Integrand size = 22, antiderivative size = 98 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1}{a^6 x}-\frac {b}{5 a^2 (a+b x)^5}-\frac {b}{2 a^3 (a+b x)^4}-\frac {b}{a^4 (a+b x)^3}-\frac {2 b}{a^5 (a+b x)^2}-\frac {5 b}{a^6 (a+b x)}-\frac {6 b \log (x)}{a^7}+\frac {6 b \log (a+b x)}{a^7} \] Output:
-1/a^6/x-1/5*b/a^2/(b*x+a)^5-1/2*b/a^3/(b*x+a)^4-b/a^4/(b*x+a)^3-2*b/a^5/( b*x+a)^2-5*b/a^6/(b*x+a)-6*b*ln(x)/a^7+6*b*ln(b*x+a)/a^7
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {a \left (10 a^5+137 a^4 b x+385 a^3 b^2 x^2+470 a^2 b^3 x^3+270 a b^4 x^4+60 b^5 x^5\right )}{x (a+b x)^5}+60 b \log (x)-60 b \log (a+b x)}{10 a^7} \] Input:
Integrate[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
-1/10*((a*(10*a^5 + 137*a^4*b*x + 385*a^3*b^2*x^2 + 470*a^2*b^3*x^3 + 270* a*b^4*x^4 + 60*b^5*x^5))/(x*(a + b*x)^5) + 60*b*Log[x] - 60*b*Log[a + b*x] )/a^7
Time = 0.45 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1098, 27, 54, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle b^6 \int \frac {1}{b^6 x^2 (a+b x)^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x^2 (a+b x)^6}dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (\frac {6 b^2}{a^7 (a+b x)}-\frac {6 b}{a^7 x}+\frac {5 b^2}{a^6 (a+b x)^2}+\frac {1}{a^6 x^2}+\frac {4 b^2}{a^5 (a+b x)^3}+\frac {3 b^2}{a^4 (a+b x)^4}+\frac {2 b^2}{a^3 (a+b x)^5}+\frac {b^2}{a^2 (a+b x)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6 b \log (x)}{a^7}+\frac {6 b \log (a+b x)}{a^7}-\frac {5 b}{a^6 (a+b x)}-\frac {1}{a^6 x}-\frac {2 b}{a^5 (a+b x)^2}-\frac {b}{a^4 (a+b x)^3}-\frac {b}{2 a^3 (a+b x)^4}-\frac {b}{5 a^2 (a+b x)^5}\) |
Input:
Int[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
-(1/(a^6*x)) - b/(5*a^2*(a + b*x)^5) - b/(2*a^3*(a + b*x)^4) - b/(a^4*(a + b*x)^3) - (2*b)/(a^5*(a + b*x)^2) - (5*b)/(a^6*(a + b*x)) - (6*b*Log[x])/ a^7 + (6*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_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.61 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {-\frac {1}{a}+\frac {30 b^{2} x^{2}}{a^{3}}+\frac {90 b^{3} x^{3}}{a^{4}}+\frac {110 b^{4} x^{4}}{a^{5}}+\frac {125 b^{5} x^{5}}{2 a^{6}}+\frac {137 b^{6} x^{6}}{10 a^{7}}}{x \left (b x +a \right )^{5}}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {6 b \ln \left (b x +a \right )}{a^{7}}\) | \(94\) |
default | \(-\frac {1}{a^{6} x}-\frac {b}{5 a^{2} \left (b x +a \right )^{5}}-\frac {b}{2 a^{3} \left (b x +a \right )^{4}}-\frac {b}{a^{4} \left (b x +a \right )^{3}}-\frac {2 b}{a^{5} \left (b x +a \right )^{2}}-\frac {5 b}{a^{6} \left (b x +a \right )}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {6 b \ln \left (b x +a \right )}{a^{7}}\) | \(95\) |
risch | \(\frac {-\frac {6 b^{5} x^{5}}{a^{6}}-\frac {27 b^{4} x^{4}}{a^{5}}-\frac {47 b^{3} x^{3}}{a^{4}}-\frac {77 b^{2} x^{2}}{2 a^{3}}-\frac {137 b x}{10 a^{2}}-\frac {1}{a}}{x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {6 b \ln \left (-b x -a \right )}{a^{7}}\) | \(111\) |
parallelrisch | \(-\frac {60 \ln \left (x \right ) x^{6} b^{6}-60 \ln \left (b x +a \right ) x^{6} b^{6}+300 \ln \left (x \right ) x^{5} a \,b^{5}-300 \ln \left (b x +a \right ) x^{5} a \,b^{5}-137 b^{6} x^{6}+600 \ln \left (x \right ) x^{4} a^{2} b^{4}-600 \ln \left (b x +a \right ) x^{4} a^{2} b^{4}-625 a \,b^{5} x^{5}+600 \ln \left (x \right ) x^{3} a^{3} b^{3}-600 \ln \left (b x +a \right ) x^{3} a^{3} b^{3}-1100 a^{2} b^{4} x^{4}+300 a^{4} b^{2} \ln \left (x \right ) x^{2}-300 \ln \left (b x +a \right ) x^{2} a^{4} b^{2}-900 a^{3} b^{3} x^{3}+60 a^{5} b \ln \left (x \right ) x -60 \ln \left (b x +a \right ) x \,a^{5} b -300 a^{4} b^{2} x^{2}+10 a^{6}}{10 a^{7} x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\) | \(252\) |
Input:
int(1/x^2/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
(-1/a+30*b^2/a^3*x^2+90*b^3/a^4*x^3+110*b^4/a^5*x^4+125/2*b^5/a^6*x^5+137/ 10*b^6/a^7*x^6)/x/(b*x+a)^5-6*b*ln(x)/a^7+6*b*ln(b*x+a)/a^7
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (94) = 188\).
Time = 0.11 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.46 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {60 \, a b^{5} x^{5} + 270 \, a^{2} b^{4} x^{4} + 470 \, a^{3} b^{3} x^{3} + 385 \, a^{4} b^{2} x^{2} + 137 \, a^{5} b x + 10 \, a^{6} - 60 \, {\left (b^{6} x^{6} + 5 \, a b^{5} x^{5} + 10 \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{3} + 5 \, a^{4} b^{2} x^{2} + a^{5} b x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 5 \, a b^{5} x^{5} + 10 \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{3} + 5 \, a^{4} b^{2} x^{2} + a^{5} b x\right )} \log \left (x\right )}{10 \, {\left (a^{7} b^{5} x^{6} + 5 \, a^{8} b^{4} x^{5} + 10 \, a^{9} b^{3} x^{4} + 10 \, a^{10} b^{2} x^{3} + 5 \, a^{11} b x^{2} + a^{12} x\right )}} \] Input:
integrate(1/x^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
-1/10*(60*a*b^5*x^5 + 270*a^2*b^4*x^4 + 470*a^3*b^3*x^3 + 385*a^4*b^2*x^2 + 137*a^5*b*x + 10*a^6 - 60*(b^6*x^6 + 5*a*b^5*x^5 + 10*a^2*b^4*x^4 + 10*a ^3*b^3*x^3 + 5*a^4*b^2*x^2 + a^5*b*x)*log(b*x + a) + 60*(b^6*x^6 + 5*a*b^5 *x^5 + 10*a^2*b^4*x^4 + 10*a^3*b^3*x^3 + 5*a^4*b^2*x^2 + 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^12*x)
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {- 10 a^{5} - 137 a^{4} b x - 385 a^{3} b^{2} x^{2} - 470 a^{2} b^{3} x^{3} - 270 a b^{4} x^{4} - 60 b^{5} x^{5}}{10 a^{11} x + 50 a^{10} b x^{2} + 100 a^{9} b^{2} x^{3} + 100 a^{8} b^{3} x^{4} + 50 a^{7} b^{4} x^{5} + 10 a^{6} b^{5} x^{6}} + \frac {6 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{7}} \] Input:
integrate(1/x**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
(-10*a**5 - 137*a**4*b*x - 385*a**3*b**2*x**2 - 470*a**2*b**3*x**3 - 270*a *b**4*x**4 - 60*b**5*x**5)/(10*a**11*x + 50*a**10*b*x**2 + 100*a**9*b**2*x **3 + 100*a**8*b**3*x**4 + 50*a**7*b**4*x**5 + 10*a**6*b**5*x**6) + 6*b*(- log(x) + log(a/b + x))/a**7
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {60 \, b^{5} x^{5} + 270 \, a b^{4} x^{4} + 470 \, a^{2} b^{3} x^{3} + 385 \, a^{3} b^{2} x^{2} + 137 \, a^{4} b x + 10 \, a^{5}}{10 \, {\left (a^{6} b^{5} x^{6} + 5 \, a^{7} b^{4} x^{5} + 10 \, a^{8} b^{3} x^{4} + 10 \, a^{9} b^{2} x^{3} + 5 \, a^{10} b x^{2} + a^{11} x\right )}} + \frac {6 \, b \log \left (b x + a\right )}{a^{7}} - \frac {6 \, b \log \left (x\right )}{a^{7}} \] Input:
integrate(1/x^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
-1/10*(60*b^5*x^5 + 270*a*b^4*x^4 + 470*a^2*b^3*x^3 + 385*a^3*b^2*x^2 + 13 7*a^4*b*x + 10*a^5)/(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) + 6*b*log(b*x + a)/a^7 - 6*b*log(x)/a^7
Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {6 \, b \log \left ({\left | b x + a \right |}\right )}{a^{7}} - \frac {6 \, b \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {60 \, a b^{5} x^{5} + 270 \, a^{2} b^{4} x^{4} + 470 \, a^{3} b^{3} x^{3} + 385 \, a^{4} b^{2} x^{2} + 137 \, a^{5} b x + 10 \, a^{6}}{10 \, {\left (b x + a\right )}^{5} a^{7} x} \] Input:
integrate(1/x^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
6*b*log(abs(b*x + a))/a^7 - 6*b*log(abs(x))/a^7 - 1/10*(60*a*b^5*x^5 + 270 *a^2*b^4*x^4 + 470*a^3*b^3*x^3 + 385*a^4*b^2*x^2 + 137*a^5*b*x + 10*a^6)/( (b*x + a)^5*a^7*x)
Time = 9.77 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {12\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^7}-\frac {\frac {1}{a}+\frac {77\,b^2\,x^2}{2\,a^3}+\frac {47\,b^3\,x^3}{a^4}+\frac {27\,b^4\,x^4}{a^5}+\frac {6\,b^5\,x^5}{a^6}+\frac {137\,b\,x}{10\,a^2}}{a^5\,x+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^3+10\,a^2\,b^3\,x^4+5\,a\,b^4\,x^5+b^5\,x^6} \] Input:
int(1/(x^2*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
Output:
(12*b*atanh((2*b*x)/a + 1))/a^7 - (1/a + (77*b^2*x^2)/(2*a^3) + (47*b^3*x^ 3)/a^4 + (27*b^4*x^4)/a^5 + (6*b^5*x^5)/a^6 + (137*b*x)/(10*a^2))/(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)
Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.81 \[ \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 \,\mathrm {log}\left (b x +a \right ) a^{5} b x +300 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} x^{2}+600 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} x^{3}+600 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} x^{4}+300 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} x^{5}+60 \,\mathrm {log}\left (b x +a \right ) b^{6} x^{6}-60 \,\mathrm {log}\left (x \right ) a^{5} b x -300 \,\mathrm {log}\left (x \right ) a^{4} b^{2} x^{2}-600 \,\mathrm {log}\left (x \right ) a^{3} b^{3} x^{3}-600 \,\mathrm {log}\left (x \right ) a^{2} b^{4} x^{4}-300 \,\mathrm {log}\left (x \right ) a \,b^{5} x^{5}-60 \,\mathrm {log}\left (x \right ) b^{6} x^{6}-10 a^{6}-125 a^{5} b x -325 a^{4} b^{2} x^{2}-350 a^{3} b^{3} x^{3}-150 a^{2} b^{4} x^{4}+12 b^{6} x^{6}}{10 a^{7} x \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right )} \] Input:
int(1/x^2/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
(60*log(a + b*x)*a**5*b*x + 300*log(a + b*x)*a**4*b**2*x**2 + 600*log(a + b*x)*a**3*b**3*x**3 + 600*log(a + b*x)*a**2*b**4*x**4 + 300*log(a + b*x)*a *b**5*x**5 + 60*log(a + b*x)*b**6*x**6 - 60*log(x)*a**5*b*x - 300*log(x)*a **4*b**2*x**2 - 600*log(x)*a**3*b**3*x**3 - 600*log(x)*a**2*b**4*x**4 - 30 0*log(x)*a*b**5*x**5 - 60*log(x)*b**6*x**6 - 10*a**6 - 125*a**5*b*x - 325* a**4*b**2*x**2 - 350*a**3*b**3*x**3 - 150*a**2*b**4*x**4 + 12*b**6*x**6)/( 10*a**7*x*(a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x**3 + 5*a *b**4*x**4 + b**5*x**5))