\(\int \frac {1}{(e+f x)^2 (a c+(b c+a d) x+b d x^2)} \, dx\) [351]

Optimal result

Integrand size = 29, antiderivative size = 136 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {f}{(b e-a f) (d e-c f) (e+f x)}+\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f)^2}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f)^2}+\frac {f (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^2 (d e-c f)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f)^2}+\frac {\frac {f (-d e+c f)}{(b e-a f) (e+f x)}+\frac {d^2 \log (c+d x)}{-b c+a d}-\frac {f (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^2}}{(d e-c f)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12, 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.

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

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

b*d*(-(f/(b*d*(b*e - a*f)*(d*e - c*f)*(e + f*x))) + (b*Log[a + b*x])/(d*(b 
*c - a*d)*(b*e - a*f)^2) - (d*Log[c + d*x])/(b*(b*c - a*d)*(d*e - c*f)^2) 
+ (f*(2*b*d*e - b*c*f - a*d*f)*Log[e + f*x])/(b*d*(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 = 1.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {b^{2} \ln \left (b x +a \right )}{\left (a f -b e \right )^{2} \left (a d -b c \right )}-\frac {f}{\left (a f -b e \right ) \left (c f -d e \right ) \left (f x +e \right )}-\frac {f \left (a d f +b c f -2 b d e \right ) \ln \left (f x +e \right )}{\left (a f -b e \right )^{2} \left (c f -d e \right )^{2}}+\frac {d^{2} \ln \left (d x +c \right )}{\left (c f -d e \right )^{2} \left (a d -b c \right )}\)	\(136\)
norman	\(\frac {f^{2} x}{e \left (a f -b e \right ) \left (c f -d e \right ) \left (f x +e \right )}+\frac {d^{2} \ln \left (d x +c \right )}{a \,c^{2} d \,f^{2}-2 a c \,d^{2} e f +a \,e^{2} d^{3}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a^{2} f^{2}-2 a b e f +b^{2} e^{2}\right ) \left (a d -b c \right )}-\frac {\left (a d f +b c f -2 b d e \right ) f \ln \left (f x +e \right )}{\left (a^{2} f^{2}-2 a b e f +b^{2} e^{2}\right ) \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}\)	\(217\)
parallelrisch	\(-\frac {2 \ln \left (d x +c \right ) x a b \,d^{2} e \,f^{3}-2 \ln \left (f x +e \right ) x a b \,d^{2} e \,f^{3}+2 \ln \left (f x +e \right ) x \,b^{2} c d e \,f^{3}-2 \ln \left (b x +a \right ) x \,b^{2} c d e \,f^{3}+\ln \left (b x +a \right ) b^{2} c^{2} e \,f^{3}+\ln \left (b x +a \right ) b^{2} d^{2} e^{3} f -\ln \left (d x +c \right ) a^{2} d^{2} e \,f^{3}-\ln \left (d x +c \right ) b^{2} d^{2} e^{3} f +a b \,d^{2} e^{2} f^{2}-b^{2} c d \,e^{2} f^{2}+a^{2} c d \,f^{4}-a^{2} d^{2} e \,f^{3}-a b \,c^{2} f^{4}+b^{2} c^{2} e \,f^{3}+\ln \left (f x +e \right ) a^{2} d^{2} e \,f^{3}-\ln \left (f x +e \right ) b^{2} c^{2} e \,f^{3}+\ln \left (b x +a \right ) x \,b^{2} c^{2} f^{4}-\ln \left (d x +c \right ) x \,a^{2} d^{2} f^{4}+\ln \left (f x +e \right ) x \,a^{2} d^{2} f^{4}-\ln \left (f x +e \right ) x \,b^{2} c^{2} f^{4}-2 \ln \left (f x +e \right ) a b \,d^{2} e^{2} f^{2}+2 \ln \left (f x +e \right ) b^{2} c d \,e^{2} f^{2}+\ln \left (b x +a \right ) x \,b^{2} d^{2} e^{2} f^{2}-\ln \left (d x +c \right ) x \,b^{2} d^{2} e^{2} f^{2}-2 \ln \left (b x +a \right ) b^{2} c d \,e^{2} f^{2}+2 \ln \left (d x +c \right ) a b \,d^{2} e^{2} f^{2}}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (a d -b c \right ) \left (a^{2} f^{2}-2 a b e f +b^{2} e^{2}\right ) \left (f x +e \right ) f}\)	\(501\)
risch	\(-\frac {f}{\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (f x +e \right )}-\frac {f^{2} \ln \left (-f x -e \right ) a d}{c^{2} f^{4} a^{2}-2 c d \,f^{3} a^{2} e +d^{2} f^{2} e^{2} a^{2}-2 b \,c^{2} f^{3} a e +4 d b c \,f^{2} a \,e^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d f \,e^{3}+b^{2} d^{2} e^{4}}-\frac {f^{2} \ln \left (-f x -e \right ) b c}{c^{2} f^{4} a^{2}-2 c d \,f^{3} a^{2} e +d^{2} f^{2} e^{2} a^{2}-2 b \,c^{2} f^{3} a e +4 d b c \,f^{2} a \,e^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d f \,e^{3}+b^{2} d^{2} e^{4}}+\frac {2 f \ln \left (-f x -e \right ) b d e}{c^{2} f^{4} a^{2}-2 c d \,f^{3} a^{2} e +d^{2} f^{2} e^{2} a^{2}-2 b \,c^{2} f^{3} a e +4 d b c \,f^{2} a \,e^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d f \,e^{3}+b^{2} d^{2} e^{4}}-\frac {b^{2} \ln \left (b x +a \right )}{d \,f^{2} a^{3}-b c \,f^{2} a^{2}-2 b d f \,a^{2} e +2 b^{2} c f a e +a \,b^{2} d \,e^{2}-b^{3} c \,e^{2}}+\frac {d^{2} \ln \left (-d x -c \right )}{a \,c^{2} d \,f^{2}-2 a c \,d^{2} e f +a \,e^{2} d^{3}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}}\)	\(539\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (136) = 272\).

Time = 45.76 (sec) , antiderivative size = 590, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {{\left (b^{2} c d - a b d^{2}\right )} e^{2} f - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f^{2} + {\left (a b c^{2} - a^{2} c d\right )} f^{3} - {\left (b^{2} d^{2} e^{3} - 2 \, b^{2} c d e^{2} f + b^{2} c^{2} e f^{2} + {\left (b^{2} d^{2} e^{2} f - 2 \, b^{2} c d e f^{2} + b^{2} c^{2} f^{3}\right )} x\right )} \log \left (b x + a\right ) + {\left (b^{2} d^{2} e^{3} - 2 \, a b d^{2} e^{2} f + a^{2} d^{2} e f^{2} + {\left (b^{2} d^{2} e^{2} f - 2 \, a b d^{2} e f^{2} + a^{2} d^{2} f^{3}\right )} x\right )} \log \left (d x + c\right ) - {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} e^{2} f - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f^{2} + {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} e f^{2} - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f^{3}\right )} x\right )} \log \left (f x + e\right )}{{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{5} - 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e^{4} f + {\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} f^{2} - 2 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} e^{2} f^{3} + {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} e f^{4} + {\left ({\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{4} f - 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e^{3} f^{2} + {\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} f^{3} - 2 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} e f^{4} + {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{5}\right )} x} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (136) = 272\).

Time = 0.05 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {b^{2} \log \left (b x + a\right )}{{\left (b^{3} c - a b^{2} d\right )} e^{2} - 2 \, {\left (a b^{2} c - a^{2} b d\right )} e f + {\left (a^{2} b c - a^{3} d\right )} f^{2}} - \frac {d^{2} \log \left (d x + c\right )}{{\left (b c d^{2} - a d^{3}\right )} e^{2} - 2 \, {\left (b c^{2} d - a c d^{2}\right )} e f + {\left (b c^{3} - a c^{2} d\right )} f^{2}} + \frac {{\left (2 \, b d e f - {\left (b c + a d\right )} f^{2}\right )} \log \left (f x + e\right )}{b^{2} d^{2} e^{4} + a^{2} c^{2} f^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} e^{3} f + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} f^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} e f^{3}} - \frac {f}{b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f + {\left (b d e^{2} f + a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} x} \] Input:

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.01 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {f^{3}}{{\left (b d e^{2} f^{2} - b c e f^{3} - a d e f^{3} + a c f^{4}\right )} {\left (f x + e\right )}} - \frac {{\left (2 \, b d e f - b c f^{2} - a d f^{2}\right )} \log \left (b d - \frac {2 \, b d e}{f x + e} + \frac {b d e^{2}}{{\left (f x + e\right )}^{2}} + \frac {b c f}{f x + e} + \frac {a d f}{f x + e} - \frac {b c e f}{{\left (f x + e\right )}^{2}} - \frac {a d e f}{{\left (f x + e\right )}^{2}} + \frac {a c f^{2}}{{\left (f x + e\right )}^{2}}\right )}{2 \, {\left (b^{2} d^{2} e^{4} - 2 \, b^{2} c d e^{3} f - 2 \, a b d^{2} e^{3} f + b^{2} c^{2} e^{2} f^{2} + 4 \, a b c d e^{2} f^{2} + a^{2} d^{2} e^{2} f^{2} - 2 \, a b c^{2} e f^{3} - 2 \, a^{2} c d e f^{3} + a^{2} c^{2} f^{4}\right )}} - \frac {i \, {\left (2 \, b^{2} d^{2} e^{2} f^{2} - 2 \, b^{2} c d e f^{3} - 2 \, a b d^{2} e f^{3} + b^{2} c^{2} f^{4} + a^{2} d^{2} f^{4}\right )} \arctan \left (\frac {-2 i \, b d e f + \frac {2 i \, b d e^{2} f}{f x + e} + i \, b c f^{2} + i \, a d f^{2} - \frac {2 i \, b c e f^{2}}{f x + e} - \frac {2 i \, a d e f^{2}}{f x + e} + \frac {2 i \, a c f^{3}}{f x + e}}{{\left | -b c f^{2} + a d f^{2} \right |}}\right )}{{\left (b^{2} d^{2} e^{4} - 2 \, b^{2} c d e^{3} f - 2 \, a b d^{2} e^{3} f + b^{2} c^{2} e^{2} f^{2} + 4 \, a b c d e^{2} f^{2} + a^{2} d^{2} e^{2} f^{2} - 2 \, a b c^{2} e f^{3} - 2 \, a^{2} c d e f^{3} + a^{2} c^{2} f^{4}\right )} {\left | -b c f^{2} + a d f^{2} \right |}} \] Input:

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

Output:

-f^3/((b*d*e^2*f^2 - b*c*e*f^3 - a*d*e*f^3 + a*c*f^4)*(f*x + e)) - 1/2*(2* 
b*d*e*f - b*c*f^2 - a*d*f^2)*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^2*d^2*e^4 - 2*b^2*c*d*e^3*f - 2*a*b*d 
^2*e^3*f + b^2*c^2*e^2*f^2 + 4*a*b*c*d*e^2*f^2 + a^2*d^2*e^2*f^2 - 2*a*b*c 
^2*e*f^3 - 2*a^2*c*d*e*f^3 + a^2*c^2*f^4) - I*(2*b^2*d^2*e^2*f^2 - 2*b^2*c 
*d*e*f^3 - 2*a*b*d^2*e*f^3 + b^2*c^2*f^4 + a^2*d^2*f^4)*arctan((-2*I*b*d*e 
*f + 2*I*b*d*e^2*f/(f*x + e) + I*b*c*f^2 + I*a*d*f^2 - 2*I*b*c*e*f^2/(f*x 
+ e) - 2*I*a*d*e*f^2/(f*x + e) + 2*I*a*c*f^3/(f*x + e))/abs(-b*c*f^2 + a*d 
*f^2))/((b^2*d^2*e^4 - 2*b^2*c*d*e^3*f - 2*a*b*d^2*e^3*f + b^2*c^2*e^2*f^2 
 + 4*a*b*c*d*e^2*f^2 + a^2*d^2*e^2*f^2 - 2*a*b*c^2*e*f^3 - 2*a^2*c*d*e*f^3 
 + a^2*c^2*f^4)*abs(-b*c*f^2 + a*d*f^2))
 

Mupad [B] (verification not implemented)

Time = 6.63 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.23 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {b^2\,\ln \left (a+b\,x\right )}{-d\,a^3\,f^2+2\,d\,a^2\,b\,e\,f+c\,a^2\,b\,f^2-d\,a\,b^2\,e^2-2\,c\,a\,b^2\,e\,f+c\,b^3\,e^2}-\frac {\ln \left (e+f\,x\right )\,\left (f^2\,\left (a\,d+b\,c\right )-2\,b\,d\,e\,f\right )}{a^2\,c^2\,f^4-2\,a^2\,c\,d\,e\,f^3+a^2\,d^2\,e^2\,f^2-2\,a\,b\,c^2\,e\,f^3+4\,a\,b\,c\,d\,e^2\,f^2-2\,a\,b\,d^2\,e^3\,f+b^2\,c^2\,e^2\,f^2-2\,b^2\,c\,d\,e^3\,f+b^2\,d^2\,e^4}-\frac {f}{\left (e+f\,x\right )\,\left (a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f\right )}+\frac {d^2\,\ln \left (c+d\,x\right )}{-b\,c^3\,f^2+2\,b\,c^2\,d\,e\,f+a\,c^2\,d\,f^2-b\,c\,d^2\,e^2-2\,a\,c\,d^2\,e\,f+a\,d^3\,e^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 789, normalized size of antiderivative = 5.80 \[ \int \frac {1}{(e+f x)^2 \left (a c+(b c+a d) x+b d x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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