Integrand size = 27, antiderivative size = 166 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {A}{3 a^4 x^3}+\frac {4 A b-a B}{2 a^5 x^2}-\frac {2 b (5 A b-2 a B)}{a^6 x}-\frac {b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac {b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}-\frac {2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac {10 b^2 (2 A b-a B) \log (x)}{a^7}+\frac {10 b^2 (2 A b-a B) \log (a+b x)}{a^7} \] Output:
-1/3*A/a^4/x^3+1/2*(4*A*b-B*a)/a^5/x^2-2*b*(5*A*b-2*B*a)/a^6/x-1/3*b^2*(A* b-B*a)/a^4/(b*x+a)^3-1/2*b^2*(4*A*b-3*B*a)/a^5/(b*x+a)^2-2*b^2*(5*A*b-3*B* a)/a^6/(b*x+a)-10*b^2*(2*A*b-B*a)*ln(x)/a^7+10*b^2*(2*A*b-B*a)*ln(b*x+a)/a ^7
Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {a \left (-120 A b^5 x^5+60 a b^4 x^4 (-5 A+B x)-a^5 (2 A+3 B x)+3 a^4 b x (2 A+5 B x)+10 a^3 b^2 x^2 (-3 A+11 B x)+10 a^2 b^3 x^3 (-22 A+15 B x)\right )}{x^3 (a+b x)^3}-60 b^2 (2 A b-a B) \log (x)+60 b^2 (2 A b-a B) \log (a+b x)}{6 a^7} \] Input:
Integrate[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
Output:
((a*(-120*A*b^5*x^5 + 60*a*b^4*x^4*(-5*A + B*x) - a^5*(2*A + 3*B*x) + 3*a^ 4*b*x*(2*A + 5*B*x) + 10*a^3*b^2*x^2*(-3*A + 11*B*x) + 10*a^2*b^3*x^3*(-22 *A + 15*B*x)))/(x^3*(a + b*x)^3) - 60*b^2*(2*A*b - a*B)*Log[x] + 60*b^2*(2 *A*b - a*B)*Log[a + b*x])/(6*a^7)
Time = 0.69 (sec) , antiderivative size = 166, 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.
| \(\displaystyle \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {A+B x}{b^4 x^4 (a+b x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{x^4 (a+b x)^4}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {10 b^3 (a B-2 A b)}{a^7 (a+b x)}+\frac {10 b^2 (a B-2 A b)}{a^7 x}-\frac {2 b^3 (3 a B-5 A b)}{a^6 (a+b x)^2}-\frac {2 b (2 a B-5 A b)}{a^6 x^2}-\frac {b^3 (3 a B-4 A b)}{a^5 (a+b x)^3}+\frac {a B-4 A b}{a^5 x^3}-\frac {b^3 (a B-A b)}{a^4 (a+b x)^4}+\frac {A}{a^4 x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^2 \log (x) (2 A b-a B)}{a^7}+\frac {10 b^2 (2 A b-a B) \log (a+b x)}{a^7}-\frac {2 b^2 (5 A b-3 a B)}{a^6 (a+b x)}-\frac {2 b (5 A b-2 a B)}{a^6 x}-\frac {b^2 (4 A b-3 a B)}{2 a^5 (a+b x)^2}+\frac {4 A b-a B}{2 a^5 x^2}-\frac {b^2 (A b-a B)}{3 a^4 (a+b x)^3}-\frac {A}{3 a^4 x^3}\) |
Input:
Int[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
Output:
-1/3*A/(a^4*x^3) + (4*A*b - a*B)/(2*a^5*x^2) - (2*b*(5*A*b - 2*a*B))/(a^6* x) - (b^2*(A*b - a*B))/(3*a^4*(a + b*x)^3) - (b^2*(4*A*b - 3*a*B))/(2*a^5* (a + b*x)^2) - (2*b^2*(5*A*b - 3*a*B))/(a^6*(a + b*x)) - (10*b^2*(2*A*b - a*B)*Log[x])/a^7 + (10*b^2*(2*A*b - a*B)*Log[a + b*x])/a^7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
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]
Time = 1.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {b^{2} \left (4 A b -3 B a \right )}{2 a^{5} \left (b x +a \right )^{2}}-\frac {b^{2} \left (A b -B a \right )}{3 a^{4} \left (b x +a \right )^{3}}+\frac {10 b^{2} \left (2 A b -B a \right ) \ln \left (b x +a \right )}{a^{7}}-\frac {2 b^{2} \left (5 A b -3 B a \right )}{a^{6} \left (b x +a \right )}-\frac {A}{3 a^{4} x^{3}}-\frac {-4 A b +B a}{2 x^{2} a^{5}}-\frac {2 b \left (5 A b -2 B a \right )}{a^{6} x}-\frac {10 b^{2} \left (2 A b -B a \right ) \ln \left (x \right )}{a^{7}}\) | \(158\) |
| norman | \(\frac {-\frac {A}{3 a}+\frac {\left (2 A b -B a \right ) x}{2 a^{2}}-\frac {5 b \left (2 A b -B a \right ) x^{2}}{2 a^{3}}+\frac {3 b \left (20 A \,b^{3}-10 B a \,b^{2}\right ) x^{4}}{a^{5}}+\frac {3 b^{2} \left (30 A \,b^{3}-15 B a \,b^{2}\right ) x^{5}}{a^{6}}+\frac {b^{3} \left (110 A \,b^{3}-55 B a \,b^{2}\right ) x^{6}}{3 a^{7}}}{x^{3} \left (b x +a \right )^{3}}-\frac {10 b^{2} \left (2 A b -B a \right ) \ln \left (x \right )}{a^{7}}+\frac {10 b^{2} \left (2 A b -B a \right ) \ln \left (b x +a \right )}{a^{7}}\) | \(168\) |
| risch | \(\frac {-\frac {10 b^{4} \left (2 A b -B a \right ) x^{5}}{a^{6}}-\frac {25 b^{3} \left (2 A b -B a \right ) x^{4}}{a^{5}}-\frac {55 b^{2} \left (2 A b -B a \right ) x^{3}}{3 a^{4}}-\frac {5 b \left (2 A b -B a \right ) x^{2}}{2 a^{3}}+\frac {\left (2 A b -B a \right ) x}{2 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}+\frac {20 b^{3} \ln \left (-b x -a \right ) A}{a^{7}}-\frac {10 b^{2} \ln \left (-b x -a \right ) B}{a^{6}}-\frac {20 b^{3} \ln \left (x \right ) A}{a^{7}}+\frac {10 b^{2} \ln \left (x \right ) B}{a^{6}}\) | \(189\) |
| parallelrisch | \(-\frac {60 B \ln \left (b x +a \right ) x^{6} a \,b^{5}-360 A \ln \left (b x +a \right ) x^{5} a \,b^{5}+180 B \ln \left (b x +a \right ) x^{5} a^{2} b^{4}-360 A \ln \left (b x +a \right ) x^{4} a^{2} b^{4}+180 B \ln \left (b x +a \right ) x^{4} a^{3} b^{3}-120 A \ln \left (b x +a \right ) x^{3} a^{3} b^{3}+60 B \ln \left (b x +a \right ) x^{3} a^{4} b^{2}+120 A \ln \left (x \right ) x^{3} a^{3} b^{3}-60 B \ln \left (x \right ) x^{6} a \,b^{5}-180 B \ln \left (x \right ) x^{5} a^{2} b^{4}-540 A a \,b^{5} x^{5}+270 B \,a^{2} b^{4} x^{5}+360 A \ln \left (x \right ) x^{5} a \,b^{5}-180 B \ln \left (x \right ) x^{4} a^{3} b^{3}-15 B \,a^{5} b \,x^{2}-6 A \,a^{5} b x +360 A \ln \left (x \right ) x^{4} a^{2} b^{4}-60 B \ln \left (x \right ) x^{3} a^{4} b^{2}+2 A \,a^{6}+120 A \ln \left (x \right ) x^{6} b^{6}-120 A \ln \left (b x +a \right ) x^{6} b^{6}+110 B a \,b^{5} x^{6}+30 A \,a^{4} b^{2} x^{2}+180 B \,a^{3} b^{3} x^{4}-360 A \,a^{2} b^{4} x^{4}+3 B \,a^{6} x -220 A \,b^{6} x^{6}}{6 a^{7} x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\) | \(385\) |
Input:
int((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*b^2*(4*A*b-3*B*a)/a^5/(b*x+a)^2-1/3*b^2*(A*b-B*a)/a^4/(b*x+a)^3+10*b^ 2*(2*A*b-B*a)*ln(b*x+a)/a^7-2*b^2*(5*A*b-3*B*a)/a^6/(b*x+a)-1/3*A/a^4/x^3- 1/2*(-4*A*b+B*a)/x^2/a^5-2*b*(5*A*b-2*B*a)/a^6/x-10*b^2*(2*A*b-B*a)*ln(x)/ a^7
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (155) = 310\).
Time = 0.08 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 \, A a^{6} - 60 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} - 150 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} - 110 \, {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3} - 15 \, {\left (B a^{5} b - 2 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (B a^{6} - 2 \, A a^{5} b\right )} x + 60 \, {\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} + {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 60 \, {\left ({\left (B a b^{5} - 2 \, A b^{6}\right )} x^{6} + 3 \, {\left (B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{5} + 3 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{4} + {\left (B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \] Input:
integrate((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
-1/6*(2*A*a^6 - 60*(B*a^2*b^4 - 2*A*a*b^5)*x^5 - 150*(B*a^3*b^3 - 2*A*a^2* b^4)*x^4 - 110*(B*a^4*b^2 - 2*A*a^3*b^3)*x^3 - 15*(B*a^5*b - 2*A*a^4*b^2)* x^2 + 3*(B*a^6 - 2*A*a^5*b)*x + 60*((B*a*b^5 - 2*A*b^6)*x^6 + 3*(B*a^2*b^4 - 2*A*a*b^5)*x^5 + 3*(B*a^3*b^3 - 2*A*a^2*b^4)*x^4 + (B*a^4*b^2 - 2*A*a^3 *b^3)*x^3)*log(b*x + a) - 60*((B*a*b^5 - 2*A*b^6)*x^6 + 3*(B*a^2*b^4 - 2*A *a*b^5)*x^5 + 3*(B*a^3*b^3 - 2*A*a^2*b^4)*x^4 + (B*a^4*b^2 - 2*A*a^3*b^3)* x^3)*log(x))/(a^7*b^3*x^6 + 3*a^8*b^2*x^5 + 3*a^9*b*x^4 + a^10*x^3)
Time = 0.50 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {- 2 A a^{5} + x^{5} \left (- 120 A b^{5} + 60 B a b^{4}\right ) + x^{4} \left (- 300 A a b^{4} + 150 B a^{2} b^{3}\right ) + x^{3} \left (- 220 A a^{2} b^{3} + 110 B a^{3} b^{2}\right ) + x^{2} \left (- 30 A a^{3} b^{2} + 15 B a^{4} b\right ) + x \left (6 A a^{4} b - 3 B a^{5}\right )}{6 a^{9} x^{3} + 18 a^{8} b x^{4} + 18 a^{7} b^{2} x^{5} + 6 a^{6} b^{3} x^{6}} + \frac {10 b^{2} \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 20 A a b^{3} + 10 B a^{2} b^{2} - 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} - \frac {10 b^{2} \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 20 A a b^{3} + 10 B a^{2} b^{2} + 10 a b^{2} \left (- 2 A b + B a\right )}{- 40 A b^{4} + 20 B a b^{3}} \right )}}{a^{7}} \] Input:
integrate((B*x+A)/x**4/(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
(-2*A*a**5 + x**5*(-120*A*b**5 + 60*B*a*b**4) + x**4*(-300*A*a*b**4 + 150* B*a**2*b**3) + x**3*(-220*A*a**2*b**3 + 110*B*a**3*b**2) + x**2*(-30*A*a** 3*b**2 + 15*B*a**4*b) + x*(6*A*a**4*b - 3*B*a**5))/(6*a**9*x**3 + 18*a**8* b*x**4 + 18*a**7*b**2*x**5 + 6*a**6*b**3*x**6) + 10*b**2*(-2*A*b + B*a)*lo g(x + (-20*A*a*b**3 + 10*B*a**2*b**2 - 10*a*b**2*(-2*A*b + B*a))/(-40*A*b* *4 + 20*B*a*b**3))/a**7 - 10*b**2*(-2*A*b + B*a)*log(x + (-20*A*a*b**3 + 1 0*B*a**2*b**2 + 10*a*b**2*(-2*A*b + B*a))/(-40*A*b**4 + 20*B*a*b**3))/a**7
Time = 0.03 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 \, A a^{5} - 60 \, {\left (B a b^{4} - 2 \, A b^{5}\right )} x^{5} - 150 \, {\left (B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{4} - 110 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{3} - 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} x}{6 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} - \frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{7}} + \frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left (x\right )}{a^{7}} \] Input:
integrate((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
-1/6*(2*A*a^5 - 60*(B*a*b^4 - 2*A*b^5)*x^5 - 150*(B*a^2*b^3 - 2*A*a*b^4)*x ^4 - 110*(B*a^3*b^2 - 2*A*a^2*b^3)*x^3 - 15*(B*a^4*b - 2*A*a^3*b^2)*x^2 + 3*(B*a^5 - 2*A*a^4*b)*x)/(a^6*b^3*x^6 + 3*a^7*b^2*x^5 + 3*a^8*b*x^4 + a^9* x^3) - 10*(B*a*b^2 - 2*A*b^3)*log(b*x + a)/a^7 + 10*(B*a*b^2 - 2*A*b^3)*lo g(x)/a^7
Time = 0.19 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {10 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {10 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{7} b} + \frac {60 \, B a b^{4} x^{5} - 120 \, A b^{5} x^{5} + 150 \, B a^{2} b^{3} x^{4} - 300 \, A a b^{4} x^{4} + 110 \, B a^{3} b^{2} x^{3} - 220 \, A a^{2} b^{3} x^{3} + 15 \, B a^{4} b x^{2} - 30 \, A a^{3} b^{2} x^{2} - 3 \, B a^{5} x + 6 \, A a^{4} b x - 2 \, A a^{5}}{6 \, {\left (b x^{2} + a x\right )}^{3} a^{6}} \] Input:
integrate((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
10*(B*a*b^2 - 2*A*b^3)*log(abs(x))/a^7 - 10*(B*a*b^3 - 2*A*b^4)*log(abs(b* x + a))/(a^7*b) + 1/6*(60*B*a*b^4*x^5 - 120*A*b^5*x^5 + 150*B*a^2*b^3*x^4 - 300*A*a*b^4*x^4 + 110*B*a^3*b^2*x^3 - 220*A*a^2*b^3*x^3 + 15*B*a^4*b*x^2 - 30*A*a^3*b^2*x^2 - 3*B*a^5*x + 6*A*a^4*b*x - 2*A*a^5)/((b*x^2 + a*x)^3* a^6)
Time = 10.56 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {20\,b^2\,\mathrm {atanh}\left (\frac {10\,b^2\,\left (2\,A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (20\,A\,b^3-10\,B\,a\,b^2\right )}\right )\,\left (2\,A\,b-B\,a\right )}{a^7}-\frac {\frac {A}{3\,a}-\frac {x\,\left (2\,A\,b-B\,a\right )}{2\,a^2}+\frac {55\,b^2\,x^3\,\left (2\,A\,b-B\,a\right )}{3\,a^4}+\frac {25\,b^3\,x^4\,\left (2\,A\,b-B\,a\right )}{a^5}+\frac {10\,b^4\,x^5\,\left (2\,A\,b-B\,a\right )}{a^6}+\frac {5\,b\,x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^3\,x^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^5+b^3\,x^6} \] Input:
int((A + B*x)/(x^4*(a^2 + b^2*x^2 + 2*a*b*x)^2),x)
Output:
(20*b^2*atanh((10*b^2*(2*A*b - B*a)*(a + 2*b*x))/(a*(20*A*b^3 - 10*B*a*b^2 )))*(2*A*b - B*a))/a^7 - (A/(3*a) - (x*(2*A*b - B*a))/(2*a^2) + (55*b^2*x^ 3*(2*A*b - B*a))/(3*a^4) + (25*b^3*x^4*(2*A*b - B*a))/a^5 + (10*b^4*x^5*(2 *A*b - B*a))/a^6 + (5*b*x^2*(2*A*b - B*a))/(2*a^3))/(a^3*x^3 + b^3*x^6 + 3 *a^2*b*x^4 + 3*a*b^2*x^5)
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {60 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} x^{3}+120 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} x^{4}+60 \,\mathrm {log}\left (b x +a \right ) b^{5} x^{5}-60 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{3}-120 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{4}-60 \,\mathrm {log}\left (x \right ) b^{5} x^{5}-2 a^{5}+5 a^{4} b x -20 a^{3} b^{2} x^{2}-60 a^{2} b^{3} x^{3}+30 b^{5} x^{5}}{6 a^{6} x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(60*log(a + b*x)*a**2*b**3*x**3 + 120*log(a + b*x)*a*b**4*x**4 + 60*log(a + b*x)*b**5*x**5 - 60*log(x)*a**2*b**3*x**3 - 120*log(x)*a*b**4*x**4 - 60* log(x)*b**5*x**5 - 2*a**5 + 5*a**4*b*x - 20*a**3*b**2*x**2 - 60*a**2*b**3* x**3 + 30*b**5*x**5)/(6*a**6*x**3*(a**2 + 2*a*b*x + b**2*x**2))