Integrand size = 24, antiderivative size = 77 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {x^2}{2 b^4}-\frac {a^4}{6 b^5 \left (a+b x^2\right )^3}+\frac {a^3}{b^5 \left (a+b x^2\right )^2}-\frac {3 a^2}{b^5 \left (a+b x^2\right )}-\frac {2 a \log \left (a+b x^2\right )}{b^5} \] Output:
1/2*x^2/b^4-1/6*a^4/b^5/(b*x^2+a)^3+a^3/b^5/(b*x^2+a)^2-3*a^2/b^5/(b*x^2+a )-2*a*ln(b*x^2+a)/b^5
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {-3 b x^2+\frac {a^2 \left (13 a^2+30 a b x^2+18 b^2 x^4\right )}{\left (a+b x^2\right )^3}+12 a \log \left (a+b x^2\right )}{6 b^5} \] Input:
Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
-1/6*(-3*b*x^2 + (a^2*(13*a^2 + 30*a*b*x^2 + 18*b^2*x^4))/(a + b*x^2)^3 + 12*a*Log[a + b*x^2])/b^5
Time = 0.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1380, 27, 243, 49, 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.
\(\displaystyle \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^4 \int \frac {x^9}{b^4 \left (b x^2+a\right )^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^9}{\left (a+b x^2\right )^4}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (b x^2+a\right )^4}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^4}{b^4 \left (b x^2+a\right )^4}-\frac {4 a^3}{b^4 \left (b x^2+a\right )^3}+\frac {6 a^2}{b^4 \left (b x^2+a\right )^2}-\frac {4 a}{b^4 \left (b x^2+a\right )}+\frac {1}{b^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^4}{3 b^5 \left (a+b x^2\right )^3}+\frac {2 a^3}{b^5 \left (a+b x^2\right )^2}-\frac {6 a^2}{b^5 \left (a+b x^2\right )}-\frac {4 a \log \left (a+b x^2\right )}{b^5}+\frac {x^2}{b^4}\right )\) |
Input:
Int[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(x^2/b^4 - a^4/(3*b^5*(a + b*x^2)^3) + (2*a^3)/(b^5*(a + b*x^2)^2) - (6*a^ 2)/(b^5*(a + b*x^2)) - (4*a*Log[a + b*x^2])/b^5)/2
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 [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[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]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84
method | result | size |
norman | \(\frac {\frac {x^{8}}{2 b}-\frac {11 a^{4}}{3 b^{5}}-\frac {6 a^{2} x^{4}}{b^{3}}-\frac {9 a^{3} x^{2}}{b^{4}}}{\left (b \,x^{2}+a \right )^{3}}-\frac {2 a \ln \left (b \,x^{2}+a \right )}{b^{5}}\) | \(65\) |
default | \(\frac {x^{2}}{2 b^{4}}-\frac {a \left (\frac {4 \ln \left (b \,x^{2}+a \right )}{b}+\frac {a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}+\frac {6 a}{b \left (b \,x^{2}+a \right )}-\frac {2 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}\right )}{2 b^{4}}\) | \(79\) |
risch | \(\frac {x^{2}}{2 b^{4}}+\frac {-3 a^{2} b \,x^{4}-5 a^{3} x^{2}-\frac {13 a^{4}}{6 b}}{b^{4} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}-\frac {2 a \ln \left (b \,x^{2}+a \right )}{b^{5}}\) | \(83\) |
parallelrisch | \(-\frac {-3 b^{4} x^{8}+12 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3}+36 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{2}+36 a^{2} b^{2} x^{4}+36 \ln \left (b \,x^{2}+a \right ) x^{2} a^{3} b +54 a^{3} b \,x^{2}+12 \ln \left (b \,x^{2}+a \right ) a^{4}+22 a^{4}}{6 b^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (b \,x^{2}+a \right )}\) | \(135\) |
Input:
int(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
(1/2/b*x^8-11/3*a^4/b^5-6*a^2/b^3*x^4-9*a^3/b^4*x^2)/(b*x^2+a)^3-2*a*ln(b* x^2+a)/b^5
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.61 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {3 \, b^{4} x^{8} + 9 \, a b^{3} x^{6} - 9 \, a^{2} b^{2} x^{4} - 27 \, a^{3} b x^{2} - 13 \, a^{4} - 12 \, {\left (a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 3 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{6 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \] Input:
integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
1/6*(3*b^4*x^8 + 9*a*b^3*x^6 - 9*a^2*b^2*x^4 - 27*a^3*b*x^2 - 13*a^4 - 12* (a*b^3*x^6 + 3*a^2*b^2*x^4 + 3*a^3*b*x^2 + a^4)*log(b*x^2 + a))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=- \frac {2 a \log {\left (a + b x^{2} \right )}}{b^{5}} + \frac {- 13 a^{4} - 30 a^{3} b x^{2} - 18 a^{2} b^{2} x^{4}}{6 a^{3} b^{5} + 18 a^{2} b^{6} x^{2} + 18 a b^{7} x^{4} + 6 b^{8} x^{6}} + \frac {x^{2}}{2 b^{4}} \] Input:
integrate(x**9/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
-2*a*log(a + b*x**2)/b**5 + (-13*a**4 - 30*a**3*b*x**2 - 18*a**2*b**2*x**4 )/(6*a**3*b**5 + 18*a**2*b**6*x**2 + 18*a*b**7*x**4 + 6*b**8*x**6) + x**2/ (2*b**4)
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {18 \, a^{2} b^{2} x^{4} + 30 \, a^{3} b x^{2} + 13 \, a^{4}}{6 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} + \frac {x^{2}}{2 \, b^{4}} - \frac {2 \, a \log \left (b x^{2} + a\right )}{b^{5}} \] Input:
integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
-1/6*(18*a^2*b^2*x^4 + 30*a^3*b*x^2 + 13*a^4)/(b^8*x^6 + 3*a*b^7*x^4 + 3*a ^2*b^6*x^2 + a^3*b^5) + 1/2*x^2/b^4 - 2*a*log(b*x^2 + a)/b^5
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {x^{2}}{2 \, b^{4}} - \frac {2 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac {22 \, a b^{3} x^{6} + 48 \, a^{2} b^{2} x^{4} + 36 \, a^{3} b x^{2} + 9 \, a^{4}}{6 \, {\left (b x^{2} + a\right )}^{3} b^{5}} \] Input:
integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
1/2*x^2/b^4 - 2*a*log(abs(b*x^2 + a))/b^5 + 1/6*(22*a*b^3*x^6 + 48*a^2*b^2 *x^4 + 36*a^3*b*x^2 + 9*a^4)/((b*x^2 + a)^3*b^5)
Time = 17.66 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {x^2}{2\,b^4}-\frac {\frac {13\,a^4}{6\,b}+5\,a^3\,x^2+3\,a^2\,b\,x^4}{a^3\,b^4+3\,a^2\,b^5\,x^2+3\,a\,b^6\,x^4+b^7\,x^6}-\frac {2\,a\,\ln \left (b\,x^2+a\right )}{b^5} \] Input:
int(x^9/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
x^2/(2*b^4) - ((13*a^4)/(6*b) + 5*a^3*x^2 + 3*a^2*b*x^4)/(a^3*b^4 + b^7*x^ 6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2) - (2*a*log(a + b*x^2))/b^5
Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74 \[ \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4}-36 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b \,x^{2}-36 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} x^{4}-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} x^{6}-10 a^{4}-18 a^{3} b \,x^{2}+12 a \,b^{3} x^{6}+3 b^{4} x^{8}}{6 b^{5} \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:
int(x^9/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
( - 12*log(a + b*x**2)*a**4 - 36*log(a + b*x**2)*a**3*b*x**2 - 36*log(a + b*x**2)*a**2*b**2*x**4 - 12*log(a + b*x**2)*a*b**3*x**6 - 10*a**4 - 18*a** 3*b*x**2 + 12*a*b**3*x**6 + 3*b**4*x**8)/(6*b**5*(a**3 + 3*a**2*b*x**2 + 3 *a*b**2*x**4 + b**3*x**6))