\(\int \frac {1}{(e+f x)^3 (a c+(b c+a d) x+b d x^2)^2} \, dx\) [361]

Optimal result

Integrand size = 29, antiderivative size = 324 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b^4}{(b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {d^4}{(b c-a d)^2 (d e-c f)^3 (c+d x)}-\frac {f^3}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 f^3 (2 b d e-b c f-a d f)}{(b e-a f)^3 (d e-c f)^3 (e+f x)}-\frac {b^4 (2 b d e+3 b c f-5 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^4}+\frac {d^4 (2 b d e-5 b c f+3 a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^4}+\frac {f^3 \left (3 a^2 d^2 f^2-2 a b d f (5 d e-2 c f)+b^2 \left (10 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^4 (d e-c f)^4} \] Output:

-b^4/(-a*d+b*c)^2/(-a*f+b*e)^3/(b*x+a)-d^4/(-a*d+b*c)^2/(-c*f+d*e)^3/(d*x+ 
c)-1/2*f^3/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)^2-2*f^3*(-a*d*f-b*c*f+2*b*d*e 
)/(-a*f+b*e)^3/(-c*f+d*e)^3/(f*x+e)-b^4*(-5*a*d*f+3*b*c*f+2*b*d*e)*ln(b*x+ 
a)/(-a*d+b*c)^3/(-a*f+b*e)^4+d^4*(3*a*d*f-5*b*c*f+2*b*d*e)*ln(d*x+c)/(-a*d 
+b*c)^3/(-c*f+d*e)^4+f^3*(3*a^2*d^2*f^2-2*a*b*d*f*(-2*c*f+5*d*e)+b^2*(3*c^ 
2*f^2-10*c*d*e*f+10*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^4/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b^4}{(b c-a d)^2 (b e-a f)^3 (a+b x)}+\frac {d^4}{(b c-a d)^2 (-d e+c f)^3 (c+d x)}-\frac {f^3}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}+\frac {2 f^3 (-2 b d e+b c f+a d f)}{(b e-a f)^3 (d e-c f)^3 (e+f x)}-\frac {b^4 (2 b d e+3 b c f-5 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^4}+\frac {d^4 (2 b d e-5 b c f+3 a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^4}+\frac {f^3 \left (3 a^2 d^2 f^2+2 a b d f (-5 d e+2 c f)+b^2 \left (10 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^4 (d e-c f)^4} \] Input:

Integrate[1/((e + f*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^2),x]
 

Output:

-(b^4/((b*c - a*d)^2*(b*e - a*f)^3*(a + b*x))) + d^4/((b*c - a*d)^2*(-(d*e 
) + c*f)^3*(c + d*x)) - f^3/(2*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^2) + 
(2*f^3*(-2*b*d*e + b*c*f + a*d*f))/((b*e - a*f)^3*(d*e - c*f)^3*(e + f*x)) 
 - (b^4*(2*b*d*e + 3*b*c*f - 5*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - 
a*f)^4) + (d^4*(2*b*d*e - 5*b*c*f + 3*a*d*f)*Log[c + d*x])/((b*c - a*d)^3* 
(d*e - c*f)^4) + (f^3*(3*a^2*d^2*f^2 + 2*a*b*d*f*(-5*d*e + 2*c*f) + b^2*(1 
0*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2))*Log[e + f*x])/((b*e - a*f)^4*(d*e - c 
*f)^4)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1141, 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.

\(\displaystyle \int \frac {1}{(e+f x)^3 \left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1141

\(\displaystyle b^2 d^2 \int \left (\frac {\left (\left (10 d^2 e^2-10 c d f e+3 c^2 f^2\right ) b^2-2 a d f (5 d e-2 c f) b+3 a^2 d^2 f^2\right ) f^4}{b^2 d^2 (b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {2 (2 b d e-b c f-a d f) f^4}{b^2 d^2 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}+\frac {f^4}{b^2 d^2 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}-\frac {b^3 (2 b d e+3 b c f-5 a d f)}{d^2 (b c-a d)^3 (b e-a f)^4 (a+b x)}+\frac {d^3 (2 b d e-5 b c f+3 a d f)}{b^2 (b c-a d)^3 (d e-c f)^4 (c+d x)}+\frac {b^3}{d^2 (b c-a d)^2 (b e-a f)^3 (a+b x)^2}+\frac {d^3}{b^2 (b c-a d)^2 (d e-c f)^3 (c+d x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle b^2 d^2 \left (\frac {f^3 \log (e+f x) \left (3 a^2 d^2 f^2-2 a b d f (5 d e-2 c f)+b^2 \left (3 c^2 f^2-10 c d e f+10 d^2 e^2\right )\right )}{b^2 d^2 (b e-a f)^4 (d e-c f)^4}-\frac {2 f^3 (-a d f-b c f+2 b d e)}{b^2 d^2 (e+f x) (b e-a f)^3 (d e-c f)^3}-\frac {f^3}{2 b^2 d^2 (e+f x)^2 (b e-a f)^2 (d e-c f)^2}-\frac {b^2}{d^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac {d^2}{b^2 (c+d x) (b c-a d)^2 (d e-c f)^3}-\frac {b^2 \log (a+b x) (-5 a d f+3 b c f+2 b d e)}{d^2 (b c-a d)^3 (b e-a f)^4}+\frac {d^2 \log (c+d x) (3 a d f-5 b c f+2 b d e)}{b^2 (b c-a d)^3 (d e-c f)^4}\right )\)

Input:

Int[1/((e + f*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^2),x]
 

Output:

b^2*d^2*(-(b^2/(d^2*(b*c - a*d)^2*(b*e - a*f)^3*(a + b*x))) - d^2/(b^2*(b* 
c - a*d)^2*(d*e - c*f)^3*(c + d*x)) - f^3/(2*b^2*d^2*(b*e - a*f)^2*(d*e - 
c*f)^2*(e + f*x)^2) - (2*f^3*(2*b*d*e - b*c*f - a*d*f))/(b^2*d^2*(b*e - a* 
f)^3*(d*e - c*f)^3*(e + f*x)) - (b^2*(2*b*d*e + 3*b*c*f - 5*a*d*f)*Log[a + 
 b*x])/(d^2*(b*c - a*d)^3*(b*e - a*f)^4) + (d^2*(2*b*d*e - 5*b*c*f + 3*a*d 
*f)*Log[c + d*x])/(b^2*(b*c - a*d)^3*(d*e - c*f)^4) + (f^3*(3*a^2*d^2*f^2 
- 2*a*b*d*f*(5*d*e - 2*c*f) + b^2*(10*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2))*L 
og[e + f*x])/(b^2*d^2*(b*e - a*f)^4*(d*e - c*f)^4))
 

Defintions of rubi rules used

rule 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((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*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 2009

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

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {b^{4}}{\left (a f -b e \right )^{3} \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {b^{4} \left (5 a d f -3 b c f -2 b d e \right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{4} \left (a d -b c \right )^{3}}-\frac {f^{3}}{2 \left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \left (f x +e \right )^{2}}+\frac {f^{3} \left (3 a^{2} d^{2} f^{2}+4 a b c d \,f^{2}-10 a b \,d^{2} e f +3 b^{2} c^{2} f^{2}-10 b^{2} c d e f +10 d^{2} e^{2} b^{2}\right ) \ln \left (f x +e \right )}{\left (c f -d e \right )^{4} \left (a f -b e \right )^{4}}+\frac {2 f^{3} \left (a d f +b c f -2 b d e \right )}{\left (c f -d e \right )^{3} \left (a f -b e \right )^{3} \left (f x +e \right )}+\frac {d^{4}}{\left (c f -d e \right )^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d^{4} \left (3 a d f -5 b c f +2 b d e \right ) \ln \left (d x +c \right )}{\left (c f -d e \right )^{4} \left (a d -b c \right )^{3}}\)	\(327\)
norman	\(\text {Expression too large to display}\)	\(2040\)
risch	\(\text {Expression too large to display}\)	\(7122\)
parallelrisch	\(\text {Expression too large to display}\)	\(8317\)

Input:

int(1/(f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

b^4/(a*f-b*e)^3/(a*d-b*c)^2/(b*x+a)-b^4*(5*a*d*f-3*b*c*f-2*b*d*e)/(a*f-b*e 
)^4/(a*d-b*c)^3*ln(b*x+a)-1/2*f^3/(c*f-d*e)^2/(a*f-b*e)^2/(f*x+e)^2+f^3*(3 
*a^2*d^2*f^2+4*a*b*c*d*f^2-10*a*b*d^2*e*f+3*b^2*c^2*f^2-10*b^2*c*d*e*f+10* 
b^2*d^2*e^2)/(c*f-d*e)^4/(a*f-b*e)^4*ln(f*x+e)+2*f^3*(a*d*f+b*c*f-2*b*d*e) 
/(c*f-d*e)^3/(a*f-b*e)^3/(f*x+e)+d^4/(c*f-d*e)^3/(a*d-b*c)^2/(d*x+c)-d^4*( 
3*a*d*f-5*b*c*f+2*b*d*e)/(c*f-d*e)^4/(a*d-b*c)^3*ln(d*x+c)
 

Fricas [F(-1)]

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

integrate(1/(f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(1/(f*x+e)**3/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4029 vs. \(2 (322) = 644\).

Time = 0.49 (sec) , antiderivative size = 4029, normalized size of antiderivative = 12.44 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")
 

Output:

-(2*b^5*d*e + (3*b^5*c - 5*a*b^4*d)*f)*log(b*x + a)/((b^7*c^3 - 3*a*b^6*c^ 
2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*e^4 - 4*(a*b^6*c^3 - 3*a^2*b^5*c^2*d 
+ 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*e^3*f + 6*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d 
+ 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*e^2*f^2 - 4*(a^3*b^4*c^3 - 3*a^4*b^3*c^2* 
d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*e*f^3 + (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 
3*a^6*b*c*d^2 - a^7*d^3)*f^4) + (2*b*d^5*e - (5*b*c*d^4 - 3*a*d^5)*f)*log( 
d*x + c)/((b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*e^4 - 
4*(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3*c*d^6)*e^3*f + 6* 
(b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)*e^2*f^2 - 
4*(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*e*f^3 + (b 
^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*f^4) + (10*b^2*d^2 
*e^2*f^3 - 10*(b^2*c*d + a*b*d^2)*e*f^4 + (3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d 
^2)*f^5)*log(f*x + e)/(b^4*d^4*e^8 + a^4*c^4*f^8 - 4*(b^4*c*d^3 + a*b^3*d^ 
4)*e^7*f + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^6*f^2 - 4*( 
b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^5*f^3 + (b^4* 
c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^4* 
f^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*e^3*f^ 
5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^2*f^6 - 4*(a^3*b*c 
^4 + a^4*c^3*d)*e*f^7) - 1/2*(2*(b^4*c*d^3 + a*b^3*d^4)*e^5 - 6*(b^4*c^2*d 
^2 + a^2*b^2*d^4)*e^4*f + 6*(b^4*c^3*d + a^3*b*d^4)*e^3*f^2 - (2*b^4*c^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2843 vs. \(2 (322) = 644\).

Time = 0.37 (sec) , antiderivative size = 2843, normalized size of antiderivative = 8.77 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
 

Output:

-(2*b^6*d*e + 3*b^6*c*f - 5*a*b^5*d*f)*log(abs(b*x + a))/(b^8*c^3*e^4 - 3* 
a*b^7*c^2*d*e^4 + 3*a^2*b^6*c*d^2*e^4 - a^3*b^5*d^3*e^4 - 4*a*b^7*c^3*e^3* 
f + 12*a^2*b^6*c^2*d*e^3*f - 12*a^3*b^5*c*d^2*e^3*f + 4*a^4*b^4*d^3*e^3*f 
+ 6*a^2*b^6*c^3*e^2*f^2 - 18*a^3*b^5*c^2*d*e^2*f^2 + 18*a^4*b^4*c*d^2*e^2* 
f^2 - 6*a^5*b^3*d^3*e^2*f^2 - 4*a^3*b^5*c^3*e*f^3 + 12*a^4*b^4*c^2*d*e*f^3 
 - 12*a^5*b^3*c*d^2*e*f^3 + 4*a^6*b^2*d^3*e*f^3 + a^4*b^4*c^3*f^4 - 3*a^5* 
b^3*c^2*d*f^4 + 3*a^6*b^2*c*d^2*f^4 - a^7*b*d^3*f^4) + (2*b*d^6*e - 5*b*c* 
d^5*f + 3*a*d^6*f)*log(abs(d*x + c))/(b^3*c^3*d^5*e^4 - 3*a*b^2*c^2*d^6*e^ 
4 + 3*a^2*b*c*d^7*e^4 - a^3*d^8*e^4 - 4*b^3*c^4*d^4*e^3*f + 12*a*b^2*c^3*d 
^5*e^3*f - 12*a^2*b*c^2*d^6*e^3*f + 4*a^3*c*d^7*e^3*f + 6*b^3*c^5*d^3*e^2* 
f^2 - 18*a*b^2*c^4*d^4*e^2*f^2 + 18*a^2*b*c^3*d^5*e^2*f^2 - 6*a^3*c^2*d^6* 
e^2*f^2 - 4*b^3*c^6*d^2*e*f^3 + 12*a*b^2*c^5*d^3*e*f^3 - 12*a^2*b*c^4*d^4* 
e*f^3 + 4*a^3*c^3*d^5*e*f^3 + b^3*c^7*d*f^4 - 3*a*b^2*c^6*d^2*f^4 + 3*a^2* 
b*c^5*d^3*f^4 - a^3*c^4*d^4*f^4) + (10*b^2*d^2*e^2*f^4 - 10*b^2*c*d*e*f^5 
- 10*a*b*d^2*e*f^5 + 3*b^2*c^2*f^6 + 4*a*b*c*d*f^6 + 3*a^2*d^2*f^6)*log(ab 
s(f*x + e))/(b^4*d^4*e^8*f - 4*b^4*c*d^3*e^7*f^2 - 4*a*b^3*d^4*e^7*f^2 + 6 
*b^4*c^2*d^2*e^6*f^3 + 16*a*b^3*c*d^3*e^6*f^3 + 6*a^2*b^2*d^4*e^6*f^3 - 4* 
b^4*c^3*d*e^5*f^4 - 24*a*b^3*c^2*d^2*e^5*f^4 - 24*a^2*b^2*c*d^3*e^5*f^4 - 
4*a^3*b*d^4*e^5*f^4 + b^4*c^4*e^4*f^5 + 16*a*b^3*c^3*d*e^4*f^5 + 36*a^2*b^ 
2*c^2*d^2*e^4*f^5 + 16*a^3*b*c*d^3*e^4*f^5 + a^4*d^4*e^4*f^5 - 4*a*b^3*...
 

Mupad [B] (verification not implemented)

Time = 13.41 (sec) , antiderivative size = 4019, normalized size of antiderivative = 12.40 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((e + f*x)^3*(a*c + x*(a*d + b*c) + b*d*x^2)^2),x)
 

Output:

(log(e + f*x)*(f^5*(3*a^2*d^2 + 3*b^2*c^2 + 4*a*b*c*d) - f^4*(10*a*b*d^2*e 
 + 10*b^2*c*d*e) + 10*b^2*d^2*e^2*f^3))/(a^4*c^4*f^8 + b^4*d^4*e^8 + a^4*d 
^4*e^4*f^4 + b^4*c^4*e^4*f^4 + 6*a^2*b^2*c^4*e^2*f^6 + 6*a^2*b^2*d^4*e^6*f 
^2 + 6*a^4*c^2*d^2*e^2*f^6 + 6*b^4*c^2*d^2*e^6*f^2 - 4*a^3*b*c^4*e*f^7 - 4 
*a*b^3*d^4*e^7*f - 4*a^4*c^3*d*e*f^7 - 4*b^4*c*d^3*e^7*f - 4*a*b^3*c^4*e^3 
*f^5 - 4*a^3*b*d^4*e^5*f^3 - 4*a^4*c*d^3*e^3*f^5 - 4*b^4*c^3*d*e^5*f^3 + 1 
6*a*b^3*c*d^3*e^6*f^2 + 16*a*b^3*c^3*d*e^4*f^4 + 16*a^3*b*c*d^3*e^4*f^4 + 
16*a^3*b*c^3*d*e^2*f^6 - 24*a*b^3*c^2*d^2*e^5*f^3 - 24*a^2*b^2*c*d^3*e^5*f 
^3 - 24*a^2*b^2*c^3*d*e^3*f^5 - 24*a^3*b*c^2*d^2*e^3*f^5 + 36*a^2*b^2*c^2* 
d^2*e^4*f^4) - (log(a + b*x)*(b^5*(3*c*f + 2*d*e) - 5*a*b^4*d*f))/(b^7*c^3 
*e^4 - a^7*d^3*f^4 - a^3*b^4*d^3*e^4 + a^4*b^3*c^3*f^4 + 6*a^2*b^5*c^3*e^2 
*f^2 - 6*a^5*b^2*d^3*e^2*f^2 - 3*a*b^6*c^2*d*e^4 + 3*a^6*b*c*d^2*f^4 - 4*a 
*b^6*c^3*e^3*f + 4*a^6*b*d^3*e*f^3 + 3*a^2*b^5*c*d^2*e^4 - 3*a^5*b^2*c^2*d 
*f^4 - 4*a^3*b^4*c^3*e*f^3 + 4*a^4*b^3*d^3*e^3*f + 12*a^2*b^5*c^2*d*e^3*f 
- 12*a^3*b^4*c*d^2*e^3*f + 12*a^4*b^3*c^2*d*e*f^3 - 12*a^5*b^2*c*d^2*e*f^3 
 - 18*a^3*b^4*c^2*d*e^2*f^2 + 18*a^4*b^3*c*d^2*e^2*f^2) - (log(c + d*x)*(d 
^5*(3*a*f + 2*b*e) - 5*b*c*d^4*f))/(a^3*d^7*e^4 - b^3*c^7*f^4 + a^3*c^4*d^ 
3*f^4 - b^3*c^3*d^4*e^4 + 6*a^3*c^2*d^5*e^2*f^2 - 6*b^3*c^5*d^2*e^2*f^2 - 
3*a^2*b*c*d^6*e^4 + 3*a*b^2*c^6*d*f^4 - 4*a^3*c*d^6*e^3*f + 4*b^3*c^6*d*e* 
f^3 + 3*a*b^2*c^2*d^5*e^4 - 3*a^2*b*c^5*d^2*f^4 - 4*a^3*c^3*d^4*e*f^3 +...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 18244, normalized size of antiderivative = 56.31 \[ \int \frac {1}{(e+f x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
 

Output:

( - 10*log(a + b*x)*a**3*b**4*c**5*d**2*e**2*f**6 - 20*log(a + b*x)*a**3*b 
**4*c**5*d**2*e*f**7*x - 10*log(a + b*x)*a**3*b**4*c**5*d**2*f**8*x**2 + 4 
0*log(a + b*x)*a**3*b**4*c**4*d**3*e**3*f**5 + 70*log(a + b*x)*a**3*b**4*c 
**4*d**3*e**2*f**6*x + 20*log(a + b*x)*a**3*b**4*c**4*d**3*e*f**7*x**2 - 1 
0*log(a + b*x)*a**3*b**4*c**4*d**3*f**8*x**3 - 60*log(a + b*x)*a**3*b**4*c 
**3*d**4*e**4*f**4 - 80*log(a + b*x)*a**3*b**4*c**3*d**4*e**3*f**5*x + 20* 
log(a + b*x)*a**3*b**4*c**3*d**4*e**2*f**6*x**2 + 40*log(a + b*x)*a**3*b** 
4*c**3*d**4*e*f**7*x**3 + 40*log(a + b*x)*a**3*b**4*c**2*d**5*e**5*f**3 + 
20*log(a + b*x)*a**3*b**4*c**2*d**5*e**4*f**4*x - 80*log(a + b*x)*a**3*b** 
4*c**2*d**5*e**3*f**5*x**2 - 60*log(a + b*x)*a**3*b**4*c**2*d**5*e**2*f**6 
*x**3 - 10*log(a + b*x)*a**3*b**4*c*d**6*e**6*f**2 + 20*log(a + b*x)*a**3* 
b**4*c*d**6*e**5*f**3*x + 70*log(a + b*x)*a**3*b**4*c*d**6*e**4*f**4*x**2 
+ 40*log(a + b*x)*a**3*b**4*c*d**6*e**3*f**5*x**3 - 10*log(a + b*x)*a**3*b 
**4*d**7*e**6*f**2*x - 20*log(a + b*x)*a**3*b**4*d**7*e**5*f**3*x**2 - 10* 
log(a + b*x)*a**3*b**4*d**7*e**4*f**4*x**3 - 4*log(a + b*x)*a**2*b**5*c**6 
*d*e**2*f**6 - 8*log(a + b*x)*a**2*b**5*c**6*d*e*f**7*x - 4*log(a + b*x)*a 
**2*b**5*c**6*d*f**8*x**2 - 14*log(a + b*x)*a**2*b**5*c**5*d**2*e**2*f**6* 
x - 28*log(a + b*x)*a**2*b**5*c**5*d**2*e*f**7*x**2 - 14*log(a + b*x)*a**2 
*b**5*c**5*d**2*f**8*x**3 + 40*log(a + b*x)*a**2*b**5*c**4*d**3*e**4*f**4 
+ 120*log(a + b*x)*a**2*b**5*c**4*d**3*e**3*f**5*x + 110*log(a + b*x)*a...