Integrand size = 24, antiderivative size = 118 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {10 a^2 x}{b^6}-\frac {4 a x^3}{3 b^5}+\frac {x^5}{5 b^4}+\frac {a^5 x}{6 b^6 \left (a+b x^2\right )^3}-\frac {31 a^4 x}{24 b^6 \left (a+b x^2\right )^2}+\frac {89 a^3 x}{16 b^6 \left (a+b x^2\right )}-\frac {231 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}} \] Output:
10*a^2*x/b^6-4/3*a*x^3/b^5+1/5*x^5/b^4+1/6*a^5*x/b^6/(b*x^2+a)^3-31/24*a^4 *x/b^6/(b*x^2+a)^2+89/16*a^3*x/b^6/(b*x^2+a)-231/16*a^(5/2)*arctan(b^(1/2) *x/a^(1/2))/b^(13/2)
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {3465 a^5 x+9240 a^4 b x^3+7623 a^3 b^2 x^5+1584 a^2 b^3 x^7-176 a b^4 x^9+48 b^5 x^{11}}{240 b^6 \left (a+b x^2\right )^3}-\frac {231 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 b^{13/2}} \] Input:
Integrate[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(3465*a^5*x + 9240*a^4*b*x^3 + 7623*a^3*b^2*x^5 + 1584*a^2*b^3*x^7 - 176*a *b^4*x^9 + 48*b^5*x^11)/(240*b^6*(a + b*x^2)^3) - (231*a^(5/2)*ArcTan[(Sqr t[b]*x)/Sqrt[a]])/(16*b^(13/2))
Time = 0.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1380, 27, 252, 252, 252, 254, 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.
\(\displaystyle \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^4 \int \frac {x^{12}}{b^4 \left (b x^2+a\right )^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^{12}}{\left (a+b x^2\right )^4}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^3}dx}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {11 \left (\frac {9 \int \frac {x^8}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \int \frac {x^6}{b x^2+a}dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \int \left (\frac {x^4}{b}-\frac {a x^2}{b^2}-\frac {a^3}{b^3 \left (b x^2+a\right )}+\frac {a^2}{b^3}\right )dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b}\right )}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\) |
Input:
Int[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
-1/6*x^11/(b*(a + b*x^2)^3) + (11*(-1/4*x^9/(b*(a + b*x^2)^2) + (9*(-1/2*x ^7/(b*(a + b*x^2)) + (7*((a^2*x)/b^3 - (a*x^3)/(3*b^2) + x^5/(5*b) - (a^(5 /2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)))/(2*b)))/(4*b)))/(6*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
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 = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {\frac {1}{5} x^{5} b^{2}-\frac {4}{3} a b \,x^{3}+10 a^{2} x}{b^{6}}-\frac {a^{3} \left (\frac {-\frac {89}{16} x^{5} b^{2}-\frac {59}{6} a b \,x^{3}-\frac {71}{16} a^{2} x}{\left (b \,x^{2}+a \right )^{3}}+\frac {231 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{b^{6}}\) | \(85\) |
risch | \(\frac {x^{5}}{5 b^{4}}-\frac {4 a \,x^{3}}{3 b^{5}}+\frac {10 a^{2} x}{b^{6}}+\frac {\frac {89}{16} a^{3} b^{2} x^{5}+\frac {59}{6} a^{4} b \,x^{3}+\frac {71}{16} x \,a^{5}}{b^{6} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}+\frac {231 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right )}{32 b^{7}}-\frac {231 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right )}{32 b^{7}}\) | \(143\) |
Input:
int(x^12/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/b^6*(1/5*x^5*b^2-4/3*a*b*x^3+10*a^2*x)-1/b^6*a^3*((-89/16*x^5*b^2-59/6*a *b*x^3-71/16*a^2*x)/(b*x^2+a)^3+231/16/(a*b)^(1/2)*arctan(b/(a*b)^(1/2)*x) )
Time = 0.07 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.73 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\left [\frac {96 \, b^{5} x^{11} - 352 \, a b^{4} x^{9} + 3168 \, a^{2} b^{3} x^{7} + 15246 \, a^{3} b^{2} x^{5} + 18480 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \, {\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{480 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac {48 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 1584 \, a^{2} b^{3} x^{7} + 7623 \, a^{3} b^{2} x^{5} + 9240 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \, {\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{240 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}\right ] \] Input:
integrate(x^12/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
[1/480*(96*b^5*x^11 - 352*a*b^4*x^9 + 3168*a^2*b^3*x^7 + 15246*a^3*b^2*x^5 + 18480*a^4*b*x^3 + 6930*a^5*x + 3465*(a^2*b^3*x^6 + 3*a^3*b^2*x^4 + 3*a^ 4*b*x^2 + a^5)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a))) /(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6), 1/240*(48*b^5*x^11 - 1 76*a*b^4*x^9 + 1584*a^2*b^3*x^7 + 7623*a^3*b^2*x^5 + 9240*a^4*b*x^3 + 3465 *a^5*x - 3465*(a^2*b^3*x^6 + 3*a^3*b^2*x^4 + 3*a^4*b*x^2 + a^5)*sqrt(a/b)* arctan(b*x*sqrt(a/b)/a))/(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6) ]
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.46 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {10 a^{2} x}{b^{6}} - \frac {4 a x^{3}}{3 b^{5}} + \frac {231 \sqrt {- \frac {a^{5}}{b^{13}}} \log {\left (x - \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} - \frac {231 \sqrt {- \frac {a^{5}}{b^{13}}} \log {\left (x + \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}}}{a^{2}} \right )}}{32} + \frac {213 a^{5} x + 472 a^{4} b x^{3} + 267 a^{3} b^{2} x^{5}}{48 a^{3} b^{6} + 144 a^{2} b^{7} x^{2} + 144 a b^{8} x^{4} + 48 b^{9} x^{6}} + \frac {x^{5}}{5 b^{4}} \] Input:
integrate(x**12/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
10*a**2*x/b**6 - 4*a*x**3/(3*b**5) + 231*sqrt(-a**5/b**13)*log(x - b**6*sq rt(-a**5/b**13)/a**2)/32 - 231*sqrt(-a**5/b**13)*log(x + b**6*sqrt(-a**5/b **13)/a**2)/32 + (213*a**5*x + 472*a**4*b*x**3 + 267*a**3*b**2*x**5)/(48*a **3*b**6 + 144*a**2*b**7*x**2 + 144*a*b**8*x**4 + 48*b**9*x**6) + x**5/(5* b**4)
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.98 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {267 \, a^{3} b^{2} x^{5} + 472 \, a^{4} b x^{3} + 213 \, a^{5} x}{48 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} - \frac {231 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{6}} + \frac {3 \, b^{2} x^{5} - 20 \, a b x^{3} + 150 \, a^{2} x}{15 \, b^{6}} \] Input:
integrate(x^12/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
1/48*(267*a^3*b^2*x^5 + 472*a^4*b*x^3 + 213*a^5*x)/(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6) - 231/16*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b ^6) + 1/15*(3*b^2*x^5 - 20*a*b*x^3 + 150*a^2*x)/b^6
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {231 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{6}} + \frac {267 \, a^{3} b^{2} x^{5} + 472 \, a^{4} b x^{3} + 213 \, a^{5} x}{48 \, {\left (b x^{2} + a\right )}^{3} b^{6}} + \frac {3 \, b^{16} x^{5} - 20 \, a b^{15} x^{3} + 150 \, a^{2} b^{14} x}{15 \, b^{20}} \] Input:
integrate(x^12/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
-231/16*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/48*(267*a^3*b^2*x^5 + 472*a^4*b*x^3 + 213*a^5*x)/((b*x^2 + a)^3*b^6) + 1/15*(3*b^16*x^5 - 20*a *b^15*x^3 + 150*a^2*b^14*x)/b^20
Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {71\,a^5\,x}{16}+\frac {59\,a^4\,b\,x^3}{6}+\frac {89\,a^3\,b^2\,x^5}{16}}{a^3\,b^6+3\,a^2\,b^7\,x^2+3\,a\,b^8\,x^4+b^9\,x^6}+\frac {x^5}{5\,b^4}-\frac {4\,a\,x^3}{3\,b^5}+\frac {10\,a^2\,x}{b^6}-\frac {231\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,b^{13/2}} \] Input:
int(x^12/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
((71*a^5*x)/16 + (59*a^4*b*x^3)/6 + (89*a^3*b^2*x^5)/16)/(a^3*b^6 + b^9*x^ 6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2) + x^5/(5*b^4) - (4*a*x^3)/(3*b^5) + (10*a ^2*x)/b^6 - (231*a^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(16*b^(13/2))
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.64 \[ \int \frac {x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5}-10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{2}-10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{4}-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{6}+3465 a^{5} b x +9240 a^{4} b^{2} x^{3}+7623 a^{3} b^{3} x^{5}+1584 a^{2} b^{4} x^{7}-176 a \,b^{5} x^{9}+48 b^{6} x^{11}}{240 b^{7} \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:
int(x^12/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
( - 3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5 - 10395*sqrt(b )*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**2 - 10395*sqrt(b)*sqrt(a )*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**4 - 3465*sqrt(b)*sqrt(a)*atan ((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*x**6 + 3465*a**5*b*x + 9240*a**4*b**2* x**3 + 7623*a**3*b**3*x**5 + 1584*a**2*b**4*x**7 - 176*a*b**5*x**9 + 48*b* *6*x**11)/(240*b**7*(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6))