\(\int \frac {1}{(a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3)^3} \, dx\) [125]

Optimal result

Integrand size = 46, antiderivative size = 495 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=-\frac {b^5}{2 (b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {3 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}+\frac {d^5}{2 (b c-a d)^3 (d e-c f)^3 (c+d x)^2}+\frac {3 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{2 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}-\frac {3 f^5 (2 b d e-b c f-a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {3 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}-\frac {3 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^5}+\frac {3 f^5 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5} \] Output:

-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
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\frac {1}{2} \left (-\frac {b^5}{(b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {6 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}-\frac {d^5}{(b c-a d)^3 (-d e+c f)^3 (c+d x)^2}+\frac {6 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{(b e-a f)^3 (d e-c f)^3 (e+f x)^2}+\frac {6 f^5 (-2 b d e+b c f+a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {6 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}+\frac {6 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (-d e+c f)^5}+\frac {6 f^5 \left (2 a^2 d^2 f^2+a b d f (-7 d e+3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5}\right ) \] Input:

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]
 

Output:

(-(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
 

Rubi [A] (verified)

Time = 1.68 (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

\(\displaystyle \frac {3 f^5 \log (e+f x) \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (2 c^2 f^2-7 c d e f+7 d^2 e^2\right )\right )}{(b e-a f)^5 (d e-c f)^5}-\frac {3 d^5 \log (c+d x) \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (7 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (d e-c f)^5}+\frac {3 b^5 \log (a+b x) \left (7 a^2 d^2 f^2-7 a b d f (c f+d e)+b^2 \left (2 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (b e-a f)^5}+\frac {3 b^5 (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d)^4 (b e-a f)^4}-\frac {b^5}{2 (a+b x)^2 (b c-a d)^3 (b e-a f)^3}+\frac {3 d^5 (a d f-2 b c f+b d e)}{(c+d x) (b c-a d)^4 (d e-c f)^4}+\frac {d^5}{2 (c+d x)^2 (b c-a d)^3 (d e-c f)^3}-\frac {3 f^5 (-a d f-b c f+2 b d e)}{(e+f x) (b e-a f)^4 (d e-c f)^4}-\frac {f^5}{2 (e+f x)^2 (b e-a f)^3 (d e-c f)^3}\)

Input:

Int[(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]
 

Output:

-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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2462

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]
 

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {d^{5}}{2 \left (a d -b c \right )^{3} \left (c f -d e \right )^{3} \left (d x +c \right )^{2}}+\frac {3 d^{5} \left (a d f -2 b c f +b d e \right )}{\left (a d -b c \right )^{4} \left (c f -d e \right )^{4} \left (d x +c \right )}-\frac {3 d^{5} \left (2 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}+3 a b \,d^{2} e f +7 b^{2} c^{2} f^{2}-7 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} \left (c f -d e \right )^{5}}-\frac {b^{5}}{2 \left (a d -b c \right )^{3} \left (a f -e b \right )^{3} \left (b x +a \right )^{2}}-\frac {3 b^{5} \left (2 a d f -b c f -b d e \right )}{\left (a d -b c \right )^{4} \left (a f -e b \right )^{4} \left (b x +a \right )}+\frac {3 b^{5} \left (7 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}+3 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} \left (a f -e b \right )^{5}}-\frac {f^{5}}{2 \left (a f -e b \right )^{3} \left (c f -d e \right )^{3} \left (f x +e \right )^{2}}+\frac {3 f^{5} \left (a d f +b c f -2 b d e \right )}{\left (a f -e b \right )^{4} \left (c f -d e \right )^{4} \left (f x +e \right )}+\frac {3 f^{5} \left (2 a^{2} d^{2} f^{2}+3 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}-7 b^{2} c d e f +7 b^{2} d^{2} e^{2}\right ) \ln \left (f x +e \right )}{\left (a f -e b \right )^{5} \left (c f -d e \right )^{5}}\)	\(514\)
norman	\(\text {Expression too large to display}\)	\(9946\)
risch	\(\text {Expression too large to display}\)	\(20863\)
parallelrisch	\(\text {Expression too large to display}\)	\(27050\)

Input:

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)
 

Output:

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)-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)/(a*d-b*c)^5/(c*f-d*e)^5*ln(d 
*x+c)-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)+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*d-b*c)^5/(a*f-b*e)^ 
5*ln(b*x+a)-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*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)/(a*f-b*e)^5/(c*f 
-d*e)^5*ln(f*x+e)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Timed out} \] Input:

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")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Timed out} \] Input:

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)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11005 vs. \(2 (489) = 978\).

Time = 0.58 (sec) , antiderivative size = 11005, normalized size of antiderivative = 22.23 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \] Input:

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")
 

Output:

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 -...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7111 vs. \(2 (489) = 978\).

Time = 0.20 (sec) , antiderivative size = 7111, normalized size of antiderivative = 14.37 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \] Input:

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")
 

Output:

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...
 

Mupad [B] (verification not implemented)

Time = 27.76 (sec) , antiderivative size = 82532, normalized size of antiderivative = 166.73 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/(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)^3,x)
 

Output:

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 - ...
 

Reduce [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 42188, normalized size of antiderivative = 85.23 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^3} \, dx =\text {Too large to display} \] Input:

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)
 

Output:

(42*log(a + b*x)*a**5*b**5*c**7*d**3*e**2*f**8 + 84*log(a + b*x)*a**5*b**5 
*c**7*d**3*e*f**9*x + 42*log(a + b*x)*a**5*b**5*c**7*d**3*f**10*x**2 - 210 
*log(a + b*x)*a**5*b**5*c**6*d**4*e**3*f**7 - 336*log(a + b*x)*a**5*b**5*c 
**6*d**4*e**2*f**8*x - 42*log(a + b*x)*a**5*b**5*c**6*d**4*e*f**9*x**2 + 8 
4*log(a + b*x)*a**5*b**5*c**6*d**4*f**10*x**3 + 420*log(a + b*x)*a**5*b**5 
*c**5*d**5*e**4*f**6 + 420*log(a + b*x)*a**5*b**5*c**5*d**5*e**3*f**7*x - 
378*log(a + b*x)*a**5*b**5*c**5*d**5*e**2*f**8*x**2 - 336*log(a + b*x)*a** 
5*b**5*c**5*d**5*e*f**9*x**3 + 42*log(a + b*x)*a**5*b**5*c**5*d**5*f**10*x 
**4 - 420*log(a + b*x)*a**5*b**5*c**4*d**6*e**5*f**5 + 1050*log(a + b*x)*a 
**5*b**5*c**4*d**6*e**3*f**7*x**2 + 420*log(a + b*x)*a**5*b**5*c**4*d**6*e 
**2*f**8*x**3 - 210*log(a + b*x)*a**5*b**5*c**4*d**6*e*f**9*x**4 + 210*log 
(a + b*x)*a**5*b**5*c**3*d**7*e**6*f**4 - 420*log(a + b*x)*a**5*b**5*c**3* 
d**7*e**5*f**5*x - 1050*log(a + b*x)*a**5*b**5*c**3*d**7*e**4*f**6*x**2 + 
420*log(a + b*x)*a**5*b**5*c**3*d**7*e**2*f**8*x**4 - 42*log(a + b*x)*a**5 
*b**5*c**2*d**8*e**7*f**3 + 336*log(a + b*x)*a**5*b**5*c**2*d**8*e**6*f**4 
*x + 378*log(a + b*x)*a**5*b**5*c**2*d**8*e**5*f**5*x**2 - 420*log(a + b*x 
)*a**5*b**5*c**2*d**8*e**4*f**6*x**3 - 420*log(a + b*x)*a**5*b**5*c**2*d** 
8*e**3*f**7*x**4 - 84*log(a + b*x)*a**5*b**5*c*d**9*e**7*f**3*x + 42*log(a 
 + b*x)*a**5*b**5*c*d**9*e**6*f**4*x**2 + 336*log(a + b*x)*a**5*b**5*c*d** 
9*e**5*f**5*x**3 + 210*log(a + b*x)*a**5*b**5*c*d**9*e**4*f**6*x**4 - 4...