\(\int \frac {1}{(e+f x)^2 (a c+(b c+a d) x+b d x^2)^3} \, dx\) [371]

Optimal result

Integrand size = 29, antiderivative size = 409 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^4}{2 (b c-a d)^3 (b e-a f)^2 (a+b x)^2}+\frac {b^4 (3 b d e+2 b c f-5 a d f)}{(b c-a d)^4 (b e-a f)^3 (a+b x)}+\frac {d^4}{2 (b c-a d)^3 (d e-c f)^2 (c+d x)^2}+\frac {d^4 (3 b d e-5 b c f+2 a d f)}{(b c-a d)^4 (d e-c f)^3 (c+d x)}-\frac {f^5}{(b e-a f)^3 (d e-c f)^3 (e+f x)}+\frac {3 b^4 \left (5 a^2 d^2 f^2-2 a b d f (3 d e+2 c f)+b^2 \left (2 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^4}-\frac {3 d^4 \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (2 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^4}+\frac {3 f^5 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^4 (d e-c f)^4} \] Output:

-1/2*b^4/(-a*d+b*c)^3/(-a*f+b*e)^2/(b*x+a)^2+b^4*(-5*a*d*f+2*b*c*f+3*b*d*e 
)/(-a*d+b*c)^4/(-a*f+b*e)^3/(b*x+a)+1/2*d^4/(-a*d+b*c)^3/(-c*f+d*e)^2/(d*x 
+c)^2+d^4*(2*a*d*f-5*b*c*f+3*b*d*e)/(-a*d+b*c)^4/(-c*f+d*e)^3/(d*x+c)-f^5/ 
(-a*f+b*e)^3/(-c*f+d*e)^3/(f*x+e)+3*b^4*(5*a^2*d^2*f^2-2*a*b*d*f*(2*c*f+3* 
d*e)+b^2*(c^2*f^2+2*c*d*e*f+2*d^2*e^2))*ln(b*x+a)/(-a*d+b*c)^5/(-a*f+b*e)^ 
4-3*d^4*(a^2*d^2*f^2+2*a*b*d*f*(-2*c*f+d*e)+b^2*(5*c^2*f^2-6*c*d*e*f+2*d^2 
*e^2))*ln(d*x+c)/(-a*d+b*c)^5/(-c*f+d*e)^4+3*f^5*(-a*d*f-b*c*f+2*b*d*e)*ln 
(f*x+e)/(-a*f+b*e)^4/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {b^4}{2 (b c-a d)^3 (b e-a f)^2 (a+b x)^2}+\frac {b^4 (3 b d e+2 b c f-5 a d f)}{(b c-a d)^4 (b e-a f)^3 (a+b x)}+\frac {d^4}{2 (b c-a d)^3 (d e-c f)^2 (c+d x)^2}+\frac {d^4 (3 b d e-5 b c f+2 a d f)}{(b c-a d)^4 (d e-c f)^3 (c+d x)}-\frac {f^5}{(b e-a f)^3 (d e-c f)^3 (e+f x)}+\frac {3 b^4 \left (5 a^2 d^2 f^2-2 a b d f (3 d e+2 c f)+b^2 \left (2 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^4}-\frac {3 d^4 \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (2 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^4}-\frac {3 f^5 (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^4 (d e-c f)^4} \] Input:

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

Output:

-1/2*b^4/((b*c - a*d)^3*(b*e - a*f)^2*(a + b*x)^2) + (b^4*(3*b*d*e + 2*b*c 
*f - 5*a*d*f))/((b*c - a*d)^4*(b*e - a*f)^3*(a + b*x)) + d^4/(2*(b*c - a*d 
)^3*(d*e - c*f)^2*(c + d*x)^2) + (d^4*(3*b*d*e - 5*b*c*f + 2*a*d*f))/((b*c 
 - a*d)^4*(d*e - c*f)^3*(c + d*x)) - f^5/((b*e - a*f)^3*(d*e - c*f)^3*(e + 
 f*x)) + (3*b^4*(5*a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e + 2*c*f) + b^2*(2*d^2*e^ 
2 + 2*c*d*e*f + c^2*f^2))*Log[a + b*x])/((b*c - a*d)^5*(b*e - a*f)^4) - (3 
*d^4*(a^2*d^2*f^2 + 2*a*b*d*f*(d*e - 2*c*f) + b^2*(2*d^2*e^2 - 6*c*d*e*f + 
 5*c^2*f^2))*Log[c + d*x])/((b*c - a*d)^5*(d*e - c*f)^4) - (3*f^5*(-2*b*d* 
e + b*c*f + a*d*f)*Log[e + f*x])/((b*e - a*f)^4*(d*e - c*f)^4)
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.06, 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.

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

\(\Big \downarrow \) 1141

\(\displaystyle b^3 d^3 \int \left (\frac {3 (2 b d e-b c f-a d f) f^6}{b^3 d^3 (b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {f^6}{b^3 d^3 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}+\frac {3 b^2 \left (\left (2 d^2 e^2+2 c d f e+c^2 f^2\right ) b^2-2 a d f (3 d e+2 c f) b+5 a^2 d^2 f^2\right )}{d^3 (b c-a d)^5 (b e-a f)^4 (a+b x)}-\frac {3 d^2 \left (\left (2 d^2 e^2-6 c d f e+5 c^2 f^2\right ) b^2+2 a d f (d e-2 c f) b+a^2 d^2 f^2\right )}{b^3 (b c-a d)^5 (d e-c f)^4 (c+d x)}-\frac {b^2 (3 b d e+2 b c f-5 a d f)}{d^3 (b c-a d)^4 (b e-a f)^3 (a+b x)^2}-\frac {d^2 (3 b d e-5 b c f+2 a d f)}{b^3 (b c-a d)^4 (d e-c f)^3 (c+d x)^2}+\frac {b^2}{d^3 (b c-a d)^3 (b e-a f)^2 (a+b x)^3}-\frac {d^2}{b^3 (b c-a d)^3 (d e-c f)^2 (c+d x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle b^3 d^3 \left (\frac {3 b \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (2 c f+3 d e)+b^2 \left (c^2 f^2+2 c d e f+2 d^2 e^2\right )\right )}{d^3 (b c-a d)^5 (b e-a f)^4}-\frac {3 d \log (c+d x) \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (5 c^2 f^2-6 c d e f+2 d^2 e^2\right )\right )}{b^3 (b c-a d)^5 (d e-c f)^4}-\frac {f^5}{b^3 d^3 (e+f x) (b e-a f)^3 (d e-c f)^3}+\frac {3 f^5 \log (e+f x) (-a d f-b c f+2 b d e)}{b^3 d^3 (b e-a f)^4 (d e-c f)^4}+\frac {d (2 a d f-5 b c f+3 b d e)}{b^3 (c+d x) (b c-a d)^4 (d e-c f)^3}+\frac {d}{2 b^3 (c+d x)^2 (b c-a d)^3 (d e-c f)^2}+\frac {b (-5 a d f+2 b c f+3 b d e)}{d^3 (a+b x) (b c-a d)^4 (b e-a f)^3}-\frac {b}{2 d^3 (a+b x)^2 (b c-a d)^3 (b e-a f)^2}\right )\)

Input:

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

Output:

b^3*d^3*(-1/2*b/(d^3*(b*c - a*d)^3*(b*e - a*f)^2*(a + b*x)^2) + (b*(3*b*d* 
e + 2*b*c*f - 5*a*d*f))/(d^3*(b*c - a*d)^4*(b*e - a*f)^3*(a + b*x)) + d/(2 
*b^3*(b*c - a*d)^3*(d*e - c*f)^2*(c + d*x)^2) + (d*(3*b*d*e - 5*b*c*f + 2* 
a*d*f))/(b^3*(b*c - a*d)^4*(d*e - c*f)^3*(c + d*x)) - f^5/(b^3*d^3*(b*e - 
a*f)^3*(d*e - c*f)^3*(e + f*x)) + (3*b*(5*a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e + 
 2*c*f) + b^2*(2*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*Log[a + b*x])/(d^3*(b*c - 
 a*d)^5*(b*e - a*f)^4) - (3*d*(a^2*d^2*f^2 + 2*a*b*d*f*(d*e - 2*c*f) + b^2 
*(2*d^2*e^2 - 6*c*d*e*f + 5*c^2*f^2))*Log[c + d*x])/(b^3*(b*c - a*d)^5*(d* 
e - c*f)^4) + (3*f^5*(2*b*d*e - b*c*f - a*d*f)*Log[e + f*x])/(b^3*d^3*(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 = 1.73 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {b^{4}}{2 \left (a f -b e \right )^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {b^{4} \left (5 a d f -2 b c f -3 b d e \right )}{\left (a f -b e \right )^{3} \left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {3 b^{4} \left (5 a^{2} d^{2} f^{2}-4 a b c d \,f^{2}-6 a b \,d^{2} e f +b^{2} c^{2} f^{2}+2 b^{2} c d e f +2 d^{2} e^{2} b^{2}\right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{4} \left (a d -b c \right )^{5}}-\frac {f^{5}}{\left (a f -b e \right )^{3} \left (c f -d e \right )^{3} \left (f x +e \right )}-\frac {3 f^{5} \left (a d f +b c f -2 b d e \right ) \ln \left (f x +e \right )}{\left (a f -b e \right )^{4} \left (c f -d e \right )^{4}}-\frac {d^{4}}{2 \left (c f -d e \right )^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {d^{4} \left (2 a d f -5 b c f +3 b d e \right )}{\left (c f -d e \right )^{3} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {3 d^{4} \left (a^{2} d^{2} f^{2}-4 a b c d \,f^{2}+2 a b \,d^{2} e f +5 b^{2} c^{2} f^{2}-6 b^{2} c d e f +2 d^{2} e^{2} b^{2}\right ) \ln \left (d x +c \right )}{\left (c f -d e \right )^{4} \left (a d -b c \right )^{5}}\)	\(420\)
norman	\(\text {Expression too large to display}\)	\(4247\)
risch	\(\text {Expression too large to display}\)	\(12744\)
parallelrisch	\(\text {Expression too large to display}\)	\(14640\)

Input:

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

Output:

1/2*b^4/(a*f-b*e)^2/(a*d-b*c)^3/(b*x+a)^2+b^4*(5*a*d*f-2*b*c*f-3*b*d*e)/(a 
*f-b*e)^3/(a*d-b*c)^4/(b*x+a)-3*b^4*(5*a^2*d^2*f^2-4*a*b*c*d*f^2-6*a*b*d^2 
*e*f+b^2*c^2*f^2+2*b^2*c*d*e*f+2*b^2*d^2*e^2)/(a*f-b*e)^4/(a*d-b*c)^5*ln(b 
*x+a)-f^5/(a*f-b*e)^3/(c*f-d*e)^3/(f*x+e)-3*f^5*(a*d*f+b*c*f-2*b*d*e)/(a*f 
-b*e)^4/(c*f-d*e)^4*ln(f*x+e)-1/2*d^4/(c*f-d*e)^2/(a*d-b*c)^3/(d*x+c)^2-d^ 
4*(2*a*d*f-5*b*c*f+3*b*d*e)/(c*f-d*e)^3/(a*d-b*c)^4/(d*x+c)+3*d^4*(a^2*d^2 
*f^2-4*a*b*c*d*f^2+2*a*b*d^2*e*f+5*b^2*c^2*f^2-6*b^2*c*d*e*f+2*b^2*d^2*e^2 
)/(c*f-d*e)^4/(a*d-b*c)^5*ln(d*x+c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7156 vs. \(2 (405) = 810\).

Time = 0.68 (sec) , antiderivative size = 7156, normalized size of antiderivative = 17.50 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3*(2*b^6*d^2*e^2 + 2*(b^6*c*d - 3*a*b^5*d^2)*e*f + (b^6*c^2 - 4*a*b^5*c*d 
+ 5*a^2*b^4*d^2)*f^2)*log(b*x + a)/((b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7* 
c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*e^4 - 4*(a*b 
^8*c^5 - 5*a^2*b^7*c^4*d + 10*a^3*b^6*c^3*d^2 - 10*a^4*b^5*c^2*d^3 + 5*a^5 
*b^4*c*d^4 - a^6*b^3*d^5)*e^3*f + 6*(a^2*b^7*c^5 - 5*a^3*b^6*c^4*d + 10*a^ 
4*b^5*c^3*d^2 - 10*a^5*b^4*c^2*d^3 + 5*a^6*b^3*c*d^4 - a^7*b^2*d^5)*e^2*f^ 
2 - 4*(a^3*b^6*c^5 - 5*a^4*b^5*c^4*d + 10*a^5*b^4*c^3*d^2 - 10*a^6*b^3*c^2 
*d^3 + 5*a^7*b^2*c*d^4 - a^8*b*d^5)*e*f^3 + (a^4*b^5*c^5 - 5*a^5*b^4*c^4*d 
 + 10*a^6*b^3*c^3*d^2 - 10*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^9*d^5)*f^4) 
 - 3*(2*b^2*d^6*e^2 - 2*(3*b^2*c*d^5 - a*b*d^6)*e*f + (5*b^2*c^2*d^4 - 4*a 
*b*c*d^5 + a^2*d^6)*f^2)*log(d*x + c)/((b^5*c^5*d^4 - 5*a*b^4*c^4*d^5 + 10 
*a^2*b^3*c^3*d^6 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9)*e^4 - 4*( 
b^5*c^6*d^3 - 5*a*b^4*c^5*d^4 + 10*a^2*b^3*c^4*d^5 - 10*a^3*b^2*c^3*d^6 + 
5*a^4*b*c^2*d^7 - a^5*c*d^8)*e^3*f + 6*(b^5*c^7*d^2 - 5*a*b^4*c^6*d^3 + 10 
*a^2*b^3*c^5*d^4 - 10*a^3*b^2*c^4*d^5 + 5*a^4*b*c^3*d^6 - a^5*c^2*d^7)*e^2 
*f^2 - 4*(b^5*c^8*d - 5*a*b^4*c^7*d^2 + 10*a^2*b^3*c^6*d^3 - 10*a^3*b^2*c^ 
5*d^4 + 5*a^4*b*c^4*d^5 - a^5*c^3*d^6)*e*f^3 + (b^5*c^9 - 5*a*b^4*c^8*d + 
10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 + 5*a^4*b*c^5*d^4 - a^5*c^4*d^5)*f 
^4) + 3*(2*b*d*e*f^5 - (b*c + a*d)*f^6)*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*...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 4279, normalized size of antiderivative = 10.46 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-f^11/((b^3*d^3*e^6*f^6 - 3*b^3*c*d^2*e^5*f^7 - 3*a*b^2*d^3*e^5*f^7 + 3*b^ 
3*c^2*d*e^4*f^8 + 9*a*b^2*c*d^2*e^4*f^8 + 3*a^2*b*d^3*e^4*f^8 - b^3*c^3*e^ 
3*f^9 - 9*a*b^2*c^2*d*e^3*f^9 - 9*a^2*b*c*d^2*e^3*f^9 - a^3*d^3*e^3*f^9 + 
3*a*b^2*c^3*e^2*f^10 + 9*a^2*b*c^2*d*e^2*f^10 + 3*a^3*c*d^2*e^2*f^10 - 3*a 
^2*b*c^3*e*f^11 - 3*a^3*c^2*d*e*f^11 + a^3*c^3*f^12)*(f*x + e)) - 3/2*(2*b 
*d*e*f^5 - b*c*f^6 - a*d*f^6)*log(b*d - 2*b*d*e/(f*x + e) + b*d*e^2/(f*x + 
 e)^2 + b*c*f/(f*x + e) + a*d*f/(f*x + e) - b*c*e*f/(f*x + e)^2 - a*d*e*f/ 
(f*x + e)^2 + a*c*f^2/(f*x + e)^2)/(b^4*d^4*e^8 - 4*b^4*c*d^3*e^7*f - 4*a* 
b^3*d^4*e^7*f + 6*b^4*c^2*d^2*e^6*f^2 + 16*a*b^3*c*d^3*e^6*f^2 + 6*a^2*b^2 
*d^4*e^6*f^2 - 4*b^4*c^3*d*e^5*f^3 - 24*a*b^3*c^2*d^2*e^5*f^3 - 24*a^2*b^2 
*c*d^3*e^5*f^3 - 4*a^3*b*d^4*e^5*f^3 + b^4*c^4*e^4*f^4 + 16*a*b^3*c^3*d*e^ 
4*f^4 + 36*a^2*b^2*c^2*d^2*e^4*f^4 + 16*a^3*b*c*d^3*e^4*f^4 + a^4*d^4*e^4* 
f^4 - 4*a*b^3*c^4*e^3*f^5 - 24*a^2*b^2*c^3*d*e^3*f^5 - 24*a^3*b*c^2*d^2*e^ 
3*f^5 - 4*a^4*c*d^3*e^3*f^5 + 6*a^2*b^2*c^4*e^2*f^6 + 16*a^3*b*c^3*d*e^2*f 
^6 + 6*a^4*c^2*d^2*e^2*f^6 - 4*a^3*b*c^4*e*f^7 - 4*a^4*c^3*d*e*f^7 + a^4*c 
^4*f^8) - 3*I*(4*b^6*d^6*e^6*f^2 - 12*b^6*c*d^5*e^5*f^3 - 12*a*b^5*d^6*e^5 
*f^3 + 10*b^6*c^2*d^4*e^4*f^4 + 40*a*b^5*c*d^5*e^4*f^4 + 10*a^2*b^4*d^6*e^ 
4*f^4 - 40*a*b^5*c^2*d^4*e^3*f^5 - 40*a^2*b^4*c*d^5*e^3*f^5 + 60*a^2*b^4*c 
^2*d^4*e^2*f^6 - 2*b^6*c^5*d*e*f^7 + 10*a*b^5*c^4*d^2*e*f^7 - 20*a^2*b^4*c 
^3*d^3*e*f^7 - 20*a^3*b^3*c^2*d^4*e*f^7 + 10*a^4*b^2*c*d^5*e*f^7 - 2*a^...
 

Mupad [B] (verification not implemented)

Time = 15.45 (sec) , antiderivative size = 57828, normalized size of antiderivative = 141.39 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((117*a^3*b^8*c^6*d^5*f^11 - 9*b^11*c^9*d^2*f^11 - 36*b^11*d^11* 
e^9*f^2 - 117*a^2*b^9*c^7*d^4*f^11 - 9*a^9*b^2*d^11*f^11 - 45*a^4*b^7*c^5* 
d^6*f^11 - 45*a^5*b^6*c^4*d^7*f^11 + 117*a^6*b^5*c^3*d^8*f^11 - 117*a^7*b^ 
4*c^2*d^9*f^11 - 180*a^2*b^9*d^11*e^7*f^4 + 36*a^3*b^8*d^11*e^6*f^5 + 63*a 
^4*b^7*d^11*e^5*f^6 - 45*a^6*b^5*d^11*e^3*f^8 + 9*a^7*b^4*d^11*e^2*f^9 - 1 
80*b^11*c^2*d^9*e^7*f^4 + 36*b^11*c^3*d^8*e^6*f^5 + 63*b^11*c^4*d^7*e^5*f^ 
6 - 45*b^11*c^6*d^5*e^3*f^8 + 9*b^11*c^7*d^4*e^2*f^9 + 54*a*b^10*c^8*d^3*f 
^11 + 54*a^8*b^3*c*d^10*f^11 + 144*a*b^10*d^11*e^8*f^3 + 18*a^8*b^3*d^11*e 
*f^10 + 144*b^11*c*d^10*e^8*f^3 + 18*b^11*c^8*d^3*e*f^10 - 648*a*b^10*c*d^ 
10*e^7*f^4 - 144*a*b^10*c^7*d^4*e*f^10 - 144*a^7*b^4*c*d^10*e*f^10 + 972*a 
*b^10*c^2*d^9*e^6*f^5 - 432*a*b^10*c^3*d^8*e^5*f^6 - 216*a*b^10*c^4*d^7*e^ 
4*f^7 + 180*a*b^10*c^5*d^6*e^3*f^8 + 81*a*b^10*c^6*d^5*e^2*f^9 + 972*a^2*b 
^9*c*d^10*e^6*f^5 + 351*a^2*b^9*c^6*d^5*e*f^10 - 432*a^3*b^8*c*d^10*e^5*f^ 
6 - 360*a^3*b^8*c^5*d^6*e*f^10 - 216*a^4*b^7*c*d^10*e^4*f^7 + 234*a^4*b^7* 
c^4*d^7*e*f^10 + 180*a^5*b^6*c*d^10*e^3*f^8 - 360*a^5*b^6*c^3*d^8*e*f^10 + 
 81*a^6*b^5*c*d^10*e^2*f^9 + 351*a^6*b^5*c^2*d^9*e*f^10 - 1782*a^2*b^9*c^2 
*d^9*e^5*f^6 + 1224*a^2*b^9*c^3*d^8*e^4*f^7 + 9*a^2*b^9*c^4*d^7*e^3*f^8 - 
405*a^2*b^9*c^5*d^6*e^2*f^9 + 1224*a^3*b^8*c^2*d^9*e^4*f^7 - 1296*a^3*b^8* 
c^3*d^8*e^3*f^8 + 459*a^3*b^8*c^4*d^7*e^2*f^9 + 9*a^4*b^7*c^2*d^9*e^3*f^8 
+ 459*a^4*b^7*c^3*d^8*e^2*f^9 - 405*a^5*b^6*c^2*d^9*e^2*f^9)/(56*a^3*b^...
 

Reduce [B] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 31106, normalized size of antiderivative = 76.05 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 60*log(a + b*x)*a**5*b**4*c**6*d**3*e*f**7 - 60*log(a + b*x)*a**5*b**4 
*c**6*d**3*f**8*x + 240*log(a + b*x)*a**5*b**4*c**5*d**4*e**2*f**6 + 120*l 
og(a + b*x)*a**5*b**4*c**5*d**4*e*f**7*x - 120*log(a + b*x)*a**5*b**4*c**5 
*d**4*f**8*x**2 - 360*log(a + b*x)*a**5*b**4*c**4*d**5*e**3*f**5 + 120*log 
(a + b*x)*a**5*b**4*c**4*d**5*e**2*f**6*x + 420*log(a + b*x)*a**5*b**4*c** 
4*d**5*e*f**7*x**2 - 60*log(a + b*x)*a**5*b**4*c**4*d**5*f**8*x**3 + 240*l 
og(a + b*x)*a**5*b**4*c**3*d**6*e**4*f**4 - 480*log(a + b*x)*a**5*b**4*c** 
3*d**6*e**3*f**5*x - 480*log(a + b*x)*a**5*b**4*c**3*d**6*e**2*f**6*x**2 + 
 240*log(a + b*x)*a**5*b**4*c**3*d**6*e*f**7*x**3 - 60*log(a + b*x)*a**5*b 
**4*c**2*d**7*e**5*f**3 + 420*log(a + b*x)*a**5*b**4*c**2*d**7*e**4*f**4*x 
 + 120*log(a + b*x)*a**5*b**4*c**2*d**7*e**3*f**5*x**2 - 360*log(a + b*x)* 
a**5*b**4*c**2*d**7*e**2*f**6*x**3 - 120*log(a + b*x)*a**5*b**4*c*d**8*e** 
5*f**3*x + 120*log(a + b*x)*a**5*b**4*c*d**8*e**4*f**4*x**2 + 240*log(a + 
b*x)*a**5*b**4*c*d**8*e**3*f**5*x**3 - 60*log(a + b*x)*a**5*b**4*d**9*e**5 
*f**3*x**2 - 60*log(a + b*x)*a**5*b**4*d**9*e**4*f**4*x**3 - 12*log(a + b* 
x)*a**4*b**5*c**7*d**2*e*f**7 - 12*log(a + b*x)*a**4*b**5*c**7*d**2*f**8*x 
 + 90*log(a + b*x)*a**4*b**5*c**6*d**3*e**2*f**6 - 54*log(a + b*x)*a**4*b* 
*5*c**6*d**3*e*f**7*x - 144*log(a + b*x)*a**4*b**5*c**6*d**3*f**8*x**2 - 2 
40*log(a + b*x)*a**4*b**5*c**5*d**4*e**3*f**5 + 420*log(a + b*x)*a**4*b**5 
*c**5*d**4*e**2*f**6*x + 408*log(a + b*x)*a**4*b**5*c**5*d**4*e*f**7*x*...