\(\int \frac {A+B x}{x^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\) [280]

Optimal result

Integrand size = 27, antiderivative size = 157 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {A}{a^6 x}-\frac {A b-a B}{5 a^2 (a+b x)^5}-\frac {2 A b-a B}{4 a^3 (a+b x)^4}-\frac {3 A b-a B}{3 a^4 (a+b x)^3}-\frac {4 A b-a B}{2 a^5 (a+b x)^2}-\frac {5 A b-a B}{a^6 (a+b x)}-\frac {(6 A b-a B) \log (x)}{a^7}+\frac {(6 A b-a B) \log (a+b x)}{a^7} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {60 a A}{x}+\frac {12 a^5 (-A b+a B)}{(a+b x)^5}+\frac {15 a^4 (-2 A b+a B)}{(a+b x)^4}+\frac {20 a^3 (-3 A b+a B)}{(a+b x)^3}+\frac {30 a^2 (-4 A b+a B)}{(a+b x)^2}+\frac {60 a (-5 A b+a B)}{a+b x}+60 (-6 A b+a B) \log (x)+60 (6 A b-a B) \log (a+b x)}{60 a^7} \] Input:

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

Output:

((-60*a*A)/x + (12*a^5*(-(A*b) + a*B))/(a + b*x)^5 + (15*a^4*(-2*A*b + a*B 
))/(a + b*x)^4 + (20*a^3*(-3*A*b + a*B))/(a + b*x)^3 + (30*a^2*(-4*A*b + a 
*B))/(a + b*x)^2 + (60*a*(-5*A*b + a*B))/(a + b*x) + 60*(-6*A*b + a*B)*Log 
[x] + 60*(6*A*b - a*B)*Log[a + b*x])/(60*a^7)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 157, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 x^2 (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{x^2 (a+b x)^6}dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {2 A b -B a}{4 a^{3} \left (b x +a \right )^{4}}-\frac {3 A b -B a}{3 a^{4} \left (b x +a \right )^{3}}-\frac {4 A b -B a}{2 a^{5} \left (b x +a \right )^{2}}-\frac {5 A b -B a}{a^{6} \left (b x +a \right )}+\frac {\left (6 A b -B a \right ) \ln \left (b x +a \right )}{a^{7}}-\frac {A b -B a}{5 a^{2} \left (b x +a \right )^{5}}-\frac {A}{a^{6} x}+\frac {\left (-6 A b +B a \right ) \ln \left (x \right )}{a^{7}}\)	\(148\)
norman	\(\frac {-\frac {A}{a}+\frac {5 b \left (6 A b -B a \right ) x^{2}}{a^{3}}+\frac {5 b^{2} \left (18 A b -3 B a \right ) x^{3}}{a^{4}}+\frac {5 b^{3} \left (66 A b -11 B a \right ) x^{4}}{3 a^{5}}+\frac {5 b^{4} \left (150 A b -25 B a \right ) x^{5}}{12 a^{6}}+\frac {b^{5} \left (822 A b -137 B a \right ) x^{6}}{60 a^{7}}}{x \left (b x +a \right )^{5}}+\frac {\left (6 A b -B a \right ) \ln \left (b x +a \right )}{a^{7}}-\frac {\left (6 A b -B a \right ) \ln \left (x \right )}{a^{7}}\)	\(153\)
risch	\(\frac {-\frac {b^{4} \left (6 A b -B a \right ) x^{5}}{a^{6}}-\frac {9 b^{3} \left (6 A b -B a \right ) x^{4}}{2 a^{5}}-\frac {47 b^{2} \left (6 A b -B a \right ) x^{3}}{6 a^{4}}-\frac {77 b \left (6 A b -B a \right ) x^{2}}{12 a^{3}}-\frac {137 \left (6 A b -B a \right ) x}{60 a^{2}}-\frac {A}{a}}{x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}+\frac {6 \ln \left (-b x -a \right ) A b}{a^{7}}-\frac {\ln \left (-b x -a \right ) B}{a^{6}}-\frac {6 \ln \left (x \right ) A b}{a^{7}}+\frac {\ln \left (x \right ) B}{a^{6}}\)	\(178\)
parallelrisch	\(-\frac {60 B \ln \left (b x +a \right ) x^{6} a \,b^{5}-1800 A \ln \left (b x +a \right ) x^{5} a \,b^{5}+300 B \ln \left (b x +a \right ) x^{5} a^{2} b^{4}-3600 A \ln \left (b x +a \right ) x^{4} a^{2} b^{4}+600 B \ln \left (b x +a \right ) x^{4} a^{3} b^{3}-3600 A \ln \left (b x +a \right ) x^{3} a^{3} b^{3}+600 B \ln \left (b x +a \right ) x^{3} a^{4} b^{2}-1800 A \ln \left (b x +a \right ) x^{2} a^{4} b^{2}+300 B \ln \left (b x +a \right ) x^{2} a^{5} b +360 A \ln \left (x \right ) x \,a^{5} b -360 A \ln \left (b x +a \right ) x \,a^{5} b +3600 A \ln \left (x \right ) x^{3} a^{3} b^{3}-60 B \ln \left (x \right ) x^{6} a \,b^{5}+1800 A \ln \left (x \right ) x^{2} a^{4} b^{2}-300 B \ln \left (x \right ) x^{2} a^{5} b -300 B \ln \left (x \right ) x^{5} a^{2} b^{4}+60 B \ln \left (b x +a \right ) x \,a^{6}-3750 A a \,b^{5} x^{5}+625 B \,a^{2} b^{4} x^{5}+1800 A \ln \left (x \right ) x^{5} a \,b^{5}-600 B \ln \left (x \right ) x^{4} a^{3} b^{3}+300 B \,a^{5} b \,x^{2}+3600 A \ln \left (x \right ) x^{4} a^{2} b^{4}-600 B \ln \left (x \right ) x^{3} a^{4} b^{2}-60 B \ln \left (x \right ) x \,a^{6}+60 A \,a^{6}+360 A \ln \left (x \right ) x^{6} b^{6}-360 A \ln \left (b x +a \right ) x^{6} b^{6}+137 B a \,b^{5} x^{6}-1800 A \,a^{4} b^{2} x^{2}+1100 B \,a^{3} b^{3} x^{4}-5400 A \,a^{3} b^{3} x^{3}+900 B \,a^{4} b^{2} x^{3}-6600 A \,a^{2} b^{4} x^{4}-822 A \,b^{6} x^{6}}{60 a^{7} x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\)	\(500\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (142) = 284\).

Time = 0.08 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.72 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {60 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \, {\left (B a^{6} - 6 \, A a^{5} b\right )} x + 60 \, {\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left ({\left (B a b^{5} - 6 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 6 \, A a^{5} b\right )} x\right )} \log \left (x\right )}{60 \, {\left (a^{7} b^{5} x^{6} + 5 \, a^{8} b^{4} x^{5} + 10 \, a^{9} b^{3} x^{4} + 10 \, a^{10} b^{2} x^{3} + 5 \, a^{11} b x^{2} + a^{12} x\right )}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (136) = 272\).

Time = 0.62 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {- 60 A a^{5} + x^{5} \left (- 360 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 1620 A a b^{4} + 270 B a^{2} b^{3}\right ) + x^{3} \left (- 2820 A a^{2} b^{3} + 470 B a^{3} b^{2}\right ) + x^{2} \left (- 2310 A a^{3} b^{2} + 385 B a^{4} b\right ) + x \left (- 822 A a^{4} b + 137 B a^{5}\right )}{60 a^{11} x + 300 a^{10} b x^{2} + 600 a^{9} b^{2} x^{3} + 600 a^{8} b^{3} x^{4} + 300 a^{7} b^{4} x^{5} + 60 a^{6} b^{5} x^{6}} + \frac {\left (- 6 A b + B a\right ) \log {\left (x + \frac {- 6 A a b + B a^{2} - a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} - \frac {\left (- 6 A b + B a\right ) \log {\left (x + \frac {- 6 A a b + B a^{2} + a \left (- 6 A b + B a\right )}{- 12 A b^{2} + 2 B a b} \right )}}{a^{7}} \] Input:

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

Output:

(-60*A*a**5 + x**5*(-360*A*b**5 + 60*B*a*b**4) + x**4*(-1620*A*a*b**4 + 27 
0*B*a**2*b**3) + x**3*(-2820*A*a**2*b**3 + 470*B*a**3*b**2) + x**2*(-2310* 
A*a**3*b**2 + 385*B*a**4*b) + x*(-822*A*a**4*b + 137*B*a**5))/(60*a**11*x 
+ 300*a**10*b*x**2 + 600*a**9*b**2*x**3 + 600*a**8*b**3*x**4 + 300*a**7*b* 
*4*x**5 + 60*a**6*b**5*x**6) + (-6*A*b + B*a)*log(x + (-6*A*a*b + B*a**2 - 
 a*(-6*A*b + B*a))/(-12*A*b**2 + 2*B*a*b))/a**7 - (-6*A*b + B*a)*log(x + ( 
-6*A*a*b + B*a**2 + a*(-6*A*b + B*a))/(-12*A*b**2 + 2*B*a*b))/a**7
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {60 \, A a^{5} - 60 \, {\left (B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 270 \, {\left (B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} - 470 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} - 385 \, {\left (B a^{4} b - 6 \, A a^{3} b^{2}\right )} x^{2} - 137 \, {\left (B a^{5} - 6 \, A a^{4} b\right )} x}{60 \, {\left (a^{6} b^{5} x^{6} + 5 \, a^{7} b^{4} x^{5} + 10 \, a^{8} b^{3} x^{4} + 10 \, a^{9} b^{2} x^{3} + 5 \, a^{10} b x^{2} + a^{11} x\right )}} - \frac {{\left (B a - 6 \, A b\right )} \log \left (b x + a\right )}{a^{7}} + \frac {{\left (B a - 6 \, A b\right )} \log \left (x\right )}{a^{7}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (B a - 6 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {{\left (B a b - 6 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{7} b} - \frac {60 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} - 270 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} - 470 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} x^{3} - 385 \, {\left (B a^{5} b - 6 \, A a^{4} b^{2}\right )} x^{2} - 137 \, {\left (B a^{6} - 6 \, A a^{5} b\right )} x}{60 \, {\left (b x + a\right )}^{5} a^{7} x} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 10.54 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (6\,A\,b-B\,a\right )}{a^7}-\frac {\frac {A}{a}+\frac {137\,x\,\left (6\,A\,b-B\,a\right )}{60\,a^2}+\frac {47\,b^2\,x^3\,\left (6\,A\,b-B\,a\right )}{6\,a^4}+\frac {9\,b^3\,x^4\,\left (6\,A\,b-B\,a\right )}{2\,a^5}+\frac {b^4\,x^5\,\left (6\,A\,b-B\,a\right )}{a^6}+\frac {77\,b\,x^2\,\left (6\,A\,b-B\,a\right )}{12\,a^3}}{a^5\,x+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^3+10\,a^2\,b^3\,x^4+5\,a\,b^4\,x^5+b^5\,x^6} \] Input:

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

Output:

(2*atanh((2*b*x)/a + 1)*(6*A*b - B*a))/a^7 - (A/a + (137*x*(6*A*b - B*a))/ 
(60*a^2) + (47*b^2*x^3*(6*A*b - B*a))/(6*a^4) + (9*b^3*x^4*(6*A*b - B*a))/ 
(2*a^5) + (b^4*x^5*(6*A*b - B*a))/a^6 + (77*b*x^2*(6*A*b - B*a))/(12*a^3)) 
/(a^5*x + b^5*x^6 + 5*a^4*b*x^2 + 5*a*b^4*x^5 + 10*a^3*b^2*x^3 + 10*a^2*b^ 
3*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 \,\mathrm {log}\left (b x +a \right ) a^{4} b x +240 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} x^{2}+360 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} x^{3}+240 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} x^{4}+60 \,\mathrm {log}\left (b x +a \right ) b^{5} x^{5}-60 \,\mathrm {log}\left (x \right ) a^{4} b x -240 \,\mathrm {log}\left (x \right ) a^{3} b^{2} x^{2}-360 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{3}-240 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{4}-60 \,\mathrm {log}\left (x \right ) b^{5} x^{5}-12 a^{5}-110 a^{4} b x -200 a^{3} b^{2} x^{2}-120 a^{2} b^{3} x^{3}+15 b^{5} x^{5}}{12 a^{6} x \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

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

Output:

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