\(\int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx\) [123]

Optimal result

Integrand size = 20, antiderivative size = 199 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x)^2}-\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}-\frac {e (B d-A e)}{2 (b d-a e)^3 (d+e x)^2}-\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (d+e x)}-\frac {3 b e (b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^5}+\frac {3 b e (b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^5} \] Output:

-1/2*b*(A*b-B*a)/(-a*e+b*d)^3/(b*x+a)^2-b*(-3*A*b*e+2*B*a*e+B*b*d)/(-a*e+b 
*d)^4/(b*x+a)-1/2*e*(-A*e+B*d)/(-a*e+b*d)^3/(e*x+d)^2-e*(-3*A*b*e+B*a*e+2* 
B*b*d)/(-a*e+b*d)^4/(e*x+d)-3*b*e*(-2*A*b*e+B*a*e+B*b*d)*ln(b*x+a)/(-a*e+b 
*d)^5+3*b*e*(-2*A*b*e+B*a*e+B*b*d)*ln(e*x+d)/(-a*e+b*d)^5
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\frac {-\frac {b (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac {2 b (b d-a e) (b B d-3 A b e+2 a B e)}{a+b x}+\frac {e (b d-a e)^2 (-B d+A e)}{(d+e x)^2}+\frac {2 e (b d-a e) (-2 b B d+3 A b e-a B e)}{d+e x}-6 b e (b B d-2 A b e+a B e) \log (a+b x)+6 b e (b B d-2 A b e+a B e) \log (d+e x)}{2 (b d-a e)^5} \] Input:

Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^3),x]
 

Output:

(-((b*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2) - (2*b*(b*d - a*e)*(b*B*d - 
3*A*b*e + 2*a*B*e))/(a + b*x) + (e*(b*d - a*e)^2*(-(B*d) + A*e))/(d + e*x) 
^2 + (2*e*(b*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x) - 6*b*e*(b*B 
*d - 2*A*b*e + a*B*e)*Log[a + b*x] + 6*b*e*(b*B*d - 2*A*b*e + a*B*e)*Log[d 
 + e*x])/(2*(b*d - a*e)^5)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.

\(\displaystyle \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5}\)

Input:

Int[(A + B*x)/((a + b*x)^3*(d + e*x)^3),x]
 

Output:

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

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {b \left (3 A b e -2 B a e -B b d \right )}{\left (a e -d b \right )^{4} \left (b x +a \right )}+\frac {\left (A b -B a \right ) b}{2 \left (a e -d b \right )^{3} \left (b x +a \right )^{2}}-\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -d b \right )^{5}}-\frac {\left (A e -B d \right ) e}{2 \left (a e -d b \right )^{3} \left (e x +d \right )^{2}}+\frac {e \left (3 A b e -B a e -2 B b d \right )}{\left (a e -d b \right )^{4} \left (e x +d \right )}+\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -d b \right )^{5}}\)	\(200\)
norman	\(\frac {\frac {\left (6 A \,b^{4} e^{4}-3 B a \,b^{3} e^{4}-3 B \,b^{4} d \,e^{3}\right ) x^{3}}{\left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b e}+\frac {\left (2 A \,a^{2} b^{3} e^{5}+14 A a \,b^{4} d \,e^{4}+2 A \,b^{5} d^{2} e^{3}-B \,a^{3} b^{2} e^{5}-8 B \,a^{2} b^{3} d \,e^{4}-8 B a \,b^{4} d^{2} e^{3}-B \,b^{5} d^{3} e^{2}\right ) x}{e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {A \,a^{3} b^{2} e^{5}-7 A \,a^{2} b^{3} d \,e^{4}-7 A a \,b^{4} d^{2} e^{3}+A \,b^{5} d^{3} e^{2}+B \,a^{3} b^{2} d \,e^{4}+10 B \,a^{2} b^{3} d^{2} e^{3}+B a \,b^{4} d^{3} e^{2}}{2 e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (18 A a \,b^{4} e^{5}+18 A \,b^{5} d \,e^{4}-9 B \,a^{2} b^{3} e^{5}-18 B a \,b^{4} d \,e^{4}-9 B \,b^{5} d^{2} e^{3}\right ) x^{2}}{2 e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{2} \left (b x +a \right )^{2}}-\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}\)	\(703\)
risch	\(\frac {\frac {3 b^{2} e^{2} \left (2 A b e -B a e -B b d \right ) x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {9 b e \left (a e +d b \right ) \left (2 A b e -B a e -B b d \right ) x^{2}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (2 A \,a^{2} b \,e^{3}+14 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e -B \,a^{3} e^{3}-8 B \,a^{2} b d \,e^{2}-8 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {a^{3} A \,e^{3}-7 A \,a^{2} b d \,e^{2}-7 A a \,b^{2} d^{2} e +A \,b^{3} d^{3}+B \,a^{3} d \,e^{2}+10 B \,a^{2} b \,d^{2} e +B a \,b^{2} d^{3}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{2} \left (b x +a \right )^{2}}-\frac {6 b^{2} e^{2} \ln \left (b x +a \right ) A}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b \,e^{2} \ln \left (b x +a \right ) B a}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b^{2} e \ln \left (b x +a \right ) B d}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {6 b^{2} e^{2} \ln \left (-e x -d \right ) A}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}-\frac {3 b \,e^{2} \ln \left (-e x -d \right ) B a}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}-\frac {3 b^{2} e \ln \left (-e x -d \right ) B d}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}\)	\(914\)
parallelrisch	\(\text {Expression too large to display}\)	\(1388\)

Input:

int((B*x+A)/(b*x+a)^3/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

b*(3*A*b*e-2*B*a*e-B*b*d)/(a*e-b*d)^4/(b*x+a)+1/2*(A*b-B*a)*b/(a*e-b*d)^3/ 
(b*x+a)^2-3*b*e*(2*A*b*e-B*a*e-B*b*d)/(a*e-b*d)^5*ln(b*x+a)-1/2*(A*e-B*d)* 
e/(a*e-b*d)^3/(e*x+d)^2+e*(3*A*b*e-B*a*e-2*B*b*d)/(a*e-b*d)^4/(e*x+d)+3*b* 
e*(2*A*b*e-B*a*e-B*b*d)/(a*e-b*d)^5*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (195) = 390\).

Time = 0.14 (sec) , antiderivative size = 1215, normalized size of antiderivative = 6.11 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^3,x, algorithm="fricas")
 

Output:

1/2*(9*B*a^3*b*d^2*e^2 + A*a^4*e^4 - (B*a*b^3 + A*b^4)*d^4 - (9*B*a^2*b^2 
- 8*A*a*b^3)*d^3*e + (B*a^4 - 8*A*a^3*b)*d*e^3 - 6*(B*b^4*d^2*e^2 - 2*A*b^ 
4*d*e^3 - (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 - 9*(B*b^4*d^3*e - B*a^2*b^2*d* 
e^3 + (B*a*b^3 - 2*A*b^4)*d^2*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 - 2*( 
B*b^4*d^4 - 12*A*a*b^3*d^2*e^2 + (7*B*a*b^3 - 2*A*b^4)*d^3*e - (7*B*a^3*b 
- 12*A*a^2*b^2)*d*e^3 - (B*a^4 - 2*A*a^3*b)*e^4)*x - 6*(B*a^2*b^2*d^3*e + 
(B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)* 
x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^ 
3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^4)*d^2*e^2 + (5*B*a^2*b^2 
- 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 
 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*d*e^3)*x)*log(b 
*x + a) + 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B*b^4*d* 
e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^4 
)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2 
*A*b^4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2 
)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b 
 - 2*A*a^2*b^2)*d*e^3)*x)*log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 1 
0*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + ( 
b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 
5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (192) = 384\).

Time = 2.88 (sec) , antiderivative size = 1431, normalized size of antiderivative = 7.19 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)**3/(e*x+d)**3,x)
 

Output:

-3*b*e*(-2*A*b*e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**3*d*e 
**2 + 3*B*a**2*b*e**3 + 6*B*a*b**2*d*e**2 + 3*B*b**3*d**2*e - 3*a**6*b*e** 
7*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 18*a**5*b**2*d*e**6*(-2*A*b* 
e + B*a*e + B*b*d)/(a*e - b*d)**5 - 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a 
*e + B*b*d)/(a*e - b*d)**5 + 60*a**3*b**4*d**3*e**4*(-2*A*b*e + B*a*e + B* 
b*d)/(a*e - b*d)**5 - 45*a**2*b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a 
*e - b*d)**5 + 18*a*b**6*d**5*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)* 
*5 - 3*b**7*d**6*e*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5)/(-12*A*b**3* 
e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e**2))/(a*e - b*d)**5 + 3*b*e*(-2*A*b* 
e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**3*d*e**2 + 3*B*a**2* 
b*e**3 + 6*B*a*b**2*d*e**2 + 3*B*b**3*d**2*e + 3*a**6*b*e**7*(-2*A*b*e + B 
*a*e + B*b*d)/(a*e - b*d)**5 - 18*a**5*b**2*d*e**6*(-2*A*b*e + B*a*e + B*b 
*d)/(a*e - b*d)**5 + 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a*e + B*b*d)/(a* 
e - b*d)**5 - 60*a**3*b**4*d**3*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d 
)**5 + 45*a**2*b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 
18*a*b**6*d**5*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 3*b**7*d** 
6*e*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5)/(-12*A*b**3*e**3 + 6*B*a*b* 
*2*e**3 + 6*B*b**3*d*e**2))/(a*e - b*d)**5 + (-A*a**3*e**3 + 7*A*a**2*b*d* 
e**2 + 7*A*a*b**2*d**2*e - A*b**3*d**3 - B*a**3*d*e**2 - 10*B*a**2*b*d**2* 
e - B*a*b**2*d**3 + x**3*(12*A*b**3*e**3 - 6*B*a*b**2*e**3 - 6*B*b**3*d...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (195) = 390\).

Time = 0.05 (sec) , antiderivative size = 745, normalized size of antiderivative = 3.74 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 7 \, A a^{2} b\right )} d e^{2} + 6 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 9 \, {\left (B b^{3} d^{2} e + 2 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B b^{3} d^{3} + 2 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d^{2} e + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \] Input:

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^3,x, algorithm="maxima")
 

Output:

-3*(B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4 
*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) + 
3*(B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4* 
e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1 
/2*(A*a^3*e^3 + (B*a*b^2 + A*b^3)*d^3 + (10*B*a^2*b - 7*A*a*b^2)*d^2*e + ( 
B*a^3 - 7*A*a^2*b)*d*e^2 + 6*(B*b^3*d*e^2 + (B*a*b^2 - 2*A*b^3)*e^3)*x^3 + 
 9*(B*b^3*d^2*e + 2*(B*a*b^2 - A*b^3)*d*e^2 + (B*a^2*b - 2*A*a*b^2)*e^3)*x 
^2 + 2*(B*b^3*d^3 + 2*(4*B*a*b^2 - A*b^3)*d^2*e + 2*(4*B*a^2*b - 7*A*a*b^2 
)*d*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*x)/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a 
^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^ 
3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^ 
5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2* 
d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 
 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a 
^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (195) = 390\).

Time = 0.13 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.67 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {6 \, B b^{3} d e^{2} x^{3} + 6 \, B a b^{2} e^{3} x^{3} - 12 \, A b^{3} e^{3} x^{3} + 9 \, B b^{3} d^{2} e x^{2} + 18 \, B a b^{2} d e^{2} x^{2} - 18 \, A b^{3} d e^{2} x^{2} + 9 \, B a^{2} b e^{3} x^{2} - 18 \, A a b^{2} e^{3} x^{2} + 2 \, B b^{3} d^{3} x + 16 \, B a b^{2} d^{2} e x - 4 \, A b^{3} d^{2} e x + 16 \, B a^{2} b d e^{2} x - 28 \, A a b^{2} d e^{2} x + 2 \, B a^{3} e^{3} x - 4 \, A a^{2} b e^{3} x + B a b^{2} d^{3} + A b^{3} d^{3} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b e x^{2} + b d x + a e x + a d\right )}^{2}} \] Input:

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^3,x, algorithm="giac")
 

Output:

-3*(B*b^3*d*e + B*a*b^2*e^2 - 2*A*b^3*e^2)*log(abs(b*x + a))/(b^6*d^5 - 5* 
a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - 
a^5*b*e^5) + 3*(B*b^2*d*e^2 + B*a*b*e^3 - 2*A*b^2*e^3)*log(abs(e*x + d))/( 
b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5* 
a^4*b*d*e^5 - a^5*e^6) - 1/2*(6*B*b^3*d*e^2*x^3 + 6*B*a*b^2*e^3*x^3 - 12*A 
*b^3*e^3*x^3 + 9*B*b^3*d^2*e*x^2 + 18*B*a*b^2*d*e^2*x^2 - 18*A*b^3*d*e^2*x 
^2 + 9*B*a^2*b*e^3*x^2 - 18*A*a*b^2*e^3*x^2 + 2*B*b^3*d^3*x + 16*B*a*b^2*d 
^2*e*x - 4*A*b^3*d^2*e*x + 16*B*a^2*b*d*e^2*x - 28*A*a*b^2*d*e^2*x + 2*B*a 
^3*e^3*x - 4*A*a^2*b*e^3*x + B*a*b^2*d^3 + A*b^3*d^3 + 10*B*a^2*b*d^2*e - 
7*A*a*b^2*d^2*e + B*a^3*d*e^2 - 7*A*a^2*b*d*e^2 + A*a^3*e^3)/((b^4*d^4 - 4 
*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(b*e*x^2 + b*d 
*x + a*e*x + a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.65 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )\,\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5\,\left (-6\,A\,b^2\,e^2+3\,B\,d\,b^2\,e+3\,B\,a\,b\,e^2\right )}\right )\,\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {B\,a^3\,d\,e^2+A\,a^3\,e^3+10\,B\,a^2\,b\,d^2\,e-7\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-7\,A\,a\,b^2\,d^2\,e+A\,b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {3\,b^2\,e^2\,x^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4} \] Input:

int((A + B*x)/((a + b*x)^3*(d + e*x)^3),x)
 

Output:

(2*atanh(((3*b*e^2*(2*A*b - B*a) - 3*B*b^2*d*e)*((a^5*e^5 + b^5*d^5 + 2*a^ 
2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3*a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a^4*e^ 
4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + 2*b*e*x 
)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) 
/((a*e - b*d)^5*(3*B*a*b*e^2 - 6*A*b^2*e^2 + 3*B*b^2*d*e)))*(3*b*e^2*(2*A* 
b - B*a) - 3*B*b^2*d*e))/(a*e - b*d)^5 - ((A*a^3*e^3 + A*b^3*d^3 + B*a*b^2 
*d^3 + B*a^3*d*e^2 - 7*A*a*b^2*d^2*e - 7*A*a^2*b*d*e^2 + 10*B*a^2*b*d^2*e) 
/(2*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3 
)) + (9*x^2*(a*b*e^2 + b^2*d*e)*(B*a*e - 2*A*b*e + B*b*d))/(2*(a^4*e^4 + b 
^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (x*(a^2*e^2 
 + b^2*d^2 + 7*a*b*d*e)*(B*a*e - 2*A*b*e + B*b*d))/(a^4*e^4 + b^4*d^4 + 6* 
a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + (3*b^2*e^2*x^3*(B*a*e - 
 2*A*b*e + B*b*d))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e 
- 4*a^3*b*d*e^3))/(x*(2*a*b*d^2 + 2*a^2*d*e) + x^2*(a^2*e^2 + b^2*d^2 + 4* 
a*b*d*e) + x^3*(2*a*b*e^2 + 2*b^2*d*e) + a^2*d^2 + b^2*e^2*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.13 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(b*x+a)^3/(e*x+d)^3,x)
                                                                                    
                                                                                    
 

Output:

( - 6*log(a + b*x)*a**2*b**2*d**2*e**2 - 12*log(a + b*x)*a**2*b**2*d*e**3* 
x - 6*log(a + b*x)*a**2*b**2*e**4*x**2 - 12*log(a + b*x)*a*b**3*d**3*e - 3 
0*log(a + b*x)*a*b**3*d**2*e**2*x - 24*log(a + b*x)*a*b**3*d*e**3*x**2 - 6 
*log(a + b*x)*a*b**3*e**4*x**3 - 12*log(a + b*x)*b**4*d**3*e*x - 24*log(a 
+ b*x)*b**4*d**2*e**2*x**2 - 12*log(a + b*x)*b**4*d*e**3*x**3 + 6*log(d + 
e*x)*a**2*b**2*d**2*e**2 + 12*log(d + e*x)*a**2*b**2*d*e**3*x + 6*log(d + 
e*x)*a**2*b**2*e**4*x**2 + 12*log(d + e*x)*a*b**3*d**3*e + 30*log(d + e*x) 
*a*b**3*d**2*e**2*x + 24*log(d + e*x)*a*b**3*d*e**3*x**2 + 6*log(d + e*x)* 
a*b**3*e**4*x**3 + 12*log(d + e*x)*b**4*d**3*e*x + 24*log(d + e*x)*b**4*d* 
*2*e**2*x**2 + 12*log(d + e*x)*b**4*d*e**3*x**3 - a**4*e**4 + 4*a**3*b*d*e 
**3 + 3*a**3*b*e**4*x + 3*a**2*b**2*d**2*e**2 - 2*a*b**3*d**3*e + 9*a*b**3 
*d**2*e**2*x - 6*a*b**3*e**4*x**3 - 4*b**4*d**4 - 12*b**4*d**3*e*x + 6*b** 
4*d*e**3*x**3)/(2*(a**6*d**2*e**5 + 2*a**6*d*e**6*x + a**6*e**7*x**2 - 2*a 
**5*b*d**3*e**4 - 3*a**5*b*d**2*e**5*x + a**5*b*e**7*x**3 - 2*a**4*b**2*d* 
*4*e**3 - 6*a**4*b**2*d**3*e**4*x - 6*a**4*b**2*d**2*e**5*x**2 - 2*a**4*b* 
*2*d*e**6*x**3 + 8*a**3*b**3*d**5*e**2 + 14*a**3*b**3*d**4*e**3*x + 4*a**3 
*b**3*d**3*e**4*x**2 - 2*a**3*b**3*d**2*e**5*x**3 - 7*a**2*b**4*d**6*e - 6 
*a**2*b**4*d**5*e**2*x + 9*a**2*b**4*d**4*e**3*x**2 + 8*a**2*b**4*d**3*e** 
4*x**3 + 2*a*b**5*d**7 - 3*a*b**5*d**6*e*x - 12*a*b**5*d**5*e**2*x**2 - 7* 
a*b**5*d**4*e**3*x**3 + 2*b**6*d**7*x + 4*b**6*d**6*e*x**2 + 2*b**6*d**...