\(\int \frac {A+B x}{x^3 (a^2+2 a b x+b^2 x^2)^3} \, dx\) [281]

Optimal result

Integrand size = 27, antiderivative size = 177 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {A}{2 a^6 x^2}+\frac {6 A b-a B}{a^7 x}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {3 b (7 A b-2 a B) \log (x)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8} \] Output:

-1/2*A/a^6/x^2+(6*A*b-B*a)/a^7/x+1/5*b*(A*b-B*a)/a^3/(b*x+a)^5+1/4*b*(3*A* 
b-2*B*a)/a^4/(b*x+a)^4+b*(2*A*b-B*a)/a^5/(b*x+a)^3+b*(5*A*b-2*B*a)/a^6/(b* 
x+a)^2+5*b*(3*A*b-B*a)/a^7/(b*x+a)+3*b*(7*A*b-2*B*a)*ln(x)/a^8-3*b*(7*A*b- 
2*B*a)*ln(b*x+a)/a^8
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {a \left (420 A b^6 x^6+5 a^3 b^3 x^3 (539 A-188 B x)+2 a^5 b x (35 A-137 B x)+7 a^4 b^2 x^2 (137 A-110 B x)+10 a^2 b^4 x^4 (329 A-54 B x)+30 a b^5 x^5 (63 A-4 B x)-10 a^6 (A+2 B x)\right )}{x^2 (a+b x)^5}+60 b (7 A b-2 a B) \log (x)+60 b (-7 A b+2 a B) \log (a+b x)}{20 a^8} \] Input:

Integrate[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

((a*(420*A*b^6*x^6 + 5*a^3*b^3*x^3*(539*A - 188*B*x) + 2*a^5*b*x*(35*A - 1 
37*B*x) + 7*a^4*b^2*x^2*(137*A - 110*B*x) + 10*a^2*b^4*x^4*(329*A - 54*B*x 
) + 30*a*b^5*x^5*(63*A - 4*B*x) - 10*a^6*(A + 2*B*x)))/(x^2*(a + b*x)^5) + 
 60*b*(7*A*b - 2*a*B)*Log[x] + 60*b*(-7*A*b + 2*a*B)*Log[a + b*x])/(20*a^8 
)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 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.

\(\displaystyle \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 x^3 (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{x^3 (a+b x)^6}dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {3 b^2 (2 a B-7 A b)}{a^8 (a+b x)}-\frac {3 b (2 a B-7 A b)}{a^8 x}+\frac {5 b^2 (a B-3 A b)}{a^7 (a+b x)^2}+\frac {a B-6 A b}{a^7 x^2}+\frac {2 b^2 (2 a B-5 A b)}{a^6 (a+b x)^3}+\frac {A}{a^6 x^3}+\frac {3 b^2 (a B-2 A b)}{a^5 (a+b x)^4}+\frac {b^2 (2 a B-3 A b)}{a^4 (a+b x)^5}+\frac {b^2 (a B-A b)}{a^3 (a+b x)^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b \log (x) (7 A b-2 a B)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8}+\frac {6 A b-a B}{a^7 x}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}-\frac {A}{2 a^6 x^2}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5}\)

Input:

Int[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/2*A/(a^6*x^2) + (6*A*b - a*B)/(a^7*x) + (b*(A*b - a*B))/(5*a^3*(a + b*x 
)^5) + (b*(3*A*b - 2*a*B))/(4*a^4*(a + b*x)^4) + (b*(2*A*b - a*B))/(a^5*(a 
 + b*x)^3) + (b*(5*A*b - 2*a*B))/(a^6*(a + b*x)^2) + (5*b*(3*A*b - a*B))/( 
a^7*(a + b*x)) + (3*b*(7*A*b - 2*a*B)*Log[x])/a^8 - (3*b*(7*A*b - 2*a*B)*L 
og[a + b*x])/a^8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2009

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

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {b \left (2 A b -B a \right )}{a^{5} \left (b x +a \right )^{3}}+\frac {b \left (3 A b -2 B a \right )}{4 a^{4} \left (b x +a \right )^{4}}+\frac {5 b \left (3 A b -B a \right )}{a^{7} \left (b x +a \right )}+\frac {b \left (5 A b -2 B a \right )}{a^{6} \left (b x +a \right )^{2}}-\frac {3 b \left (7 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{8}}+\frac {b \left (A b -B a \right )}{5 a^{3} \left (b x +a \right )^{5}}-\frac {A}{2 a^{6} x^{2}}-\frac {-6 A b +B a}{x \,a^{7}}+\frac {3 b \left (7 A b -2 B a \right ) \ln \left (x \right )}{a^{8}}\)	\(172\)
norman	\(\frac {-\frac {A}{2 a}+\frac {\left (7 A b -2 B a \right ) x}{2 a^{2}}-\frac {5 b \left (21 b^{2} A -6 a b B \right ) x^{3}}{a^{4}}-\frac {5 b^{2} \left (63 b^{2} A -18 a b B \right ) x^{4}}{a^{5}}-\frac {5 b^{3} \left (77 b^{2} A -22 a b B \right ) x^{5}}{a^{6}}-\frac {5 b^{4} \left (175 b^{2} A -50 a b B \right ) x^{6}}{4 a^{7}}-\frac {b^{5} \left (959 b^{2} A -274 a b B \right ) x^{7}}{20 a^{8}}}{x^{2} \left (b x +a \right )^{5}}+\frac {3 b \left (7 A b -2 B a \right ) \ln \left (x \right )}{a^{8}}-\frac {3 b \left (7 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{8}}\)	\(186\)
risch	\(\frac {\frac {3 b^{5} \left (7 A b -2 B a \right ) x^{6}}{a^{7}}+\frac {27 \left (7 A b -2 B a \right ) b^{4} x^{5}}{2 a^{6}}+\frac {47 b^{3} \left (7 A b -2 B a \right ) x^{4}}{2 a^{5}}+\frac {77 b^{2} \left (7 A b -2 B a \right ) x^{3}}{4 a^{4}}+\frac {137 b \left (7 A b -2 B a \right ) x^{2}}{20 a^{3}}+\frac {\left (7 A b -2 B a \right ) x}{2 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}+\frac {21 b^{2} \ln \left (-x \right ) A}{a^{8}}-\frac {6 b \ln \left (-x \right ) B}{a^{7}}-\frac {21 b^{2} \ln \left (b x +a \right ) A}{a^{8}}+\frac {6 b \ln \left (b x +a \right ) B}{a^{7}}\)	\(203\)
parallelrisch	\(\frac {2100 A \ln \left (x \right ) x^{3} a^{4} b^{3}-2100 A \ln \left (b x +a \right ) x^{3} a^{4} b^{3}-600 B \ln \left (x \right ) x^{3} a^{5} b^{2}+600 B \ln \left (b x +a \right ) x^{3} a^{5} b^{2}+420 A \ln \left (x \right ) x^{2} a^{5} b^{2}+70 A \,a^{6} b x -1200 B \ln \left (x \right ) x^{5} a^{3} b^{4}-420 A \ln \left (b x +a \right ) x^{7} b^{7}+420 A \ln \left (x \right ) x^{7} b^{7}+1200 B \ln \left (b x +a \right ) x^{5} a^{3} b^{4}+4200 A \ln \left (x \right ) x^{4} a^{3} b^{4}-4200 A \ln \left (b x +a \right ) x^{4} a^{3} b^{4}-1200 B \ln \left (x \right ) x^{4} a^{4} b^{3}+1200 B \ln \left (b x +a \right ) x^{4} a^{4} b^{3}+274 B a \,b^{6} x^{7}+2100 A \ln \left (x \right ) x^{6} a \,b^{6}-2100 A \ln \left (b x +a \right ) x^{6} a \,b^{6}-600 B \ln \left (x \right ) x^{6} a^{2} b^{5}+600 B \ln \left (b x +a \right ) x^{6} a^{2} b^{5}+4200 A \ln \left (x \right ) x^{5} a^{2} b^{5}-4200 A \ln \left (b x +a \right ) x^{5} a^{2} b^{5}-2100 A \,a^{4} b^{3} x^{3}+600 B \,a^{5} b^{2} x^{3}-6300 A \,a^{3} b^{4} x^{4}+1800 B \,a^{4} b^{3} x^{4}-7700 A \,a^{2} b^{5} x^{5}+2200 B \,a^{3} b^{4} x^{5}-4375 A a \,b^{6} x^{6}+1250 B \,a^{2} b^{5} x^{6}+120 B \ln \left (b x +a \right ) x^{2} a^{6} b -120 B \ln \left (x \right ) x^{7} a \,b^{6}+120 B \ln \left (b x +a \right ) x^{7} a \,b^{6}-420 A \ln \left (b x +a \right ) x^{2} a^{5} b^{2}-120 B \ln \left (x \right ) x^{2} a^{6} b -10 A \,a^{7}-20 B \,a^{7} x -959 A \,b^{7} x^{7}}{20 a^{8} x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\)	\(535\)

Input:

int((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

b*(2*A*b-B*a)/a^5/(b*x+a)^3+1/4*b*(3*A*b-2*B*a)/a^4/(b*x+a)^4+5*b*(3*A*b-B 
*a)/a^7/(b*x+a)+b*(5*A*b-2*B*a)/a^6/(b*x+a)^2-3*b*(7*A*b-2*B*a)*ln(b*x+a)/ 
a^8+1/5*b*(A*b-B*a)/a^3/(b*x+a)^5-1/2*A/a^6/x^2-(-6*A*b+B*a)/x/a^7+3*b*(7* 
A*b-2*B*a)*ln(x)/a^8
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (171) = 342\).

Time = 0.08 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x - 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{20 \, {\left (a^{8} b^{5} x^{7} + 5 \, a^{9} b^{4} x^{6} + 10 \, a^{10} b^{3} x^{5} + 10 \, a^{11} b^{2} x^{4} + 5 \, a^{12} b x^{3} + a^{13} x^{2}\right )}} \] Input:

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

-1/20*(10*A*a^7 + 60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270*(2*B*a^3*b^4 - 7* 
A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 385*(2*B*a^5*b^2 - 
7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 10*(2*B*a^7 - 7*A*a 
^6*b)*x - 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 
+ 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 
+ 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)*log(b 
*x + a) + 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 
+ 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 
+ 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)*log(x 
))/(a^8*b^5*x^7 + 5*a^9*b^4*x^6 + 10*a^10*b^3*x^5 + 10*a^11*b^2*x^4 + 5*a^ 
12*b*x^3 + a^13*x^2)
 

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.89 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {- 10 A a^{6} + x^{6} \cdot \left (420 A b^{6} - 120 B a b^{5}\right ) + x^{5} \cdot \left (1890 A a b^{5} - 540 B a^{2} b^{4}\right ) + x^{4} \cdot \left (3290 A a^{2} b^{4} - 940 B a^{3} b^{3}\right ) + x^{3} \cdot \left (2695 A a^{3} b^{3} - 770 B a^{4} b^{2}\right ) + x^{2} \cdot \left (959 A a^{4} b^{2} - 274 B a^{5} b\right ) + x \left (70 A a^{5} b - 20 B a^{6}\right )}{20 a^{12} x^{2} + 100 a^{11} b x^{3} + 200 a^{10} b^{2} x^{4} + 200 a^{9} b^{3} x^{5} + 100 a^{8} b^{4} x^{6} + 20 a^{7} b^{5} x^{7}} - \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b - 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} + \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b + 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} \] Input:

integrate((B*x+A)/x**3/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

(-10*A*a**6 + x**6*(420*A*b**6 - 120*B*a*b**5) + x**5*(1890*A*a*b**5 - 540 
*B*a**2*b**4) + x**4*(3290*A*a**2*b**4 - 940*B*a**3*b**3) + x**3*(2695*A*a 
**3*b**3 - 770*B*a**4*b**2) + x**2*(959*A*a**4*b**2 - 274*B*a**5*b) + x*(7 
0*A*a**5*b - 20*B*a**6))/(20*a**12*x**2 + 100*a**11*b*x**3 + 200*a**10*b** 
2*x**4 + 200*a**9*b**3*x**5 + 100*a**8*b**4*x**6 + 20*a**7*b**5*x**7) - 3* 
b*(-7*A*b + 2*B*a)*log(x + (-21*A*a*b**2 + 6*B*a**2*b - 3*a*b*(-7*A*b + 2* 
B*a))/(-42*A*b**3 + 12*B*a*b**2))/a**8 + 3*b*(-7*A*b + 2*B*a)*log(x + (-21 
*A*a*b**2 + 6*B*a**2*b + 3*a*b*(-7*A*b + 2*B*a))/(-42*A*b**3 + 12*B*a*b**2 
))/a**8
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {10 \, A a^{6} + 60 \, {\left (2 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{6} - 7 \, A a^{5} b\right )} x}{20 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}} + \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{8}} - \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (x\right )}{a^{8}} \] Input:

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

-1/20*(10*A*a^6 + 60*(2*B*a*b^5 - 7*A*b^6)*x^6 + 270*(2*B*a^2*b^4 - 7*A*a* 
b^5)*x^5 + 470*(2*B*a^3*b^3 - 7*A*a^2*b^4)*x^4 + 385*(2*B*a^4*b^2 - 7*A*a^ 
3*b^3)*x^3 + 137*(2*B*a^5*b - 7*A*a^4*b^2)*x^2 + 10*(2*B*a^6 - 7*A*a^5*b)* 
x)/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^1 
1*b*x^3 + a^12*x^2) + 3*(2*B*a*b - 7*A*b^2)*log(b*x + a)/a^8 - 3*(2*B*a*b 
- 7*A*b^2)*log(x)/a^8
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {3 \, {\left (2 \, B a b^{2} - 7 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{8} b} - \frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x}{20 \, {\left (b x + a\right )}^{5} a^{8} x^{2}} \] Input:

integrate((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

-3*(2*B*a*b - 7*A*b^2)*log(abs(x))/a^8 + 3*(2*B*a*b^2 - 7*A*b^3)*log(abs(b 
*x + a))/(a^8*b) - 1/20*(10*A*a^7 + 60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270 
*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 3 
85*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 1 
0*(2*B*a^7 - 7*A*a^6*b)*x)/((b*x + a)^5*a^8*x^2)
 

Mupad [B] (verification not implemented)

Time = 10.56 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {x\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {77\,b^2\,x^3\,\left (7\,A\,b-2\,B\,a\right )}{4\,a^4}+\frac {47\,b^3\,x^4\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^5}+\frac {27\,b^4\,x^5\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^6}+\frac {3\,b^5\,x^6\,\left (7\,A\,b-2\,B\,a\right )}{a^7}+\frac {137\,b\,x^2\,\left (7\,A\,b-2\,B\,a\right )}{20\,a^3}}{a^5\,x^2+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^5+5\,a\,b^4\,x^6+b^5\,x^7}-\frac {6\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (7\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (21\,A\,b^2-6\,B\,a\,b\right )}\right )\,\left (7\,A\,b-2\,B\,a\right )}{a^8} \] Input:

int((A + B*x)/(x^3*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
 

Output:

((x*(7*A*b - 2*B*a))/(2*a^2) - A/(2*a) + (77*b^2*x^3*(7*A*b - 2*B*a))/(4*a 
^4) + (47*b^3*x^4*(7*A*b - 2*B*a))/(2*a^5) + (27*b^4*x^5*(7*A*b - 2*B*a))/ 
(2*a^6) + (3*b^5*x^6*(7*A*b - 2*B*a))/a^7 + (137*b*x^2*(7*A*b - 2*B*a))/(2 
0*a^3))/(a^5*x^2 + b^5*x^7 + 5*a^4*b*x^3 + 5*a*b^4*x^6 + 10*a^3*b^2*x^4 + 
10*a^2*b^3*x^5) - (6*b*atanh((3*b*(7*A*b - 2*B*a)*(a + 2*b*x))/(a*(21*A*b^ 
2 - 6*B*a*b)))*(7*A*b - 2*B*a))/a^8
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} x^{2}-240 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} x^{3}-360 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} x^{4}-240 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} x^{5}-60 \,\mathrm {log}\left (b x +a \right ) b^{6} x^{6}+60 \,\mathrm {log}\left (x \right ) a^{4} b^{2} x^{2}+240 \,\mathrm {log}\left (x \right ) a^{3} b^{3} x^{3}+360 \,\mathrm {log}\left (x \right ) a^{2} b^{4} x^{4}+240 \,\mathrm {log}\left (x \right ) a \,b^{5} x^{5}+60 \,\mathrm {log}\left (x \right ) b^{6} x^{6}-2 a^{6}+12 a^{5} b x +110 a^{4} b^{2} x^{2}+200 a^{3} b^{3} x^{3}+120 a^{2} b^{4} x^{4}-15 b^{6} x^{6}}{4 a^{7} x^{2} \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

int((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

( - 60*log(a + b*x)*a**4*b**2*x**2 - 240*log(a + b*x)*a**3*b**3*x**3 - 360 
*log(a + b*x)*a**2*b**4*x**4 - 240*log(a + b*x)*a*b**5*x**5 - 60*log(a + b 
*x)*b**6*x**6 + 60*log(x)*a**4*b**2*x**2 + 240*log(x)*a**3*b**3*x**3 + 360 
*log(x)*a**2*b**4*x**4 + 240*log(x)*a*b**5*x**5 + 60*log(x)*b**6*x**6 - 2* 
a**6 + 12*a**5*b*x + 110*a**4*b**2*x**2 + 200*a**3*b**3*x**3 + 120*a**2*b* 
*4*x**4 - 15*b**6*x**6)/(4*a**7*x**2*(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 
 + 4*a*b**3*x**3 + b**4*x**4))