Integrand size = 22, antiderivative size = 85 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{5 a (a+b x)^5}+\frac {1}{4 a^2 (a+b x)^4}+\frac {1}{3 a^3 (a+b x)^3}+\frac {1}{2 a^4 (a+b x)^2}+\frac {1}{a^5 (a+b x)}+\frac {\log (x)}{a^6}-\frac {\log (a+b x)}{a^6} \] Output:
1/5/a/(b*x+a)^5+1/4/a^2/(b*x+a)^4+1/3/a^3/(b*x+a)^3+1/2/a^4/(b*x+a)^2+1/a^ 5/(b*x+a)+ln(x)/a^6-ln(b*x+a)/a^6
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {a \left (137 a^4+385 a^3 b x+470 a^2 b^2 x^2+270 a b^3 x^3+60 b^4 x^4\right )}{(a+b x)^5}+60 \log (x)-60 \log (a+b x)}{60 a^6} \] Input:
Integrate[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
((a*(137*a^4 + 385*a^3*b*x + 470*a^2*b^2*x^2 + 270*a*b^3*x^3 + 60*b^4*x^4) )/(a + b*x)^5 + 60*Log[x] - 60*Log[a + b*x])/(60*a^6)
Time = 0.41 (sec) , antiderivative size = 85, 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.
\(\displaystyle \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle b^6 \int \frac {1}{b^6 x (a+b x)^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x (a+b x)^6}dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {b}{a^6 (a+b x)}+\frac {1}{a^6 x}-\frac {b}{a^5 (a+b x)^2}-\frac {b}{a^4 (a+b x)^3}-\frac {b}{a^3 (a+b x)^4}-\frac {b}{a^2 (a+b x)^5}-\frac {b}{a (a+b x)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log (a+b x)}{a^6}+\frac {\log (x)}{a^6}+\frac {1}{a^5 (a+b x)}+\frac {1}{2 a^4 (a+b x)^2}+\frac {1}{3 a^3 (a+b x)^3}+\frac {1}{4 a^2 (a+b x)^4}+\frac {1}{5 a (a+b x)^5}\) |
Input:
Int[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
Output:
1/(5*a*(a + b*x)^5) + 1/(4*a^2*(a + b*x)^4) + 1/(3*a^3*(a + b*x)^3) + 1/(2 *a^4*(a + b*x)^2) + 1/(a^5*(a + b*x)) + Log[x]/a^6 - Log[a + b*x]/a^6
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 = 0.60 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {1}{5 a \left (b x +a \right )^{5}}+\frac {1}{4 a^{2} \left (b x +a \right )^{4}}+\frac {1}{3 a^{3} \left (b x +a \right )^{3}}+\frac {1}{2 a^{4} \left (b x +a \right )^{2}}+\frac {1}{a^{5} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{6}}-\frac {\ln \left (b x +a \right )}{a^{6}}\) | \(78\) |
norman | \(\frac {-\frac {5 b x}{a^{2}}-\frac {15 b^{2} x^{2}}{a^{3}}-\frac {55 b^{3} x^{3}}{3 a^{4}}-\frac {125 b^{4} x^{4}}{12 a^{5}}-\frac {137 b^{5} x^{5}}{60 a^{6}}}{\left (b x +a \right )^{5}}+\frac {\ln \left (x \right )}{a^{6}}-\frac {\ln \left (b x +a \right )}{a^{6}}\) | \(79\) |
risch | \(\frac {\frac {b^{4} x^{4}}{a^{5}}+\frac {9 b^{3} x^{3}}{2 a^{4}}+\frac {47 b^{2} x^{2}}{6 a^{3}}+\frac {77 b x}{12 a^{2}}+\frac {137}{60 a}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}+\frac {\ln \left (-x \right )}{a^{6}}-\frac {\ln \left (b x +a \right )}{a^{6}}\) | \(92\) |
parallelrisch | \(\frac {60 \ln \left (x \right ) x^{5} b^{5}-60 \ln \left (b x +a \right ) x^{5} b^{5}+300 \ln \left (x \right ) x^{4} a \,b^{4}-300 \ln \left (b x +a \right ) x^{4} a \,b^{4}-137 b^{5} x^{5}+600 \ln \left (x \right ) x^{3} a^{2} b^{3}-600 \ln \left (b x +a \right ) x^{3} a^{2} b^{3}-625 a \,b^{4} x^{4}+600 \ln \left (x \right ) x^{2} a^{3} b^{2}-600 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}-1100 a^{2} b^{3} x^{3}+300 \ln \left (x \right ) x \,a^{4} b -300 \ln \left (b x +a \right ) x \,a^{4} b -900 a^{3} b^{2} x^{2}+60 \ln \left (x \right ) a^{5}-60 \ln \left (b x +a \right ) a^{5}-300 a^{4} b x}{60 a^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\) | \(228\) |
Input:
int(1/x/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/5/a/(b*x+a)^5+1/4/a^2/(b*x+a)^4+1/3/a^3/(b*x+a)^3+1/2/a^4/(b*x+a)^2+1/a^ 5/(b*x+a)+ln(x)/a^6-ln(b*x+a)/a^6
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (77) = 154\).
Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 \, a b^{4} x^{4} + 270 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 385 \, a^{4} b x + 137 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \log \left (x\right )}{60 \, {\left (a^{6} b^{5} x^{5} + 5 \, a^{7} b^{4} x^{4} + 10 \, a^{8} b^{3} x^{3} + 10 \, a^{9} b^{2} x^{2} + 5 \, a^{10} b x + a^{11}\right )}} \] Input:
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/60*(60*a*b^4*x^4 + 270*a^2*b^3*x^3 + 470*a^3*b^2*x^2 + 385*a^4*b*x + 137 *a^5 - 60*(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)*log(b*x + a) + 60*(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)*log(x))/(a^6*b^5*x^5 + 5*a^7*b^4*x^4 + 10* a^8*b^3*x^3 + 10*a^9*b^2*x^2 + 5*a^10*b*x + a^11)
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {137 a^{4} + 385 a^{3} b x + 470 a^{2} b^{2} x^{2} + 270 a b^{3} x^{3} + 60 b^{4} x^{4}}{60 a^{10} + 300 a^{9} b x + 600 a^{8} b^{2} x^{2} + 600 a^{7} b^{3} x^{3} + 300 a^{6} b^{4} x^{4} + 60 a^{5} b^{5} x^{5}} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{6}} \] Input:
integrate(1/x/(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
(137*a**4 + 385*a**3*b*x + 470*a**2*b**2*x**2 + 270*a*b**3*x**3 + 60*b**4* x**4)/(60*a**10 + 300*a**9*b*x + 600*a**8*b**2*x**2 + 600*a**7*b**3*x**3 + 300*a**6*b**4*x**4 + 60*a**5*b**5*x**5) + (log(x) - log(a/b + x))/a**6
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {60 \, b^{4} x^{4} + 270 \, a b^{3} x^{3} + 470 \, a^{2} b^{2} x^{2} + 385 \, a^{3} b x + 137 \, a^{4}}{60 \, {\left (a^{5} b^{5} x^{5} + 5 \, a^{6} b^{4} x^{4} + 10 \, a^{7} b^{3} x^{3} + 10 \, a^{8} b^{2} x^{2} + 5 \, a^{9} b x + a^{10}\right )}} - \frac {\log \left (b x + a\right )}{a^{6}} + \frac {\log \left (x\right )}{a^{6}} \] Input:
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
1/60*(60*b^4*x^4 + 270*a*b^3*x^3 + 470*a^2*b^2*x^2 + 385*a^3*b*x + 137*a^4 )/(a^5*b^5*x^5 + 5*a^6*b^4*x^4 + 10*a^7*b^3*x^3 + 10*a^8*b^2*x^2 + 5*a^9*b *x + a^10) - log(b*x + a)/a^6 + log(x)/a^6
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac {\log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, a b^{4} x^{4} + 270 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 385 \, a^{4} b x + 137 \, a^{5}}{60 \, {\left (b x + a\right )}^{5} a^{6}} \] Input:
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
-log(abs(b*x + a))/a^6 + log(abs(x))/a^6 + 1/60*(60*a*b^4*x^4 + 270*a^2*b^ 3*x^3 + 470*a^3*b^2*x^2 + 385*a^4*b*x + 137*a^5)/((b*x + a)^5*a^6)
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {137}{60\,a}+\frac {47\,b^2\,x^2}{6\,a^3}+\frac {9\,b^3\,x^3}{2\,a^4}+\frac {b^4\,x^4}{a^5}+\frac {77\,b\,x}{12\,a^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \] Input:
int(1/(x*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
Output:
(137/(60*a) + (47*b^2*x^2)/(6*a^3) + (9*b^3*x^3)/(2*a^4) + (b^4*x^4)/a^5 + (77*b*x)/(12*a^2))/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2 *b^3*x^3 + 5*a^4*b*x) - (2*atanh((2*b*x)/a + 1))/a^6
Time = 0.31 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.93 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{5}-300 \,\mathrm {log}\left (b x +a \right ) a^{4} b x -600 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} x^{2}-600 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} x^{3}-300 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} x^{4}-60 \,\mathrm {log}\left (b x +a \right ) b^{5} x^{5}+60 \,\mathrm {log}\left (x \right ) a^{5}+300 \,\mathrm {log}\left (x \right ) a^{4} b x +600 \,\mathrm {log}\left (x \right ) a^{3} b^{2} x^{2}+600 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{3}+300 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{4}+60 \,\mathrm {log}\left (x \right ) b^{5} x^{5}+125 a^{5}+325 a^{4} b x +350 a^{3} b^{2} x^{2}+150 a^{2} b^{3} x^{3}-12 b^{5} x^{5}}{60 a^{6} \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right )} \] Input:
int(1/x/(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
( - 60*log(a + b*x)*a**5 - 300*log(a + b*x)*a**4*b*x - 600*log(a + b*x)*a* *3*b**2*x**2 - 600*log(a + b*x)*a**2*b**3*x**3 - 300*log(a + b*x)*a*b**4*x **4 - 60*log(a + b*x)*b**5*x**5 + 60*log(x)*a**5 + 300*log(x)*a**4*b*x + 6 00*log(x)*a**3*b**2*x**2 + 600*log(x)*a**2*b**3*x**3 + 300*log(x)*a*b**4*x **4 + 60*log(x)*b**5*x**5 + 125*a**5 + 325*a**4*b*x + 350*a**3*b**2*x**2 + 150*a**2*b**3*x**3 - 12*b**5*x**5)/(60*a**6*(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))