\(\int \frac {1}{(e+f x) (a c+(b c+a d) x+b d x^2)^2} \, dx\) [359]

Optimal result

Integrand size = 29, antiderivative size = 186 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b^2}{(b c-a d)^2 (b e-a f) (a+b x)}-\frac {d^2}{(b c-a d)^2 (d e-c f) (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^2}+\frac {d^2 (2 b d e-3 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^2}+\frac {f^3 \log (e+f x)}{(b e-a f)^2 (d e-c f)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {b^2}{(b c-a d)^2 (b e-a f) (a+b x)}+\frac {d^2}{(b c-a d)^2 (-d e+c f) (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^2}+\frac {d^2 (2 b d e-3 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^2}+\frac {f^3 \log (e+f x)}{(b e-a f)^2 (d e-c f)^2} \] Input:

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

Output:

-(b^2/((b*c - a*d)^2*(b*e - a*f)*(a + b*x))) + d^2/((b*c - a*d)^2*(-(d*e) 
+ c*f)*(c + d*x)) - (b^2*(2*b*d*e + b*c*f - 3*a*d*f)*Log[a + b*x])/((b*c - 
 a*d)^3*(b*e - a*f)^2) + (d^2*(2*b*d*e - 3*b*c*f + a*d*f)*Log[c + d*x])/(( 
b*c - a*d)^3*(d*e - c*f)^2) + (f^3*Log[e + f*x])/((b*e - a*f)^2*(d*e - c*f 
)^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.

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

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {b^{2}}{\left (a f -b e \right ) \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {b^{2} \left (3 a d f -b c f -2 b d e \right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{2} \left (a d -b c \right )^{3}}+\frac {f^{3} \ln \left (f x +e \right )}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{2}}+\frac {d^{2}}{\left (c f -d e \right ) \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d^{2} \left (a d f -3 b c f +2 b d e \right ) \ln \left (d x +c \right )}{\left (c f -d e \right )^{2} \left (a d -b c \right )^{3}}\)	\(187\)
norman	\(\frac {\frac {a^{2} b \,d^{3} f -a \,b^{2} d^{3} e +b^{3} c^{2} d f -b^{3} c \,d^{2} e}{d b \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a \,b^{2} d^{3} f +b^{3} c \,d^{2} f -2 d^{3} e \,b^{3}\right ) x}{d b \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {f^{3} \ln \left (f x +e \right )}{c^{2} f^{4} a^{2}-2 c d \,f^{3} a^{2} e +a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 d b c \,f^{2} a \,e^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}}-\frac {b^{2} \left (3 a d f -b c f -2 b d e \right ) \ln \left (b x +a \right )}{\left (a^{2} f^{2}-2 a b e f +b^{2} e^{2}\right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (a d f -3 b c f +2 b d e \right ) d^{2} \ln \left (d x +c \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (c^{2} f^{2}-2 c e f d +d^{2} e^{2}\right )}\)	\(490\)
risch	\(\text {Expression too large to display}\)	\(1454\)
parallelrisch	\(\text {Expression too large to display}\)	\(1706\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(1/(f*x+e)/(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. 873 vs. \(2 (186) = 372\).

Time = 0.09 (sec) , antiderivative size = 873, normalized size of antiderivative = 4.69 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (186) = 372\).

Time = 0.34 (sec) , antiderivative size = 794, normalized size of antiderivative = 4.27 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^{4} \log \left ({\left | f x + e \right |}\right )}{b^{2} d^{2} e^{4} f - 2 \, b^{2} c d e^{3} f^{2} - 2 \, a b d^{2} e^{3} f^{2} + b^{2} c^{2} e^{2} f^{3} + 4 \, a b c d e^{2} f^{3} + a^{2} d^{2} e^{2} f^{3} - 2 \, a b c^{2} e f^{4} - 2 \, a^{2} c d e f^{4} + a^{2} c^{2} f^{5}} - \frac {{\left (2 \, b^{4} d e + b^{4} c f - 3 \, a b^{3} d f\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{3} e^{2} - 3 \, a b^{5} c^{2} d e^{2} + 3 \, a^{2} b^{4} c d^{2} e^{2} - a^{3} b^{3} d^{3} e^{2} - 2 \, a b^{5} c^{3} e f + 6 \, a^{2} b^{4} c^{2} d e f - 6 \, a^{3} b^{3} c d^{2} e f + 2 \, a^{4} b^{2} d^{3} e f + a^{2} b^{4} c^{3} f^{2} - 3 \, a^{3} b^{3} c^{2} d f^{2} + 3 \, a^{4} b^{2} c d^{2} f^{2} - a^{5} b d^{3} f^{2}} + \frac {{\left (2 \, b d^{4} e - 3 \, b c d^{3} f + a d^{4} f\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{3} e^{2} - 3 \, a b^{2} c^{2} d^{4} e^{2} + 3 \, a^{2} b c d^{5} e^{2} - a^{3} d^{6} e^{2} - 2 \, b^{3} c^{4} d^{2} e f + 6 \, a b^{2} c^{3} d^{3} e f - 6 \, a^{2} b c^{2} d^{4} e f + 2 \, a^{3} c d^{5} e f + b^{3} c^{5} d f^{2} - 3 \, a b^{2} c^{4} d^{2} f^{2} + 3 \, a^{2} b c^{3} d^{3} f^{2} - a^{3} c^{2} d^{4} f^{2}} - \frac {b^{3} c d^{2} e^{3} + a b^{2} d^{3} e^{3} - 2 \, b^{3} c^{2} d e^{2} f - 2 \, a b^{2} c d^{2} e^{2} f - 2 \, a^{2} b d^{3} e^{2} f + b^{3} c^{3} e f^{2} + 2 \, a b^{2} c^{2} d e f^{2} + 2 \, a^{2} b c d^{2} e f^{2} + a^{3} d^{3} e f^{2} - a b^{2} c^{3} f^{3} - a^{3} c d^{2} f^{3} + {\left (2 \, b^{3} d^{3} e^{3} - 3 \, b^{3} c d^{2} e^{2} f - 3 \, a b^{2} d^{3} e^{2} f + b^{3} c^{2} d e f^{2} + 4 \, a b^{2} c d^{2} e f^{2} + a^{2} b d^{3} e f^{2} - a b^{2} c^{2} d f^{3} - a^{2} b c d^{2} f^{3}\right )} x}{{\left (b c - a d\right )}^{2} {\left (b e - a f\right )}^{2} {\left (d e - c f\right )}^{2} {\left (b x + a\right )} {\left (d x + c\right )}} \] Input:

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

Output:

f^4*log(abs(f*x + e))/(b^2*d^2*e^4*f - 2*b^2*c*d*e^3*f^2 - 2*a*b*d^2*e^3*f 
^2 + b^2*c^2*e^2*f^3 + 4*a*b*c*d*e^2*f^3 + a^2*d^2*e^2*f^3 - 2*a*b*c^2*e*f 
^4 - 2*a^2*c*d*e*f^4 + a^2*c^2*f^5) - (2*b^4*d*e + b^4*c*f - 3*a*b^3*d*f)* 
log(abs(b*x + a))/(b^6*c^3*e^2 - 3*a*b^5*c^2*d*e^2 + 3*a^2*b^4*c*d^2*e^2 - 
 a^3*b^3*d^3*e^2 - 2*a*b^5*c^3*e*f + 6*a^2*b^4*c^2*d*e*f - 6*a^3*b^3*c*d^2 
*e*f + 2*a^4*b^2*d^3*e*f + a^2*b^4*c^3*f^2 - 3*a^3*b^3*c^2*d*f^2 + 3*a^4*b 
^2*c*d^2*f^2 - a^5*b*d^3*f^2) + (2*b*d^4*e - 3*b*c*d^3*f + a*d^4*f)*log(ab 
s(d*x + c))/(b^3*c^3*d^3*e^2 - 3*a*b^2*c^2*d^4*e^2 + 3*a^2*b*c*d^5*e^2 - a 
^3*d^6*e^2 - 2*b^3*c^4*d^2*e*f + 6*a*b^2*c^3*d^3*e*f - 6*a^2*b*c^2*d^4*e*f 
 + 2*a^3*c*d^5*e*f + b^3*c^5*d*f^2 - 3*a*b^2*c^4*d^2*f^2 + 3*a^2*b*c^3*d^3 
*f^2 - a^3*c^2*d^4*f^2) - (b^3*c*d^2*e^3 + a*b^2*d^3*e^3 - 2*b^3*c^2*d*e^2 
*f - 2*a*b^2*c*d^2*e^2*f - 2*a^2*b*d^3*e^2*f + b^3*c^3*e*f^2 + 2*a*b^2*c^2 
*d*e*f^2 + 2*a^2*b*c*d^2*e*f^2 + a^3*d^3*e*f^2 - a*b^2*c^3*f^3 - a^3*c*d^2 
*f^3 + (2*b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f - 3*a*b^2*d^3*e^2*f + b^3*c^2*d* 
e*f^2 + 4*a*b^2*c*d^2*e*f^2 + a^2*b*d^3*e*f^2 - a*b^2*c^2*d*f^3 - a^2*b*c* 
d^2*f^3)*x)/((b*c - a*d)^2*(b*e - a*f)^2*(d*e - c*f)^2*(b*x + a)*(d*x + c) 
)
 

Mupad [B] (verification not implemented)

Time = 7.88 (sec) , antiderivative size = 801, normalized size of antiderivative = 4.31 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {\frac {f\,a^2\,d^2-e\,a\,b\,d^2+f\,b^2\,c^2-e\,b^2\,c\,d}{a^3\,c\,d^2\,f^2-a^3\,d^3\,e\,f-2\,a^2\,b\,c^2\,d\,f^2+a^2\,b\,c\,d^2\,e\,f+a^2\,b\,d^3\,e^2+a\,b^2\,c^3\,f^2+a\,b^2\,c^2\,d\,e\,f-2\,a\,b^2\,c\,d^2\,e^2-b^3\,c^3\,e\,f+b^3\,c^2\,d\,e^2}+\frac {x\,\left (-2\,e\,b^2\,d^2+c\,f\,b^2\,d+a\,f\,b\,d^2\right )}{a^3\,c\,d^2\,f^2-a^3\,d^3\,e\,f-2\,a^2\,b\,c^2\,d\,f^2+a^2\,b\,c\,d^2\,e\,f+a^2\,b\,d^3\,e^2+a\,b^2\,c^3\,f^2+a\,b^2\,c^2\,d\,e\,f-2\,a\,b^2\,c\,d^2\,e^2-b^3\,c^3\,e\,f+b^3\,c^2\,d\,e^2}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}+\frac {f^3\,\ln \left (e+f\,x\right )}{a^2\,c^2\,f^4-2\,a^2\,c\,d\,e\,f^3+a^2\,d^2\,e^2\,f^2-2\,a\,b\,c^2\,e\,f^3+4\,a\,b\,c\,d\,e^2\,f^2-2\,a\,b\,d^2\,e^3\,f+b^2\,c^2\,e^2\,f^2-2\,b^2\,c\,d\,e^3\,f+b^2\,d^2\,e^4}-\frac {\ln \left (a+b\,x\right )\,\left (b^3\,\left (c\,f+2\,d\,e\right )-3\,a\,b^2\,d\,f\right )}{-a^5\,d^3\,f^2+3\,a^4\,b\,c\,d^2\,f^2+2\,a^4\,b\,d^3\,e\,f-3\,a^3\,b^2\,c^2\,d\,f^2-6\,a^3\,b^2\,c\,d^2\,e\,f-a^3\,b^2\,d^3\,e^2+a^2\,b^3\,c^3\,f^2+6\,a^2\,b^3\,c^2\,d\,e\,f+3\,a^2\,b^3\,c\,d^2\,e^2-2\,a\,b^4\,c^3\,e\,f-3\,a\,b^4\,c^2\,d\,e^2+b^5\,c^3\,e^2}-\frac {\ln \left (c+d\,x\right )\,\left (d^3\,\left (a\,f+2\,b\,e\right )-3\,b\,c\,d^2\,f\right )}{a^3\,c^2\,d^3\,f^2-2\,a^3\,c\,d^4\,e\,f+a^3\,d^5\,e^2-3\,a^2\,b\,c^3\,d^2\,f^2+6\,a^2\,b\,c^2\,d^3\,e\,f-3\,a^2\,b\,c\,d^4\,e^2+3\,a\,b^2\,c^4\,d\,f^2-6\,a\,b^2\,c^3\,d^2\,e\,f+3\,a\,b^2\,c^2\,d^3\,e^2-b^3\,c^5\,f^2+2\,b^3\,c^4\,d\,e\,f-b^3\,c^3\,d^2\,e^2} \] Input:

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

Output:

((a^2*d^2*f + b^2*c^2*f - a*b*d^2*e - b^2*c*d*e)/(a*b^2*c^3*f^2 + a^2*b*d^ 
3*e^2 + a^3*c*d^2*f^2 + b^3*c^2*d*e^2 - a^3*d^3*e*f - b^3*c^3*e*f - 2*a*b^ 
2*c*d^2*e^2 - 2*a^2*b*c^2*d*f^2 + a*b^2*c^2*d*e*f + a^2*b*c*d^2*e*f) + (x* 
(a*b*d^2*f - 2*b^2*d^2*e + b^2*c*d*f))/(a*b^2*c^3*f^2 + a^2*b*d^3*e^2 + a^ 
3*c*d^2*f^2 + b^3*c^2*d*e^2 - a^3*d^3*e*f - b^3*c^3*e*f - 2*a*b^2*c*d^2*e^ 
2 - 2*a^2*b*c^2*d*f^2 + a*b^2*c^2*d*e*f + a^2*b*c*d^2*e*f))/(a*c + x*(a*d 
+ b*c) + b*d*x^2) + (f^3*log(e + f*x))/(a^2*c^2*f^4 + b^2*d^2*e^4 + a^2*d^ 
2*e^2*f^2 + b^2*c^2*e^2*f^2 - 2*a*b*c^2*e*f^3 - 2*a*b*d^2*e^3*f - 2*a^2*c* 
d*e*f^3 - 2*b^2*c*d*e^3*f + 4*a*b*c*d*e^2*f^2) - (log(a + b*x)*(b^3*(c*f + 
 2*d*e) - 3*a*b^2*d*f))/(b^5*c^3*e^2 - a^5*d^3*f^2 + a^2*b^3*c^3*f^2 - a^3 
*b^2*d^3*e^2 - 2*a*b^4*c^3*e*f + 2*a^4*b*d^3*e*f - 3*a*b^4*c^2*d*e^2 + 3*a 
^4*b*c*d^2*f^2 + 3*a^2*b^3*c*d^2*e^2 - 3*a^3*b^2*c^2*d*f^2 + 6*a^2*b^3*c^2 
*d*e*f - 6*a^3*b^2*c*d^2*e*f) - (log(c + d*x)*(d^3*(a*f + 2*b*e) - 3*b*c*d 
^2*f))/(a^3*d^5*e^2 - b^3*c^5*f^2 + a^3*c^2*d^3*f^2 - b^3*c^3*d^2*e^2 - 2* 
a^3*c*d^4*e*f + 2*b^3*c^4*d*e*f - 3*a^2*b*c*d^4*e^2 + 3*a*b^2*c^4*d*f^2 + 
3*a*b^2*c^2*d^3*e^2 - 3*a^2*b*c^3*d^2*f^2 - 6*a*b^2*c^3*d^2*e*f + 6*a^2*b* 
c^2*d^3*e*f)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 3635, normalized size of antiderivative = 19.54 \[ \int \frac {1}{(e+f x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*log(a + b*x)*a**3*b**2*c**3*d**2*f**3 + 6*log(a + b*x)*a**3*b**2*c** 
2*d**3*e*f**2 - 3*log(a + b*x)*a**3*b**2*c**2*d**3*f**3*x - 3*log(a + b*x) 
*a**3*b**2*c*d**4*e**2*f + 6*log(a + b*x)*a**3*b**2*c*d**4*e*f**2*x - 3*lo 
g(a + b*x)*a**3*b**2*d**5*e**2*f*x - 2*log(a + b*x)*a**2*b**3*c**4*d*f**3 
+ 6*log(a + b*x)*a**2*b**3*c**3*d**2*e*f**2 - 5*log(a + b*x)*a**2*b**3*c** 
3*d**2*f**3*x - 6*log(a + b*x)*a**2*b**3*c**2*d**3*e**2*f + 12*log(a + b*x 
)*a**2*b**3*c**2*d**3*e*f**2*x - 3*log(a + b*x)*a**2*b**3*c**2*d**3*f**3*x 
**2 + 2*log(a + b*x)*a**2*b**3*c*d**4*e**3 - 9*log(a + b*x)*a**2*b**3*c*d* 
*4*e**2*f*x + 6*log(a + b*x)*a**2*b**3*c*d**4*e*f**2*x**2 + 2*log(a + b*x) 
*a**2*b**3*d**5*e**3*x - 3*log(a + b*x)*a**2*b**3*d**5*e**2*f*x**2 + log(a 
 + b*x)*a*b**4*c**5*f**3 - log(a + b*x)*a*b**4*c**4*d*f**3*x - 3*log(a + b 
*x)*a*b**4*c**3*d**2*e**2*f + 6*log(a + b*x)*a*b**4*c**3*d**2*e*f**2*x - 2 
*log(a + b*x)*a*b**4*c**3*d**2*f**3*x**2 + 2*log(a + b*x)*a*b**4*c**2*d**3 
*e**3 - 9*log(a + b*x)*a*b**4*c**2*d**3*e**2*f*x + 6*log(a + b*x)*a*b**4*c 
**2*d**3*e*f**2*x**2 + 4*log(a + b*x)*a*b**4*c*d**4*e**3*x - 6*log(a + b*x 
)*a*b**4*c*d**4*e**2*f*x**2 + 2*log(a + b*x)*a*b**4*d**5*e**3*x**2 + log(a 
 + b*x)*b**5*c**5*f**3*x + log(a + b*x)*b**5*c**4*d*f**3*x**2 - 3*log(a + 
b*x)*b**5*c**3*d**2*e**2*f*x + 2*log(a + b*x)*b**5*c**2*d**3*e**3*x - 3*lo 
g(a + b*x)*b**5*c**2*d**3*e**2*f*x**2 + 2*log(a + b*x)*b**5*c*d**4*e**3*x* 
*2 - log(c + d*x)*a**5*c*d**4*f**3 - log(c + d*x)*a**5*d**5*f**3*x + 2*...