {\left (4 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f^{2} g + 2 \, {\left (c^{3} d^{2} - 3 \, b c^{2} d e + 2 \, b^{2} c e^{2}\right )} f g^{2} - {\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} g^{3} + {\left (3 \, c^{3} e^{2} f^{3} + {\left (2 \, c^{3} d e - 7 \, b c^{2} e^{2}\right )} f^{2} g - {\left (c^{3} d^{2} + 2 \, b c^{2} d e - 5 \, b^{2} c e^{2}\right )} f g^{2} + {\left (b c^{2} d^{2} - b^{3} e^{2}\right )} g^{3} - {\left (3 \, c^{3} e^{2} f^{2} g + 2 \, {\left (c^{3} d e - 2 \, b c^{2} e^{2}\right )} f g^{2} - {\left (c^{3} d^{2} - b^{2} c e^{2}\right )} g^{3}\right )} x\right )} \log \left (c g x - c f + b g\right ) + {\left (c^{3} e^{2} f^{3} - b c^{2} d^{2} g^{3} - {\left (2 \, c^{3} d e + b c^{2} e^{2}\right )} f^{2} g + {\left (c^{3} d^{2} + 2 \, b c^{2} d e\right )} f g^{2} - {\left (c^{3} e^{2} f^{2} g - 2 \, c^{3} d e f g^{2} + c^{3} d^{2} g^{3}\right )} x\right )} \log \left (g x + f\right )}{4 \, c^{5} f^{3} g^{3} - 8 \, b c^{4} f^{2} g^{4} + 5 \, b^{2} c^{3} f g^{5} - b^{3} c^{2} g^{6} - {\left (4 \, c^{5} f^{2} g^{4} - 4 \, b c^{4} f g^{5} + b^{2} c^{3} g^{6}\right )} x} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*g^2*x^2-b*g^2*x-b*f*g+c*f^2)^2,x, algorith m="fricas")
Output:
(2*c^3*e^2*f^3 + (4*c^3*d*e - 5*b*c^2*e^2)*f^2*g + 2*(c^3*d^2 - 3*b*c^2*d* e + 2*b^2*c*e^2)*f*g^2 - (b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*g^3 + (3*c^3* e^2*f^3 + (2*c^3*d*e - 7*b*c^2*e^2)*f^2*g - (c^3*d^2 + 2*b*c^2*d*e - 5*b^2 *c*e^2)*f*g^2 + (b*c^2*d^2 - b^3*e^2)*g^3 - (3*c^3*e^2*f^2*g + 2*(c^3*d*e - 2*b*c^2*e^2)*f*g^2 - (c^3*d^2 - b^2*c*e^2)*g^3)*x)*log(c*g*x - c*f + b*g ) + (c^3*e^2*f^3 - b*c^2*d^2*g^3 - (2*c^3*d*e + b*c^2*e^2)*f^2*g + (c^3*d^ 2 + 2*b*c^2*d*e)*f*g^2 - (c^3*e^2*f^2*g - 2*c^3*d*e*f*g^2 + c^3*d^2*g^3)*x )*log(g*x + f))/(4*c^5*f^3*g^3 - 8*b*c^4*f^2*g^4 + 5*b^2*c^3*f*g^5 - b^3*c ^2*g^6 - (4*c^5*f^2*g^4 - 4*b*c^4*f*g^5 + b^2*c^3*g^6)*x)
Timed out. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**2*(g*x+f)/(-c*g**2*x**2-b*g**2*x-b*f*g+c*f**2)**2,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.87 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {{\left (3 \, c^{2} e^{2} f^{2} + 2 \, {\left (c^{2} d e - 2 \, b c e^{2}\right )} f g - {\left (c^{2} d^{2} - b^{2} e^{2}\right )} g^{2}\right )} \log \left (c g x - c f + b g\right )}{4 \, c^{4} f^{2} g^{3} - 4 \, b c^{3} f g^{4} + b^{2} c^{2} g^{5}} + \frac {{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (g x + f\right )}{4 \, c^{2} f^{2} g^{3} - 4 \, b c f g^{4} + b^{2} g^{5}} + \frac {c^{2} e^{2} f^{2} + 2 \, {\left (c^{2} d e - b c e^{2}\right )} f g + {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} g^{2}}{2 \, c^{4} f^{2} g^{3} - 3 \, b c^{3} f g^{4} + b^{2} c^{2} g^{5} - {\left (2 \, c^{4} f g^{4} - b c^{3} g^{5}\right )} x} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*g^2*x^2-b*g^2*x-b*f*g+c*f^2)^2,x, algorith m="maxima")
Output:
(3*c^2*e^2*f^2 + 2*(c^2*d*e - 2*b*c*e^2)*f*g - (c^2*d^2 - b^2*e^2)*g^2)*lo g(c*g*x - c*f + b*g)/(4*c^4*f^2*g^3 - 4*b*c^3*f*g^4 + b^2*c^2*g^5) + (e^2* f^2 - 2*d*e*f*g + d^2*g^2)*log(g*x + f)/(4*c^2*f^2*g^3 - 4*b*c*f*g^4 + b^2 *g^5) + (c^2*e^2*f^2 + 2*(c^2*d*e - b*c*e^2)*f*g + (c^2*d^2 - 2*b*c*d*e + b^2*e^2)*g^2)/(2*c^4*f^2*g^3 - 3*b*c^3*f*g^4 + b^2*c^2*g^5 - (2*c^4*f*g^4 - b*c^3*g^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (143) = 286\).
Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {{\left (3 \, c^{2} e^{2} f^{2} + 2 \, c^{2} d e f g - 4 \, b c e^{2} f g - c^{2} d^{2} g^{2} + b^{2} e^{2} g^{2}\right )} \log \left ({\left | c g x - c f + b g \right |}\right )}{4 \, c^{4} f^{2} g^{3} - 4 \, b c^{3} f g^{4} + b^{2} c^{2} g^{5}} + \frac {{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | g x + f \right |}\right )}{4 \, c^{2} f^{2} g^{3} - 4 \, b c f g^{4} + b^{2} g^{5}} - \frac {2 \, c^{3} e^{2} f^{3} + 4 \, c^{3} d e f^{2} g - 5 \, b c^{2} e^{2} f^{2} g + 2 \, c^{3} d^{2} f g^{2} - 6 \, b c^{2} d e f g^{2} + 4 \, b^{2} c e^{2} f g^{2} - b c^{2} d^{2} g^{3} + 2 \, b^{2} c d e g^{3} - b^{3} e^{2} g^{3}}{{\left (c g x - c f + b g\right )} {\left (2 \, c f - b g\right )}^{2} c^{2} g^{3}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(-c*g^2*x^2-b*g^2*x-b*f*g+c*f^2)^2,x, algorith m="giac")
Output:
(3*c^2*e^2*f^2 + 2*c^2*d*e*f*g - 4*b*c*e^2*f*g - c^2*d^2*g^2 + b^2*e^2*g^2 )*log(abs(c*g*x - c*f + b*g))/(4*c^4*f^2*g^3 - 4*b*c^3*f*g^4 + b^2*c^2*g^5 ) + (e^2*f^2 - 2*d*e*f*g + d^2*g^2)*log(abs(g*x + f))/(4*c^2*f^2*g^3 - 4*b *c*f*g^4 + b^2*g^5) - (2*c^3*e^2*f^3 + 4*c^3*d*e*f^2*g - 5*b*c^2*e^2*f^2*g + 2*c^3*d^2*f*g^2 - 6*b*c^2*d*e*f*g^2 + 4*b^2*c*e^2*f*g^2 - b*c^2*d^2*g^3 + 2*b^2*c*d*e*g^3 - b^3*e^2*g^3)/((c*g*x - c*f + b*g)*(2*c*f - b*g)^2*c^2 *g^3)
Time = 11.39 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.57 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx=\frac {\ln \left (f+g\,x\right )\,\left (d^2\,g^2-2\,d\,e\,f\,g+e^2\,f^2\right )}{b^2\,g^5-4\,b\,c\,f\,g^4+4\,c^2\,f^2\,g^3}+\frac {b^2\,e^2\,g^2-2\,b\,c\,d\,e\,g^2-2\,b\,c\,e^2\,f\,g+c^2\,d^2\,g^2+2\,c^2\,d\,e\,f\,g+c^2\,e^2\,f^2}{c^2\,g^3\,\left (b\,g-2\,c\,f\right )\,\left (b\,g-c\,f+c\,g\,x\right )}+\frac {\ln \left (b\,g-c\,f+c\,g\,x\right )\,\left (c^2\,\left (-d^2\,g^2+2\,d\,e\,f\,g+3\,e^2\,f^2\right )+b^2\,e^2\,g^2-4\,b\,c\,e^2\,f\,g\right )}{c^2\,g^3\,{\left (b\,g-2\,c\,f\right )}^2} \] Input:
int(((f + g*x)*(d + e*x)^2)/(c*g^2*x^2 - c*f^2 + b*f*g + b*g^2*x)^2,x)
Output:
(log(f + g*x)*(d^2*g^2 + e^2*f^2 - 2*d*e*f*g))/(b^2*g^5 + 4*c^2*f^2*g^3 - 4*b*c*f*g^4) + (b^2*e^2*g^2 + c^2*d^2*g^2 + c^2*e^2*f^2 - 2*b*c*d*e*g^2 - 2*b*c*e^2*f*g + 2*c^2*d*e*f*g)/(c^2*g^3*(b*g - 2*c*f)*(b*g - c*f + c*g*x)) + (log(b*g - c*f + c*g*x)*(c^2*(3*e^2*f^2 - d^2*g^2 + 2*d*e*f*g) + b^2*e^ 2*g^2 - 4*b*c*e^2*f*g))/(c^2*g^3*(b*g - 2*c*f)^2)
Time = 0.20 (sec) , antiderivative size = 1017, normalized size of antiderivative = 7.11 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c f^2-b f g-b g^2 x-c g^2 x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^2*(g*x+f)/(-c*g^2*x^2-b*g^2*x-b*f*g+c*f^2)^2,x)
Output:
(log(b*g - c*f + c*g*x)*b**4*e**2*g**4 - 6*log(b*g - c*f + c*g*x)*b**3*c*e **2*f*g**3 + log(b*g - c*f + c*g*x)*b**3*c*e**2*g**4*x - log(b*g - c*f + c *g*x)*b**2*c**2*d**2*g**4 + 2*log(b*g - c*f + c*g*x)*b**2*c**2*d*e*f*g**3 + 12*log(b*g - c*f + c*g*x)*b**2*c**2*e**2*f**2*g**2 - 5*log(b*g - c*f + c *g*x)*b**2*c**2*e**2*f*g**3*x + 2*log(b*g - c*f + c*g*x)*b*c**3*d**2*f*g** 3 - log(b*g - c*f + c*g*x)*b*c**3*d**2*g**4*x - 4*log(b*g - c*f + c*g*x)*b *c**3*d*e*f**2*g**2 + 2*log(b*g - c*f + c*g*x)*b*c**3*d*e*f*g**3*x - 10*lo g(b*g - c*f + c*g*x)*b*c**3*e**2*f**3*g + 7*log(b*g - c*f + c*g*x)*b*c**3* e**2*f**2*g**2*x - log(b*g - c*f + c*g*x)*c**4*d**2*f**2*g**2 + log(b*g - c*f + c*g*x)*c**4*d**2*f*g**3*x + 2*log(b*g - c*f + c*g*x)*c**4*d*e*f**3*g - 2*log(b*g - c*f + c*g*x)*c**4*d*e*f**2*g**2*x + 3*log(b*g - c*f + c*g*x )*c**4*e**2*f**4 - 3*log(b*g - c*f + c*g*x)*c**4*e**2*f**3*g*x + log(f + g *x)*b**2*c**2*d**2*g**4 - 2*log(f + g*x)*b**2*c**2*d*e*f*g**3 + log(f + g* x)*b**2*c**2*e**2*f**2*g**2 - 2*log(f + g*x)*b*c**3*d**2*f*g**3 + log(f + g*x)*b*c**3*d**2*g**4*x + 4*log(f + g*x)*b*c**3*d*e*f**2*g**2 - 2*log(f + g*x)*b*c**3*d*e*f*g**3*x - 2*log(f + g*x)*b*c**3*e**2*f**3*g + log(f + g*x )*b*c**3*e**2*f**2*g**2*x + log(f + g*x)*c**4*d**2*f**2*g**2 - log(f + g*x )*c**4*d**2*f*g**3*x - 2*log(f + g*x)*c**4*d*e*f**3*g + 2*log(f + g*x)*c** 4*d*e*f**2*g**2*x + log(f + g*x)*c**4*e**2*f**4 - log(f + g*x)*c**4*e**2*f **3*g*x - b**3*c*e**2*g**4*x + 2*b**2*c**2*d*e*g**4*x + 4*b**2*c**2*e**...