Integrand size = 24, antiderivative size = 109 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac {5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac {5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac {5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac {5 a}{2 b^6 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^6} \] Output:
1/10*a^5/b^6/(b*x^2+a)^5-5/8*a^4/b^6/(b*x^2+a)^4+5/3*a^3/b^6/(b*x^2+a)^3-5 /2*a^2/b^6/(b*x^2+a)^2+5/2*a/b^6/(b*x^2+a)+1/2*ln(b*x^2+a)/b^6
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.66 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {a \left (137 a^4+625 a^3 b x^2+1100 a^2 b^2 x^4+900 a b^3 x^6+300 b^4 x^8\right )}{\left (a+b x^2\right )^5}+60 \log \left (a+b x^2\right )}{120 b^6} \] Input:
Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((a*(137*a^4 + 625*a^3*b*x^2 + 1100*a^2*b^2*x^4 + 900*a*b^3*x^6 + 300*b^4* x^8))/(a + b*x^2)^5 + 60*Log[a + b*x^2])/(120*b^6)
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97, 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.
\(\displaystyle \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^6 \int \frac {x^{11}}{b^6 \left (b x^2+a\right )^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^{11}}{\left (a+b x^2\right )^6}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^{10}}{\left (b x^2+a\right )^6}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {a^5}{b^5 \left (b x^2+a\right )^6}+\frac {5 a^4}{b^5 \left (b x^2+a\right )^5}-\frac {10 a^3}{b^5 \left (b x^2+a\right )^4}+\frac {10 a^2}{b^5 \left (b x^2+a\right )^3}-\frac {5 a}{b^5 \left (b x^2+a\right )^2}+\frac {1}{b^5 \left (b x^2+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^5}{5 b^6 \left (a+b x^2\right )^5}-\frac {5 a^4}{4 b^6 \left (a+b x^2\right )^4}+\frac {10 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac {5 a^2}{b^6 \left (a+b x^2\right )^2}+\frac {5 a}{b^6 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{b^6}\right )\) |
Input:
Int[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(a^5/(5*b^6*(a + b*x^2)^5) - (5*a^4)/(4*b^6*(a + b*x^2)^4) + (10*a^3)/(3*b ^6*(a + b*x^2)^3) - (5*a^2)/(b^6*(a + b*x^2)^2) + (5*a)/(b^6*(a + b*x^2)) + Log[a + b*x^2]/b^6)/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 = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {\frac {137 a^{5}}{120 b^{6}}+\frac {5 a \,x^{8}}{2 b^{2}}+\frac {15 a^{2} x^{6}}{2 b^{3}}+\frac {55 a^{3} x^{4}}{6 b^{4}}+\frac {125 a^{4} x^{2}}{24 b^{5}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{6}}\) | \(76\) |
risch | \(\frac {\frac {137 a^{5}}{120 b^{6}}+\frac {5 a \,x^{8}}{2 b^{2}}+\frac {15 a^{2} x^{6}}{2 b^{3}}+\frac {55 a^{3} x^{4}}{6 b^{4}}+\frac {125 a^{4} x^{2}}{24 b^{5}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{6}}\) | \(96\) |
default | \(\frac {a^{5}}{10 b^{6} \left (b \,x^{2}+a \right )^{5}}-\frac {5 a^{4}}{8 b^{6} \left (b \,x^{2}+a \right )^{4}}+\frac {5 a^{3}}{3 b^{6} \left (b \,x^{2}+a \right )^{3}}-\frac {5 a^{2}}{2 b^{6} \left (b \,x^{2}+a \right )^{2}}+\frac {5 a}{2 b^{6} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{6}}\) | \(98\) |
parallelrisch | \(\frac {60 \ln \left (b \,x^{2}+a \right ) x^{10} b^{5}+300 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{4}+300 a \,x^{8} b^{4}+600 \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{3}+900 a^{2} x^{6} b^{3}+600 \ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b^{2}+1100 a^{3} x^{4} b^{2}+300 \ln \left (b \,x^{2}+a \right ) x^{2} a^{4} b +625 x^{2} a^{4} b +60 \ln \left (b \,x^{2}+a \right ) a^{5}+137 a^{5}}{120 b^{6} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}\) | \(182\) |
Input:
int(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
(137/120*a^5/b^6+5/2*a/b^2*x^8+15/2*a^2/b^3*x^6+55/6*a^3/b^4*x^4+125/24*a^ 4/b^5*x^2)/(b*x^2+a)^5+1/2*ln(b*x^2+a)/b^6
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.54 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5} + 60 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{120 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \] Input:
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*a^5 + 60*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9 *x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)
Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {137 a^{5} + 625 a^{4} b x^{2} + 1100 a^{3} b^{2} x^{4} + 900 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 a^{5} b^{6} + 600 a^{4} b^{7} x^{2} + 1200 a^{3} b^{8} x^{4} + 1200 a^{2} b^{9} x^{6} + 600 a b^{10} x^{8} + 120 b^{11} x^{10}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{6}} \] Input:
integrate(x**11/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
(137*a**5 + 625*a**4*b*x**2 + 1100*a**3*b**2*x**4 + 900*a**2*b**3*x**6 + 3 00*a*b**4*x**8)/(120*a**5*b**6 + 600*a**4*b**7*x**2 + 1200*a**3*b**8*x**4 + 1200*a**2*b**9*x**6 + 600*a*b**10*x**8 + 120*b**11*x**10) + log(a + b*x* *2)/(2*b**6)
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5}}{120 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{6}} \] Input:
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*a^5)/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5 *a^4*b^7*x^2 + a^5*b^6) + 1/2*log(b*x^2 + a)/b^6
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac {137 \, b^{4} x^{10} + 385 \, a b^{3} x^{8} + 470 \, a^{2} b^{2} x^{6} + 270 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, {\left (b x^{2} + a\right )}^{5} b^{5}} \] Input:
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/2*log(abs(b*x^2 + a))/b^6 - 1/120*(137*b^4*x^10 + 385*a*b^3*x^8 + 470*a^ 2*b^2*x^6 + 270*a^3*b*x^4 + 60*a^4*x^2)/((b*x^2 + a)^5*b^5)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {137\,a^5}{120\,b^6}+\frac {5\,a\,x^8}{2\,b^2}+\frac {15\,a^2\,x^6}{2\,b^3}+\frac {55\,a^3\,x^4}{6\,b^4}+\frac {125\,a^4\,x^2}{24\,b^5}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}}+\frac {\ln \left (b\,x^2+a\right )}{2\,b^6} \] Input:
int(x^11/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((137*a^5)/(120*b^6) + (5*a*x^8)/(2*b^2) + (15*a^2*x^6)/(2*b^3) + (55*a^3* x^4)/(6*b^4) + (125*a^4*x^2)/(24*b^5))/(a^5 + b^5*x^10 + 5*a^4*b*x^2 + 5*a *b^4*x^8 + 10*a^3*b^2*x^4 + 10*a^2*b^3*x^6) + log(a + b*x^2)/(2*b^6)
Time = 0.15 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.87 \[ \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {60 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5}+300 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b \,x^{2}+600 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{2} x^{4}+600 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{3} x^{6}+300 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{4} x^{8}+60 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{5} x^{10}+77 a^{5}+325 a^{4} b \,x^{2}+500 a^{3} b^{2} x^{4}+300 a^{2} b^{3} x^{6}-60 b^{5} x^{10}}{120 b^{6} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:
int(x^11/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(60*log(a + b*x**2)*a**5 + 300*log(a + b*x**2)*a**4*b*x**2 + 600*log(a + b *x**2)*a**3*b**2*x**4 + 600*log(a + b*x**2)*a**2*b**3*x**6 + 300*log(a + b *x**2)*a*b**4*x**8 + 60*log(a + b*x**2)*b**5*x**10 + 77*a**5 + 325*a**4*b* x**2 + 500*a**3*b**2*x**4 + 300*a**2*b**3*x**6 - 60*b**5*x**10)/(120*b**6* (a**5 + 5*a**4*b*x**2 + 10*a**3*b**2*x**4 + 10*a**2*b**3*x**6 + 5*a*b**4*x **8 + b**5*x**10))