\(\int \frac {A+B x}{(d+e x) (b x+c x^2)^2} \, dx\) [42]

Optimal result

Integrand size = 24, antiderivative size = 147 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d x}+\frac {c (b B-A c)}{b^2 (c d-b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \] Output:

-A/b^2/d/x+c*(-A*c+B*b)/b^2/(-b*e+c*d)/(c*x+b)+(-A*b*e-2*A*c*d+B*b*d)*ln(x 
)/b^3/d^2+c*(2*A*c^2*d+2*b^2*B*e-b*c*(3*A*e+B*d))*ln(c*x+b)/b^3/(-b*e+c*d) 
^2-e^2*(-A*e+B*d)*ln(e*x+d)/d^2/(-b*e+c*d)^2
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d x}+\frac {c (-b B+A c)}{b^2 (-c d+b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}+\frac {e^2 (-B d+A e) \log (d+e x)}{d^2 (c d-b e)^2} \] Input:

Integrate[(A + B*x)/((d + e*x)*(b*x + c*x^2)^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1206, 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.

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

\(\Big \downarrow \) 1206

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x)/((d + e*x)*(b*x + c*x^2)^2),x]
 

Output:

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

Defintions of rubi rules used

rule 1206

Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.) 
*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[x^p*(d + e*x)^m*(f + g*x)^n 
*(b + c*x)^p, x], x] /; FreeQ[{b, c, d, e, f, g}, x] && ILtQ[p, -1] && Inte 
gersQ[m, n]
 

rule 2009

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {c \left (3 A b c e -2 A \,c^{2} d -2 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{2}}+\frac {\left (A c -B b \right ) c}{b^{2} \left (b e -c d \right ) \left (c x +b \right )}+\frac {\left (A e -B d \right ) e^{2} \ln \left (e x +d \right )}{d^{2} \left (b e -c d \right )^{2}}-\frac {A}{b^{2} d x}+\frac {\left (-A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{2}}\)	\(147\)
norman	\(\frac {\frac {\left (A b c e -2 A \,c^{2} d +B b c d \right ) c \,x^{2}}{d \,b^{3} \left (b e -c d \right )}-\frac {A}{b d}}{x \left (c x +b \right )}+\frac {e^{2} \left (A e -B d \right ) \ln \left (e x +d \right )}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (A b e +2 A c d -B b d \right ) \ln \left (x \right )}{b^{3} d^{2}}-\frac {c \left (3 A b c e -2 A \,c^{2} d -2 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}\)	\(192\)
risch	\(\frac {-\frac {c \left (A b e -2 A c d +B b d \right ) x}{b^{2} d \left (b e -c d \right )}-\frac {A}{b d}}{x \left (c x +b \right )}-\frac {3 c^{2} \ln \left (c x +b \right ) A e}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2}}+\frac {2 c^{3} \ln \left (c x +b \right ) A d}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}+\frac {2 c \ln \left (c x +b \right ) B e}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b}-\frac {c^{2} \ln \left (c x +b \right ) B d}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2}}+\frac {e^{3} \ln \left (-e x -d \right ) A}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {e^{2} \ln \left (-e x -d \right ) B}{d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\ln \left (-x \right ) A e}{b^{2} d^{2}}-\frac {2 \ln \left (-x \right ) A c}{b^{3} d}+\frac {\ln \left (-x \right ) B}{b^{2} d}\)	\(332\)
parallelrisch	\(-\frac {A \,b^{2} c^{3} d^{3}-2 A \ln \left (c x +b \right ) x^{2} c^{5} d^{3}+3 A \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{2} e -B \ln \left (x \right ) x \,b^{4} c d \,e^{2}+2 B \ln \left (x \right ) x \,b^{3} c^{2} d^{2} e +B \ln \left (e x +d \right ) x^{2} b^{3} c^{2} d \,e^{2}-2 B \ln \left (c x +b \right ) x \,b^{3} c^{2} d^{2} e +B \ln \left (e x +d \right ) x \,b^{4} c d \,e^{2}+A \,b^{4} c d \,e^{2}-2 A \,b^{3} c^{2} d^{2} e -3 A \ln \left (x \right ) x^{2} b \,c^{4} d^{2} e +3 A \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{2} e -B \ln \left (x \right ) x^{2} b^{3} c^{2} d \,e^{2}+2 B \ln \left (x \right ) x^{2} b^{2} c^{3} d^{2} e -2 B \ln \left (c x +b \right ) x^{2} b^{2} c^{3} d^{2} e -3 A \ln \left (x \right ) x \,b^{2} c^{3} d^{2} e +2 A \ln \left (x \right ) x^{2} c^{5} d^{3}+2 A x b \,c^{4} d^{3}-B x \,b^{2} c^{3} d^{3}+A x \,b^{3} c^{2} d \,e^{2}-3 A x \,b^{2} c^{3} d^{2} e +B x \,b^{3} c^{2} d^{2} e +A \ln \left (x \right ) x \,b^{4} c \,e^{3}-A \ln \left (e x +d \right ) x \,b^{4} c \,e^{3}+A \ln \left (x \right ) x^{2} b^{3} c^{2} e^{3}-A \ln \left (e x +d \right ) x^{2} b^{3} c^{2} e^{3}-2 A \ln \left (c x +b \right ) x b \,c^{4} d^{3}-B \ln \left (x \right ) x \,b^{2} c^{3} d^{3}+B \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{3}-B \ln \left (x \right ) x^{2} b \,c^{4} d^{3}+B \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{3}+2 A \ln \left (x \right ) x b \,c^{4} d^{3}}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \left (c x +b \right ) x \,b^{3} c \,d^{2}}\)	\(556\)

Input:

int((B*x+A)/(e*x+d)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

-c*(3*A*b*c*e-2*A*c^2*d-2*B*b^2*e+B*b*c*d)/b^3/(b*e-c*d)^2*ln(c*x+b)+(A*c- 
B*b)*c/b^2/(b*e-c*d)/(c*x+b)+(A*e-B*d)*e^2/d^2/(b*e-c*d)^2*ln(e*x+d)-A/b^2 
/d/x+(-A*b*e-2*A*c*d+B*b*d)*ln(x)/b^3/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (147) = 294\).

Time = 18.10 (sec) , antiderivative size = 450, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A b^{2} c^{2} d^{3} - 2 \, A b^{3} c d^{2} e + A b^{4} d e^{2} + {\left (A b^{3} c d e^{2} - {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} + {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x + {\left ({\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3}\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x\right )} \log \left (e x + d\right ) - {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3} + {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} + {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \] Input:

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((B*x+A)/(e*x+d)/(c*x**2+b*x)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {{\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} - \frac {A b c d - A b^{2} e - {\left (A b c e + {\left (B b c - 2 \, A c^{2}\right )} d\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} + {\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac {{\left (A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3} d^{2}} \] Input:

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^2,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (B b c^{3} d - 2 \, A c^{4} d - 2 \, B b^{2} c^{2} e + 3 \, A b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} - \frac {{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} + \frac {{\left (B b d - 2 \, A c d - A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} - {\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2} {\left (c x + b\right )} b^{2} d^{2} x} \] Input:

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^2,x, algorithm="giac")
 

Output:

-(B*b*c^3*d - 2*A*c^4*d - 2*B*b^2*c^2*e + 3*A*b*c^3*e)*log(abs(c*x + b))/( 
b^3*c^3*d^2 - 2*b^4*c^2*d*e + b^5*c*e^2) - (B*d*e^3 - A*e^4)*log(abs(e*x + 
 d))/(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3) + (B*b*d - 2*A*c*d - A*b*e) 
*log(abs(x))/(b^3*d^2) - (A*b*c^2*d^3 - 2*A*b^2*c*d^2*e + A*b^3*d*e^2 - (B 
*b*c^2*d^3 - 2*A*c^3*d^3 - B*b^2*c*d^2*e + 3*A*b*c^2*d^2*e - A*b^2*c*d*e^2 
)*x)/((c*d - b*e)^2*(c*x + b)*b^2*d^2*x)
 

Mupad [B] (verification not implemented)

Time = 11.33 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {\ln \left (b+c\,x\right )\,\left (d\,\left (2\,A\,c^3-B\,b\,c^2\right )-3\,A\,b\,c^2\,e+2\,B\,b^2\,c\,e\right )}{b^5\,e^2-2\,b^4\,c\,d\,e+b^3\,c^2\,d^2}-\frac {\frac {A}{b\,d}+\frac {x\,\left (A\,b\,c\,e-2\,A\,c^2\,d+B\,b\,c\,d\right )}{b^2\,d\,\left (b\,e-c\,d\right )}}{c\,x^2+b\,x}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^3-B\,d\,e^2\right )}{b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4}-\frac {\ln \left (x\right )\,\left (b\,\left (A\,e-B\,d\right )+2\,A\,c\,d\right )}{b^3\,d^2} \] Input:

int((A + B*x)/((b*x + c*x^2)^2*(d + e*x)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 553, normalized size of antiderivative = 3.76 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {\mathrm {log}\left (x \right ) b^{4} c d \,e^{2} x^{2}+\mathrm {log}\left (x \right ) b^{3} c^{2} d^{3} x +\mathrm {log}\left (x \right ) b^{2} c^{3} d^{3} x^{2}+a \,b^{2} c^{2} d \,e^{2} x^{2}+b^{3} c^{2} d^{2} e \,x^{2}+2 \,\mathrm {log}\left (c x +b \right ) a b \,c^{3} d^{3} x +2 \,\mathrm {log}\left (c x +b \right ) b^{4} c \,d^{2} e x +2 \,\mathrm {log}\left (c x +b \right ) b^{3} c^{2} d^{2} e \,x^{2}-2 \,\mathrm {log}\left (x \right ) a b \,c^{3} d^{3} x -a \,b^{4} d \,e^{2}-a \,b^{2} c^{2} d^{3}-b^{2} c^{3} d^{3} x^{2}+2 a \,c^{4} d^{3} x^{2}+2 \,\mathrm {log}\left (c x +b \right ) a \,c^{4} d^{3} x^{2}-2 \,\mathrm {log}\left (x \right ) a \,c^{4} d^{3} x^{2}+2 a \,b^{3} c \,d^{2} e -\mathrm {log}\left (x \right ) a \,b^{3} c \,e^{3} x^{2}+\mathrm {log}\left (x \right ) b^{5} d \,e^{2} x -\mathrm {log}\left (c x +b \right ) b^{2} c^{3} d^{3} x^{2}-2 \,\mathrm {log}\left (x \right ) b^{4} c \,d^{2} e x -2 \,\mathrm {log}\left (x \right ) b^{3} c^{2} d^{2} e \,x^{2}-3 a b \,c^{3} d^{2} e \,x^{2}-3 \,\mathrm {log}\left (c x +b \right ) a \,b^{2} c^{2} d^{2} e x -3 \,\mathrm {log}\left (c x +b \right ) a b \,c^{3} d^{2} e \,x^{2}+3 \,\mathrm {log}\left (x \right ) a \,b^{2} c^{2} d^{2} e x +3 \,\mathrm {log}\left (x \right ) a b \,c^{3} d^{2} e \,x^{2}-\mathrm {log}\left (c x +b \right ) b^{3} c^{2} d^{3} x +\mathrm {log}\left (e x +d \right ) a \,b^{4} e^{3} x +\mathrm {log}\left (e x +d \right ) a \,b^{3} c \,e^{3} x^{2}-\mathrm {log}\left (e x +d \right ) b^{5} d \,e^{2} x -\mathrm {log}\left (e x +d \right ) b^{4} c d \,e^{2} x^{2}-\mathrm {log}\left (x \right ) a \,b^{4} e^{3} x}{b^{3} d^{2} x \left (b^{2} c \,e^{2} x -2 b \,c^{2} d e x +c^{3} d^{2} x +b^{3} e^{2}-2 b^{2} c d e +b \,c^{2} d^{2}\right )} \] Input:

int((B*x+A)/(e*x+d)/(c*x^2+b*x)^2,x)
 

Output:

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