Integrand size = 46, antiderivative size = 495 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=-\frac {b^5}{2 (b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {3 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}+\frac {d^5}{2 (b c-a d)^3 (d e-c f)^3 (c+d x)^2}+\frac {3 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{2 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}-\frac {3 f^5 (2 b d e-b c f-a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {3 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}-\frac {3 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^5}+\frac {3 f^5 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5} \]
-1/2*b^5/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)^2+3*b^5*(-2*a*d*f+b*c*f+b*d*e)/ (-a*d+b*c)^4/(-a*f+b*e)^4/(b*x+a)+1/2*d^5/(-a*d+b*c)^3/(-c*f+d*e)^3/(d*x+c )^2+3*d^5*(a*d*f-2*b*c*f+b*d*e)/(-a*d+b*c)^4/(-c*f+d*e)^4/(d*x+c)-1/2*f^5/ (-a*f+b*e)^3/(-c*f+d*e)^3/(f*x+e)^2-3*f^5*(-a*d*f-b*c*f+2*b*d*e)/(-a*f+b*e )^4/(-c*f+d*e)^4/(f*x+e)+3*b^5*(7*a^2*d^2*f^2-7*a*b*d*f*(c*f+d*e)+b^2*(2*c ^2*f^2+3*c*d*e*f+2*d^2*e^2))*ln(b*x+a)/(-a*d+b*c)^5/(-a*f+b*e)^5-3*d^5*(2* a^2*d^2*f^2+a*b*d*f*(-7*c*f+3*d*e)+b^2*(7*c^2*f^2-7*c*d*e*f+2*d^2*e^2))*ln (d*x+c)/(-a*d+b*c)^5/(-c*f+d*e)^5+3*f^5*(2*a^2*d^2*f^2-a*b*d*f*(-3*c*f+7*d *e)+b^2*(2*c^2*f^2-7*c*d*e*f+7*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^5/(-c*f+d*e) ^5
Time = 0.72 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\frac {1}{2} \left (-\frac {b^5}{(b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {6 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}-\frac {d^5}{(b c-a d)^3 (-d e+c f)^3 (c+d x)^2}+\frac {6 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{(b e-a f)^3 (d e-c f)^3 (e+f x)^2}+\frac {6 f^5 (-2 b d e+b c f+a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {6 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}+\frac {6 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (-d e+c f)^5}+\frac {6 f^5 \left (2 a^2 d^2 f^2+a b d f (-7 d e+3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5}\right ) \]
Integrate[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^(-3),x]
(-(b^5/((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^2)) + (6*b^5*(b*d*e + b*c*f - 2*a*d*f))/((b*c - a*d)^4*(b*e - a*f)^4*(a + b*x)) - d^5/((b*c - a*d)^3*( -(d*e) + c*f)^3*(c + d*x)^2) + (6*d^5*(b*d*e - 2*b*c*f + a*d*f))/((b*c - a *d)^4*(d*e - c*f)^4*(c + d*x)) - f^5/((b*e - a*f)^3*(d*e - c*f)^3*(e + f*x )^2) + (6*f^5*(-2*b*d*e + b*c*f + a*d*f))/((b*e - a*f)^4*(d*e - c*f)^4*(e + f*x)) + (6*b^5*(7*a^2*d^2*f^2 - 7*a*b*d*f*(d*e + c*f) + b^2*(2*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2))*Log[a + b*x])/((b*c - a*d)^5*(b*e - a*f)^5) + (6* d^5*(2*a^2*d^2*f^2 + a*b*d*f*(3*d*e - 7*c*f) + b^2*(2*d^2*e^2 - 7*c*d*e*f + 7*c^2*f^2))*Log[c + d*x])/((b*c - a*d)^5*(-(d*e) + c*f)^5) + (6*f^5*(2*a ^2*d^2*f^2 + a*b*d*f*(-7*d*e + 3*c*f) + b^2*(7*d^2*e^2 - 7*c*d*e*f + 2*c^2 *f^2))*Log[e + f*x])/((b*e - a*f)^5*(d*e - c*f)^5))/2
Time = 1.60 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2462, 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.
| \(\displaystyle \int \frac {1}{\left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {3 f^6 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (2 c^2 f^2-7 c d e f+7 d^2 e^2\right )\right )}{(e+f x) (b e-a f)^5 (d e-c f)^5}+\frac {3 d^6 \left (-2 a^2 d^2 f^2-a b d f (3 d e-7 c f)-\left (b^2 \left (7 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right )\right )}{(c+d x) (b c-a d)^5 (d e-c f)^5}+\frac {3 b^6 \left (7 a^2 d^2 f^2-7 a b d f (c f+d e)+b^2 \left (2 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right )}{(a+b x) (b c-a d)^5 (b e-a f)^5}-\frac {3 b^6 (-2 a d f+b c f+b d e)}{(a+b x)^2 (b c-a d)^4 (b e-a f)^4}+\frac {b^6}{(a+b x)^3 (b c-a d)^3 (b e-a f)^3}-\frac {3 d^6 (a d f-2 b c f+b d e)}{(c+d x)^2 (b c-a d)^4 (c f-d e)^4}+\frac {d^6}{(c+d x)^3 (b c-a d)^3 (c f-d e)^3}-\frac {3 f^6 (a d f+b c f-2 b d e)}{(e+f x)^2 (b e-a f)^4 (d e-c f)^4}+\frac {f^6}{(e+f x)^3 (b e-a f)^3 (d e-c f)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 f^5 \log (e+f x) \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (2 c^2 f^2-7 c d e f+7 d^2 e^2\right )\right )}{(b e-a f)^5 (d e-c f)^5}-\frac {3 d^5 \log (c+d x) \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (7 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (d e-c f)^5}+\frac {3 b^5 \log (a+b x) \left (7 a^2 d^2 f^2-7 a b d f (c f+d e)+b^2 \left (2 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (b e-a f)^5}+\frac {3 b^5 (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d)^4 (b e-a f)^4}-\frac {b^5}{2 (a+b x)^2 (b c-a d)^3 (b e-a f)^3}+\frac {3 d^5 (a d f-2 b c f+b d e)}{(c+d x) (b c-a d)^4 (d e-c f)^4}+\frac {d^5}{2 (c+d x)^2 (b c-a d)^3 (d e-c f)^3}-\frac {3 f^5 (-a d f-b c f+2 b d e)}{(e+f x) (b e-a f)^4 (d e-c f)^4}-\frac {f^5}{2 (e+f x)^2 (b e-a f)^3 (d e-c f)^3}\) |
-1/2*b^5/((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^2) + (3*b^5*(b*d*e + b*c*f - 2*a*d*f))/((b*c - a*d)^4*(b*e - a*f)^4*(a + b*x)) + d^5/(2*(b*c - a*d)^ 3*(d*e - c*f)^3*(c + d*x)^2) + (3*d^5*(b*d*e - 2*b*c*f + a*d*f))/((b*c - a *d)^4*(d*e - c*f)^4*(c + d*x)) - f^5/(2*(b*e - a*f)^3*(d*e - c*f)^3*(e + f *x)^2) - (3*f^5*(2*b*d*e - b*c*f - a*d*f))/((b*e - a*f)^4*(d*e - c*f)^4*(e + f*x)) + (3*b^5*(7*a^2*d^2*f^2 - 7*a*b*d*f*(d*e + c*f) + b^2*(2*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2))*Log[a + b*x])/((b*c - a*d)^5*(b*e - a*f)^5) - (3 *d^5*(2*a^2*d^2*f^2 + a*b*d*f*(3*d*e - 7*c*f) + b^2*(2*d^2*e^2 - 7*c*d*e*f + 7*c^2*f^2))*Log[c + d*x])/((b*c - a*d)^5*(d*e - c*f)^5) + (3*f^5*(2*a^2 *d^2*f^2 - a*b*d*f*(7*d*e - 3*c*f) + b^2*(7*d^2*e^2 - 7*c*d*e*f + 2*c^2*f^ 2))*Log[e + f*x])/((b*e - a*f)^5*(d*e - c*f)^5)
3.1.20.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 2.37 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(-\frac {f^{5}}{2 \left (c f -e d \right )^{3} \left (a f -b e \right )^{3} \left (f x +e \right )^{2}}+\frac {3 f^{5} \left (a d f +f b c -2 b d e \right )}{\left (c f -e d \right )^{4} \left (a f -b e \right )^{4} \left (f x +e \right )}+\frac {3 f^{5} \left (2 a^{2} d^{2} f^{2}+3 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}-7 b^{2} c d e f +7 b^{2} d^{2} e^{2}\right ) \ln \left (f x +e \right )}{\left (c f -e d \right )^{5} \left (a f -b e \right )^{5}}+\frac {d^{5}}{2 \left (c f -e d \right )^{3} \left (d a -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {3 d^{5} \left (a d f -2 f b c +b d e \right )}{\left (c f -e d \right )^{4} \left (d a -b c \right )^{4} \left (d x +c \right )}-\frac {3 d^{5} \left (2 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}+3 a b \,d^{2} e f +7 b^{2} c^{2} f^{2}-7 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (d x +c \right )}{\left (c f -e d \right )^{5} \left (d a -b c \right )^{5}}-\frac {b^{5}}{2 \left (a f -b e \right )^{3} \left (d a -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {3 b^{5} \left (2 a d f -f b c -b d e \right )}{\left (a f -b e \right )^{4} \left (d a -b c \right )^{4} \left (b x +a \right )}+\frac {3 b^{5} \left (7 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}+3 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{5} \left (d a -b c \right )^{5}}\) | \(514\) |
| norman | \(\text {Expression too large to display}\) | \(9946\) |
| risch | \(\text {Expression too large to display}\) | \(20863\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(27050\) |
int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^3,x, method=_RETURNVERBOSE)
-1/2*f^5/(c*f-d*e)^3/(a*f-b*e)^3/(f*x+e)^2+3*f^5*(a*d*f+b*c*f-2*b*d*e)/(c* f-d*e)^4/(a*f-b*e)^4/(f*x+e)+3*f^5*(2*a^2*d^2*f^2+3*a*b*c*d*f^2-7*a*b*d^2* e*f+2*b^2*c^2*f^2-7*b^2*c*d*e*f+7*b^2*d^2*e^2)/(c*f-d*e)^5/(a*f-b*e)^5*ln( f*x+e)+1/2*d^5/(c*f-d*e)^3/(a*d-b*c)^3/(d*x+c)^2+3*d^5*(a*d*f-2*b*c*f+b*d* e)/(c*f-d*e)^4/(a*d-b*c)^4/(d*x+c)-3*d^5*(2*a^2*d^2*f^2-7*a*b*c*d*f^2+3*a* b*d^2*e*f+7*b^2*c^2*f^2-7*b^2*c*d*e*f+2*b^2*d^2*e^2)/(c*f-d*e)^5/(a*d-b*c) ^5*ln(d*x+c)-1/2*b^5/(a*f-b*e)^3/(a*d-b*c)^3/(b*x+a)^2-3*b^5*(2*a*d*f-b*c* f-b*d*e)/(a*f-b*e)^4/(a*d-b*c)^4/(b*x+a)+3*b^5*(7*a^2*d^2*f^2-7*a*b*c*d*f^ 2-7*a*b*d^2*e*f+2*b^2*c^2*f^2+3*b^2*c*d*e*f+2*b^2*d^2*e^2)/(a*f-b*e)^5/(a* d-b*c)^5*ln(b*x+a)
Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Timed out} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^3,x, algorithm="fricas")
Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 11005 vs. \(2 (489) = 978\).
Time = 0.74 (sec) , antiderivative size = 11005, normalized size of antiderivative = 22.23 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^3,x, algorithm="maxima")
3*(2*b^7*d^2*e^2 + (3*b^7*c*d - 7*a*b^6*d^2)*e*f + (2*b^7*c^2 - 7*a*b^6*c* d + 7*a^2*b^5*d^2)*f^2)*log(b*x + a)/((b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*b ^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5)*e^5 - 5*( a*b^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^3*b^7*c^3*d^2 - 10*a^4*b^6*c^2*d^3 + 5* a^5*b^5*c*d^4 - a^6*b^4*d^5)*e^4*f + 10*(a^2*b^8*c^5 - 5*a^3*b^7*c^4*d + 1 0*a^4*b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 - a^7*b^3*d^5)*e^ 3*f^2 - 10*(a^3*b^7*c^5 - 5*a^4*b^6*c^4*d + 10*a^5*b^5*c^3*d^2 - 10*a^6*b^ 4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*e^2*f^3 + 5*(a^4*b^6*c^5 - 5*a^ 5*b^5*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5)*e*f^4 - (a^5*b^5*c^5 - 5*a^6*b^4*c^4*d + 10*a^7*b^3*c^3*d^2 - 1 0*a^8*b^2*c^2*d^3 + 5*a^9*b*c*d^4 - a^10*d^5)*f^5) - 3*(2*b^2*d^7*e^2 - (7 *b^2*c*d^6 - 3*a*b*d^7)*e*f + (7*b^2*c^2*d^5 - 7*a*b*c*d^6 + 2*a^2*d^7)*f^ 2)*log(d*x + c)/((b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10* a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9 - a^5*d^10)*e^5 - 5*(b^5*c^6*d^4 - 5*a*b^4 *c^5*d^5 + 10*a^2*b^3*c^4*d^6 - 10*a^3*b^2*c^3*d^7 + 5*a^4*b*c^2*d^8 - a^5 *c*d^9)*e^4*f + 10*(b^5*c^7*d^3 - 5*a*b^4*c^6*d^4 + 10*a^2*b^3*c^5*d^5 - 1 0*a^3*b^2*c^4*d^6 + 5*a^4*b*c^3*d^7 - a^5*c^2*d^8)*e^3*f^2 - 10*(b^5*c^8*d ^2 - 5*a*b^4*c^7*d^3 + 10*a^2*b^3*c^6*d^4 - 10*a^3*b^2*c^5*d^5 + 5*a^4*b*c ^4*d^6 - a^5*c^3*d^7)*e^2*f^3 + 5*(b^5*c^9*d - 5*a*b^4*c^8*d^2 + 10*a^2*b^ 3*c^7*d^3 - 10*a^3*b^2*c^6*d^4 + 5*a^4*b*c^5*d^5 - a^5*c^4*d^6)*e*f^4 -...
Leaf count of result is larger than twice the leaf count of optimal. 7111 vs. \(2 (489) = 978\).
Time = 0.40 (sec) , antiderivative size = 7111, normalized size of antiderivative = 14.37 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \]
integrate(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3 )^3,x, algorithm="giac")
3*(2*b^8*d^2*e^2 + 3*b^8*c*d*e*f - 7*a*b^7*d^2*e*f + 2*b^8*c^2*f^2 - 7*a*b ^7*c*d*f^2 + 7*a^2*b^6*d^2*f^2)*log(abs(b*x + a))/(b^11*c^5*e^5 - 5*a*b^10 *c^4*d*e^5 + 10*a^2*b^9*c^3*d^2*e^5 - 10*a^3*b^8*c^2*d^3*e^5 + 5*a^4*b^7*c *d^4*e^5 - a^5*b^6*d^5*e^5 - 5*a*b^10*c^5*e^4*f + 25*a^2*b^9*c^4*d*e^4*f - 50*a^3*b^8*c^3*d^2*e^4*f + 50*a^4*b^7*c^2*d^3*e^4*f - 25*a^5*b^6*c*d^4*e^ 4*f + 5*a^6*b^5*d^5*e^4*f + 10*a^2*b^9*c^5*e^3*f^2 - 50*a^3*b^8*c^4*d*e^3* f^2 + 100*a^4*b^7*c^3*d^2*e^3*f^2 - 100*a^5*b^6*c^2*d^3*e^3*f^2 + 50*a^6*b ^5*c*d^4*e^3*f^2 - 10*a^7*b^4*d^5*e^3*f^2 - 10*a^3*b^8*c^5*e^2*f^3 + 50*a^ 4*b^7*c^4*d*e^2*f^3 - 100*a^5*b^6*c^3*d^2*e^2*f^3 + 100*a^6*b^5*c^2*d^3*e^ 2*f^3 - 50*a^7*b^4*c*d^4*e^2*f^3 + 10*a^8*b^3*d^5*e^2*f^3 + 5*a^4*b^7*c^5* e*f^4 - 25*a^5*b^6*c^4*d*e*f^4 + 50*a^6*b^5*c^3*d^2*e*f^4 - 50*a^7*b^4*c^2 *d^3*e*f^4 + 25*a^8*b^3*c*d^4*e*f^4 - 5*a^9*b^2*d^5*e*f^4 - a^5*b^6*c^5*f^ 5 + 5*a^6*b^5*c^4*d*f^5 - 10*a^7*b^4*c^3*d^2*f^5 + 10*a^8*b^3*c^2*d^3*f^5 - 5*a^9*b^2*c*d^4*f^5 + a^10*b*d^5*f^5) - 3*(2*b^2*d^8*e^2 - 7*b^2*c*d^7*e *f + 3*a*b*d^8*e*f + 7*b^2*c^2*d^6*f^2 - 7*a*b*c*d^7*f^2 + 2*a^2*d^8*f^2)* log(abs(d*x + c))/(b^5*c^5*d^6*e^5 - 5*a*b^4*c^4*d^7*e^5 + 10*a^2*b^3*c^3* d^8*e^5 - 10*a^3*b^2*c^2*d^9*e^5 + 5*a^4*b*c*d^10*e^5 - a^5*d^11*e^5 - 5*b ^5*c^6*d^5*e^4*f + 25*a*b^4*c^5*d^6*e^4*f - 50*a^2*b^3*c^4*d^7*e^4*f + 50* a^3*b^2*c^3*d^8*e^4*f - 25*a^4*b*c^2*d^9*e^4*f + 5*a^5*c*d^10*e^4*f + 10*b ^5*c^7*d^4*e^3*f^2 - 50*a*b^4*c^6*d^5*e^3*f^2 + 100*a^2*b^3*c^5*d^6*e^3...
Time = 23.45 (sec) , antiderivative size = 82532, normalized size of antiderivative = 166.73 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \]
symsum(log(root(756756*a^10*b^10*c^10*d^10*e^10*f^10*z^3 + 573300*a^12*b^8 *c^9*d^11*e^9*f^11*z^3 + 573300*a^11*b^9*c^11*d^9*e^8*f^12*z^3 + 573300*a^ 11*b^9*c^8*d^12*e^11*f^9*z^3 + 573300*a^9*b^11*c^12*d^8*e^9*f^11*z^3 + 573 300*a^9*b^11*c^9*d^11*e^12*f^8*z^3 + 573300*a^8*b^12*c^11*d^9*e^11*f^9*z^3 - 343980*a^11*b^9*c^10*d^10*e^9*f^11*z^3 - 343980*a^11*b^9*c^9*d^11*e^10* f^10*z^3 - 343980*a^10*b^10*c^11*d^9*e^9*f^11*z^3 - 343980*a^10*b^10*c^9*d ^11*e^11*f^9*z^3 - 343980*a^9*b^11*c^11*d^9*e^10*f^10*z^3 - 343980*a^9*b^1 1*c^10*d^10*e^11*f^9*z^3 + 326340*a^13*b^7*c^10*d^10*e^7*f^13*z^3 + 326340 *a^13*b^7*c^7*d^13*e^10*f^10*z^3 + 326340*a^10*b^10*c^13*d^7*e^7*f^13*z^3 + 326340*a^10*b^10*c^7*d^13*e^13*f^7*z^3 + 326340*a^7*b^13*c^13*d^7*e^10*f ^10*z^3 + 326340*a^7*b^13*c^10*d^10*e^13*f^7*z^3 - 267540*a^12*b^8*c^10*d^ 10*e^8*f^12*z^3 - 267540*a^12*b^8*c^8*d^12*e^10*f^10*z^3 - 267540*a^10*b^1 0*c^12*d^8*e^8*f^12*z^3 - 267540*a^10*b^10*c^8*d^12*e^12*f^8*z^3 - 267540* a^8*b^12*c^12*d^8*e^10*f^10*z^3 - 267540*a^8*b^12*c^10*d^10*e^12*f^8*z^3 + 245700*a^14*b^6*c^8*d^12*e^8*f^12*z^3 + 245700*a^12*b^8*c^12*d^8*e^6*f^14 *z^3 + 245700*a^12*b^8*c^6*d^14*e^12*f^8*z^3 + 245700*a^8*b^12*c^14*d^6*e^ 8*f^12*z^3 + 245700*a^8*b^12*c^8*d^12*e^14*f^6*z^3 + 245700*a^6*b^14*c^12* d^8*e^12*f^8*z^3 - 191100*a^13*b^7*c^9*d^11*e^8*f^12*z^3 - 191100*a^13*b^7 *c^8*d^12*e^9*f^11*z^3 - 191100*a^12*b^8*c^11*d^9*e^7*f^13*z^3 - 191100*a^ 12*b^8*c^7*d^13*e^11*f^9*z^3 - 191100*a^11*b^9*c^12*d^8*e^7*f^13*z^3 - ...