\(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^2} \, dx\) [43]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (c d-b e)^2 (b+c x)}+\frac {e^2 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {(b B d-2 A c d-2 A b e) \log (x)}{b^3 d^3}+\frac {c^2 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e)) \log (d+e x)}{d^3 (c d-b e)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (c d-b e)^2 (b+c x)}+\frac {e^2 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {(b B d-2 A c d-2 A b e) \log (x)}{b^3 d^3}-\frac {c^2 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right ) \log (b+c x)}{b^3 (-c d+b e)^3}-\frac {e^2 (B d (3 c d-b e)+2 A e (-2 c d+b e)) \log (d+e x)}{d^3 (c d-b e)^3} \] Input:

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

Output:

-(A/(b^2*d^2*x)) + (c^2*(b*B - A*c))/(b^2*(c*d - b*e)^2*(b + c*x)) + (e^2* 
(B*d - A*e))/(d^2*(c*d - b*e)^2*(d + e*x)) + ((b*B*d - 2*A*c*d - 2*A*b*e)* 
Log[x])/(b^3*d^3) - (c^2*(2*A*c^2*d + 3*b^2*B*e - b*c*(B*d + 4*A*e))*Log[b 
 + c*x])/(b^3*(-(c*d) + b*e)^3) - (e^2*(B*d*(3*c*d - b*e) + 2*A*e*(-2*c*d 
+ b*e))*Log[d + e*x])/(d^3*(c*d - b*e)^3)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 201, 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.

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

\(\Big \downarrow \) 1206

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}+\frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac {A}{b^2 d^2 x}+\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (c d-b e)^3}+\frac {e^2 \log (d+e x) (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2}\)

Input:

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

Output:

-(A/(b^2*d^2*x)) + (c^2*(b*B - A*c))/(b^2*(c*d - b*e)^2*(b + c*x)) + (e^2* 
(B*d - A*e))/(d^2*(c*d - b*e)^2*(d + e*x)) + ((b*B*d - 2*A*c*d - 2*A*b*e)* 
Log[x])/(b^3*d^3) + (c^2*(2*A*c^2*d + 3*b^2*B*e - b*c*(B*d + 4*A*e))*Log[b 
 + c*x])/(b^3*(c*d - b*e)^3) + (e^2*(2*A*e*(2*c*d - b*e) - B*d*(3*c*d - b* 
e))*Log[d + e*x])/(d^3*(c*d - b*e)^3)
 

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 = 203, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {c^{2} \left (4 A b c e -2 A \,c^{2} d -3 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{3}}-\frac {\left (A c -B b \right ) c^{2}}{b^{2} \left (b e -c d \right )^{2} \left (c x +b \right )}+\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}-\frac {\left (A e -B d \right ) e^{2}}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )}-\frac {A}{b^{2} d^{2} x}+\frac {\left (-2 A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{3}}\)	\(203\)
norman	\(\frac {\frac {\left (2 A \,b^{4} e^{4}-A \,b^{3} c d \,e^{3}-A b \,c^{3} d^{3} e +2 A \,c^{4} d^{4}-B \,b^{4} d \,e^{3}-B b \,c^{3} d^{4}\right ) x^{2}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (2 A \,b^{3} e^{3}-A \,b^{2} c d \,e^{2}-A b \,c^{2} d^{2} e +2 A \,c^{3} d^{3}-B \,b^{3} d \,e^{2}-B b \,c^{2} d^{3}\right ) c e \,x^{3}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {A}{b d}}{\left (c x +b \right ) x \left (e x +d \right )}+\frac {c^{2} \left (4 A b c e -2 A \,c^{2} d -3 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{3}}+\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}-\frac {\left (2 A b e +2 A c d -B b d \right ) \ln \left (x \right )}{b^{3} d^{3}}\)	\(401\)
risch	\(\frac {-\frac {c e \left (2 A \,b^{2} e^{2}-2 A b c d e +2 A \,c^{2} d^{2}-B \,b^{2} d e -B b c \,d^{2}\right ) x^{2}}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (2 A \,b^{3} e^{3}-A \,b^{2} c d \,e^{2}-A b \,c^{2} d^{2} e +2 A \,c^{3} d^{3}-B \,b^{3} d \,e^{2}-B b \,c^{2} d^{3}\right ) x}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {A}{b d}}{\left (c x +b \right ) x \left (e x +d \right )}-\frac {2 \ln \left (-x \right ) A e}{b^{2} d^{3}}-\frac {2 \ln \left (-x \right ) A c}{b^{3} d^{2}}+\frac {\ln \left (-x \right ) B}{b^{2} d^{2}}+\frac {2 e^{4} \ln \left (-e x -d \right ) A b}{d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}-\frac {4 e^{3} \ln \left (-e x -d \right ) A c}{d^{2} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}-\frac {e^{3} \ln \left (-e x -d \right ) B b}{d^{2} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}+\frac {3 e^{2} \ln \left (-e x -d \right ) B c}{d \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}+\frac {4 c^{3} \ln \left (c x +b \right ) A e}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{2}}-\frac {2 c^{4} \ln \left (c x +b \right ) A d}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{3}}-\frac {3 c^{2} \ln \left (c x +b \right ) B e}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b}+\frac {c^{3} \ln \left (c x +b \right ) B d}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{2}}\)	\(684\)
parallelrisch	\(\text {Expression too large to display}\)	\(1371\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (201) = 402\).

Time = 67.87 (sec) , antiderivative size = 1034, normalized size of antiderivative = 5.14 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-(A*b^2*c^3*d^5 - 3*A*b^3*c^2*d^4*e + 3*A*b^4*c*d^3*e^2 - A*b^5*d^2*e^3 - 
(4*A*b^2*c^3*d^3*e^2 + 2*A*b^4*c*d*e^4 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*e - ( 
B*b^4*c + 4*A*b^3*c^2)*d^2*e^3)*x^2 - (B*b^4*c*d^3*e^2 + 2*A*b^5*d*e^4 + ( 
B*b^2*c^3 - 2*A*b*c^4)*d^5 - (B*b^3*c^2 - 3*A*b^2*c^3)*d^4*e - (B*b^5 + 3* 
A*b^4*c)*d^2*e^3)*x + (((B*b*c^4 - 2*A*c^5)*d^4*e - (3*B*b^2*c^3 - 4*A*b*c 
^4)*d^3*e^2)*x^3 + ((B*b*c^4 - 2*A*c^5)*d^5 - 2*(B*b^2*c^3 - A*b*c^4)*d^4* 
e - (3*B*b^3*c^2 - 4*A*b^2*c^3)*d^3*e^2)*x^2 + ((B*b^2*c^3 - 2*A*b*c^4)*d^ 
5 - (3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e)*x)*log(c*x + b) + ((3*B*b^3*c^2*d^2 
*e^3 + 2*A*b^4*c*e^5 - (B*b^4*c + 4*A*b^3*c^2)*d*e^4)*x^3 + (3*B*b^3*c^2*d 
^3*e^2 + 2*A*b^5*e^5 + 2*(B*b^4*c - 2*A*b^3*c^2)*d^2*e^3 - (B*b^5 + 2*A*b^ 
4*c)*d*e^4)*x^2 + (3*B*b^4*c*d^3*e^2 + 2*A*b^5*d*e^4 - (B*b^5 + 4*A*b^4*c) 
*d^2*e^3)*x)*log(e*x + d) - ((3*B*b^3*c^2*d^2*e^3 + 2*A*b^4*c*e^5 + (B*b*c 
^4 - 2*A*c^5)*d^4*e - (3*B*b^2*c^3 - 4*A*b*c^4)*d^3*e^2 - (B*b^4*c + 4*A*b 
^3*c^2)*d*e^4)*x^3 + (4*A*b^2*c^3*d^3*e^2 + 2*A*b^5*e^5 + (B*b*c^4 - 2*A*c 
^5)*d^5 - 2*(B*b^2*c^3 - A*b*c^4)*d^4*e + 2*(B*b^4*c - 2*A*b^3*c^2)*d^2*e^ 
3 - (B*b^5 + 2*A*b^4*c)*d*e^4)*x^2 + (3*B*b^4*c*d^3*e^2 + 2*A*b^5*d*e^4 + 
(B*b^2*c^3 - 2*A*b*c^4)*d^5 - (3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e - (B*b^5 + 
 4*A*b^4*c)*d^2*e^3)*x)*log(x))/((b^3*c^4*d^6*e - 3*b^4*c^3*d^5*e^2 + 3*b^ 
5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x^3 + (b^3*c^4*d^7 - 2*b^4*c^3*d^6*e + 2*b^ 
6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*c^3*d^7 - 3*b^5*c^2*d^6*e + 3*b^6...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (201) = 402\).

Time = 0.05 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.32 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d - {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} - \frac {{\left (3 \, B c d^{2} e^{2} + 2 \, A b e^{4} - {\left (B b + 4 \, A c\right )} d e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} + {\left (2 \, A b^{2} c e^{3} - {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} e - {\left (B b^{2} c + 2 \, A b c^{2}\right )} d e^{2}\right )} x^{2} - {\left (A b c^{2} d^{2} e - 2 \, A b^{3} e^{3} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{3} + {\left (B b^{3} + A b^{2} c\right )} d e^{2}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {{\left (2 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3} d^{3}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (201) = 402\).

Time = 0.27 (sec) , antiderivative size = 678, normalized size of antiderivative = 3.37 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=\frac {{\left (3 \, B c d^{2} e^{2} - B b d e^{3} - 4 \, A c d e^{3} + 2 \, A b e^{4}\right )} \log \left ({\left | c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}} \right |}\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}} + \frac {\frac {B d e^{6}}{e x + d} - \frac {A e^{7}}{e x + d}}{c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}} + \frac {{\left (2 \, B b c^{3} d^{4} e^{2} - 4 \, A c^{4} d^{4} e^{2} - 6 \, B b^{2} c^{2} d^{3} e^{3} + 8 \, A b c^{3} d^{3} e^{3} + 3 \, B b^{3} c d^{2} e^{4} - B b^{4} d e^{5} - 4 \, A b^{3} c d e^{5} + 2 \, A b^{4} e^{6}\right )} \log \left (\frac {{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{e x + d} - b e^{2} + \frac {2 \, b d e^{2}}{e x + d} - e^{2} {\left | b \right |} \right |}}{{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{e x + d} - b e^{2} + \frac {2 \, b d e^{2}}{e x + d} + e^{2} {\left | b \right |} \right |}}\right )}{2 \, {\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} e^{2} {\left | b \right |}} + \frac {\frac {B b c^{3} d^{3} e - 2 \, A c^{4} d^{3} e + 3 \, A b c^{3} d^{2} e^{2} - 3 \, A b^{2} c^{2} d e^{3} + A b^{3} c e^{4}}{c d^{2} - b d e} - \frac {B b c^{3} d^{4} e^{2} - 2 \, A c^{4} d^{4} e^{2} + 4 \, A b c^{3} d^{3} e^{3} - 6 \, A b^{2} c^{2} d^{2} e^{4} + 4 \, A b^{3} c d e^{5} - A b^{4} e^{6}}{{\left (c d^{2} - b d e\right )} {\left (e x + d\right )} e}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}\right )} d^{2}} \] Input:

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

Output:

1/2*(3*B*c*d^2*e^2 - B*b*d*e^3 - 4*A*c*d*e^3 + 2*A*b*e^4)*log(abs(c - 2*c* 
d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2))/(c^3 
*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3) + (B*d*e^6/(e*x + d) 
 - A*e^7/(e*x + d))/(c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2*d^2*e^6) + 1/2*(2*B 
*b*c^3*d^4*e^2 - 4*A*c^4*d^4*e^2 - 6*B*b^2*c^2*d^3*e^3 + 8*A*b*c^3*d^3*e^3 
 + 3*B*b^3*c*d^2*e^4 - B*b^4*d*e^5 - 4*A*b^3*c*d*e^5 + 2*A*b^4*e^6)*log(ab 
s(2*c*d*e - 2*c*d^2*e/(e*x + d) - b*e^2 + 2*b*d*e^2/(e*x + d) - e^2*abs(b) 
)/abs(2*c*d*e - 2*c*d^2*e/(e*x + d) - b*e^2 + 2*b*d*e^2/(e*x + d) + e^2*ab 
s(b)))/((b^2*c^3*d^6 - 3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*e^ 
2*abs(b)) + ((B*b*c^3*d^3*e - 2*A*c^4*d^3*e + 3*A*b*c^3*d^2*e^2 - 3*A*b^2* 
c^2*d*e^3 + A*b^3*c*e^4)/(c*d^2 - b*d*e) - (B*b*c^3*d^4*e^2 - 2*A*c^4*d^4* 
e^2 + 4*A*b*c^3*d^3*e^3 - 6*A*b^2*c^2*d^2*e^4 + 4*A*b^3*c*d*e^5 - A*b^4*e^ 
6)/((c*d^2 - b*d*e)*(e*x + d)*e))/((c*d - b*e)^2*b^2*(c - 2*c*d/(e*x + d) 
+ c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2)*d^2)
 

Mupad [B] (verification not implemented)

Time = 12.03 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.04 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx=\frac {\frac {x\,\left (B\,b^3\,d\,e^2-2\,A\,b^3\,e^3+A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}-\frac {A}{b\,d}+\frac {x^2\,\left (B\,b^2\,c\,d\,e^2-2\,A\,b^2\,c\,e^3+B\,b\,c^2\,d^2\,e+2\,A\,b\,c^2\,d\,e^2-2\,A\,c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (3\,B\,b^2\,c^2-4\,A\,b\,c^3\right )+d\,\left (2\,A\,c^4-B\,b\,c^3\right )\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (3\,B\,d^2\,e^2-4\,A\,d\,e^3\right )+b\,\left (2\,A\,e^4-B\,d\,e^3\right )\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {\ln \left (x\right )\,\left (d\,\left (2\,A\,c-B\,b\right )+2\,A\,b\,e\right )}{b^3\,d^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1745, normalized size of antiderivative = 8.68 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*log(b + c*x)*a*b**3*c**3*d**4*e**2*x + 4*log(b + c*x)*a*b**3*c**3*d**3* 
e**3*x**2 + 2*log(b + c*x)*a*b**2*c**4*d**5*e*x + 6*log(b + c*x)*a*b**2*c* 
*4*d**4*e**2*x**2 + 4*log(b + c*x)*a*b**2*c**4*d**3*e**3*x**3 - 2*log(b + 
c*x)*a*b*c**5*d**6*x + 2*log(b + c*x)*a*b*c**5*d**4*e**2*x**3 - 2*log(b + 
c*x)*a*c**6*d**6*x**2 - 2*log(b + c*x)*a*c**6*d**5*e*x**3 - 3*log(b + c*x) 
*b**5*c**2*d**4*e**2*x - 3*log(b + c*x)*b**5*c**2*d**3*e**3*x**2 - 2*log(b 
 + c*x)*b**4*c**3*d**5*e*x - 5*log(b + c*x)*b**4*c**3*d**4*e**2*x**2 - 3*l 
og(b + c*x)*b**4*c**3*d**3*e**3*x**3 + log(b + c*x)*b**3*c**4*d**6*x - log 
(b + c*x)*b**3*c**4*d**5*e*x**2 - 2*log(b + c*x)*b**3*c**4*d**4*e**2*x**3 
+ log(b + c*x)*b**2*c**5*d**6*x**2 + log(b + c*x)*b**2*c**5*d**5*e*x**3 + 
2*log(d + e*x)*a*b**6*d*e**5*x + 2*log(d + e*x)*a*b**6*e**6*x**2 - 2*log(d 
 + e*x)*a*b**5*c*d**2*e**4*x + 2*log(d + e*x)*a*b**5*c*e**6*x**3 - 4*log(d 
 + e*x)*a*b**4*c**2*d**3*e**3*x - 6*log(d + e*x)*a*b**4*c**2*d**2*e**4*x** 
2 - 2*log(d + e*x)*a*b**4*c**2*d*e**5*x**3 - 4*log(d + e*x)*a*b**3*c**3*d* 
*3*e**3*x**2 - 4*log(d + e*x)*a*b**3*c**3*d**2*e**4*x**3 - log(d + e*x)*b* 
*7*d**2*e**4*x - log(d + e*x)*b**7*d*e**5*x**2 + 2*log(d + e*x)*b**6*c*d** 
3*e**3*x + log(d + e*x)*b**6*c*d**2*e**4*x**2 - log(d + e*x)*b**6*c*d*e**5 
*x**3 + 3*log(d + e*x)*b**5*c**2*d**4*e**2*x + 5*log(d + e*x)*b**5*c**2*d* 
*3*e**3*x**2 + 2*log(d + e*x)*b**5*c**2*d**2*e**4*x**3 + 3*log(d + e*x)*b* 
*4*c**3*d**4*e**2*x**2 + 3*log(d + e*x)*b**4*c**3*d**3*e**3*x**3 - 2*lo...