Integrand size = 13, antiderivative size = 93 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=-\frac {1}{8 a^2 x^8}+\frac {b}{3 a^3 x^6}-\frac {3 b^2}{4 a^4 x^4}+\frac {2 b^3}{a^5 x^2}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log \left (a+b x^2\right )}{2 a^6} \] Output:
-1/8/a^2/x^8+1/3*b/a^3/x^6-3/4*b^2/a^4/x^4+2*b^3/a^5/x^2+1/2*b^4/a^5/(b*x^ 2+a)+5*b^4*ln(x)/a^6-5/2*b^4*ln(b*x^2+a)/a^6
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {a \left (-\frac {3 a^3}{x^8}+\frac {8 a^2 b}{x^6}-\frac {18 a b^2}{x^4}+12 b^3 \left (\frac {4}{x^2}+\frac {b}{a+b x^2}\right )\right )+120 b^4 \log (x)-60 b^4 \log \left (a+b x^2\right )}{24 a^6} \] Input:
Integrate[1/(x^9*(a + b*x^2)^2),x]
Output:
(a*((-3*a^3)/x^8 + (8*a^2*b)/x^6 - (18*a*b^2)/x^4 + 12*b^3*(4/x^2 + b/(a + b*x^2))) + 120*b^4*Log[x] - 60*b^4*Log[a + b*x^2])/(24*a^6)
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 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.
\(\displaystyle \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^{10} \left (b x^2+a\right )^2}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {5 b^5}{a^6 \left (b x^2+a\right )}-\frac {b^5}{a^5 \left (b x^2+a\right )^2}+\frac {5 b^4}{a^6 x^2}-\frac {4 b^3}{a^5 x^4}+\frac {3 b^2}{a^4 x^6}-\frac {2 b}{a^3 x^8}+\frac {1}{a^2 x^{10}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {5 b^4 \log \left (x^2\right )}{a^6}-\frac {5 b^4 \log \left (a+b x^2\right )}{a^6}+\frac {b^4}{a^5 \left (a+b x^2\right )}+\frac {4 b^3}{a^5 x^2}-\frac {3 b^2}{2 a^4 x^4}+\frac {2 b}{3 a^3 x^6}-\frac {1}{4 a^2 x^8}\right )\) |
Input:
Int[1/(x^9*(a + b*x^2)^2),x]
Output:
(-1/4*1/(a^2*x^8) + (2*b)/(3*a^3*x^6) - (3*b^2)/(2*a^4*x^4) + (4*b^3)/(a^5 *x^2) + b^4/(a^5*(a + b*x^2)) + (5*b^4*Log[x^2])/a^6 - (5*b^4*Log[a + b*x^ 2])/a^6)/2
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {b^{5} \left (-\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {5 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{6}}-\frac {1}{8 a^{2} x^{8}}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{2}}{4 a^{4} x^{4}}+\frac {b}{3 a^{3} x^{6}}\) | \(88\) |
norman | \(\frac {-\frac {1}{8 a}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{5} x^{10}}{2 a^{6}}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\) | \(89\) |
risch | \(\frac {\frac {5 b^{4} x^{8}}{2 a^{5}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {1}{8 a}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\) | \(89\) |
parallelrisch | \(\frac {120 \ln \left (x \right ) x^{10} b^{5}-60 \ln \left (b \,x^{2}+a \right ) x^{10} b^{5}-60 b^{5} x^{10}+120 a \,b^{4} \ln \left (x \right ) x^{8}-60 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{4}+30 a^{2} b^{3} x^{6}-10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}-3 a^{5}}{24 a^{6} x^{8} \left (b \,x^{2}+a \right )}\) | \(117\) |
Input:
int(1/x^9/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*b^5/a^6*(-a/b/(b*x^2+a)+5*ln(b*x^2+a)/b)-1/8/a^2/x^8+5*b^4*ln(x)/a^6+ 2*b^3/a^5/x^2-3/4*b^2/a^4/x^4+1/3*b/a^3/x^6
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {60 \, a b^{4} x^{8} + 30 \, a^{2} b^{3} x^{6} - 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (x\right )}{24 \, {\left (a^{6} b x^{10} + a^{7} x^{8}\right )}} \] Input:
integrate(1/x^9/(b*x^2+a)^2,x, algorithm="fricas")
Output:
1/24*(60*a*b^4*x^8 + 30*a^2*b^3*x^6 - 10*a^3*b^2*x^4 + 5*a^4*b*x^2 - 3*a^5 - 60*(b^5*x^10 + a*b^4*x^8)*log(b*x^2 + a) + 120*(b^5*x^10 + a*b^4*x^8)*l og(x))/(a^6*b*x^10 + a^7*x^8)
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {- 3 a^{4} + 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} + 30 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{6} x^{8} + 24 a^{5} b x^{10}} + \frac {5 b^{4} \log {\left (x \right )}}{a^{6}} - \frac {5 b^{4} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{6}} \] Input:
integrate(1/x**9/(b*x**2+a)**2,x)
Output:
(-3*a**4 + 5*a**3*b*x**2 - 10*a**2*b**2*x**4 + 30*a*b**3*x**6 + 60*b**4*x* *8)/(24*a**6*x**8 + 24*a**5*b*x**10) + 5*b**4*log(x)/a**6 - 5*b**4*log(a/b + x**2)/(2*a**6)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {60 \, b^{4} x^{8} + 30 \, a b^{3} x^{6} - 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, {\left (a^{5} b x^{10} + a^{6} x^{8}\right )}} - \frac {5 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} \] Input:
integrate(1/x^9/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/24*(60*b^4*x^8 + 30*a*b^3*x^6 - 10*a^2*b^2*x^4 + 5*a^3*b*x^2 - 3*a^4)/(a ^5*b*x^10 + a^6*x^8) - 5/2*b^4*log(b*x^2 + a)/a^6 + 5/2*b^4*log(x^2)/a^6
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac {5 \, b^{5} x^{2} + 6 \, a b^{4}}{2 \, {\left (b x^{2} + a\right )} a^{6}} - \frac {125 \, b^{4} x^{8} - 48 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 8 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{6} x^{8}} \] Input:
integrate(1/x^9/(b*x^2+a)^2,x, algorithm="giac")
Output:
5/2*b^4*log(x^2)/a^6 - 5/2*b^4*log(abs(b*x^2 + a))/a^6 + 1/2*(5*b^5*x^2 + 6*a*b^4)/((b*x^2 + a)*a^6) - 1/24*(125*b^4*x^8 - 48*a*b^3*x^6 + 18*a^2*b^2 *x^4 - 8*a^3*b*x^2 + 3*a^4)/(a^6*x^8)
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {\frac {5\,b\,x^2}{24\,a^2}-\frac {1}{8\,a}-\frac {5\,b^2\,x^4}{12\,a^3}+\frac {5\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{b\,x^{10}+a\,x^8}-\frac {5\,b^4\,\ln \left (b\,x^2+a\right )}{2\,a^6}+\frac {5\,b^4\,\ln \left (x\right )}{a^6} \] Input:
int(1/(x^9*(a + b*x^2)^2),x)
Output:
((5*b*x^2)/(24*a^2) - 1/(8*a) - (5*b^2*x^4)/(12*a^3) + (5*b^3*x^6)/(4*a^4) + (5*b^4*x^8)/(2*a^5))/(a*x^8 + b*x^10) - (5*b^4*log(a + b*x^2))/(2*a^6) + (5*b^4*log(x))/a^6
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^9 \left (a+b x^2\right )^2} \, dx=\frac {-60 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{4} x^{8}-60 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{5} x^{10}+120 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{8}+120 \,\mathrm {log}\left (x \right ) b^{5} x^{10}-3 a^{5}+5 a^{4} b \,x^{2}-10 a^{3} b^{2} x^{4}+30 a^{2} b^{3} x^{6}-60 b^{5} x^{10}}{24 a^{6} x^{8} \left (b \,x^{2}+a \right )} \] Input:
int(1/x^9/(b*x^2+a)^2,x)
Output:
( - 60*log(a + b*x**2)*a*b**4*x**8 - 60*log(a + b*x**2)*b**5*x**10 + 120*l og(x)*a*b**4*x**8 + 120*log(x)*b**5*x**10 - 3*a**5 + 5*a**4*b*x**2 - 10*a* *3*b**2*x**4 + 30*a**2*b**3*x**6 - 60*b**5*x**10)/(24*a**6*x**8*(a + b*x** 2))