Integrand size = 20, antiderivative size = 113 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x}{10 a \left (a+b x^2\right )^5}+\frac {9 x}{80 a^2 \left (a+b x^2\right )^4}+\frac {21 x}{160 a^3 \left (a+b x^2\right )^3}+\frac {21 x}{128 a^4 \left (a+b x^2\right )^2}+\frac {63 x}{256 a^5 \left (a+b x^2\right )}+\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{11/2} \sqrt {b}} \] Output:
1/10*x/a/(b*x^2+a)^5+9/80*x/a^2/(b*x^2+a)^4+21/160*x/a^3/(b*x^2+a)^3+21/12 8*x/a^4/(b*x^2+a)^2+63/256*x/a^5/(b*x^2+a)+63/256*arctan(b^(1/2)*x/a^(1/2) )/a^(11/2)/b^(1/2)
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {\sqrt {a} x \left (965 a^4+2370 a^3 b x^2+2688 a^2 b^2 x^4+1470 a b^3 x^6+315 b^4 x^8\right )}{\left (a+b x^2\right )^5}+\frac {315 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}{1280 a^{11/2}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3),x]
Output:
((Sqrt[a]*x*(965*a^4 + 2370*a^3*b*x^2 + 2688*a^2*b^2*x^4 + 1470*a*b^3*x^6 + 315*b^4*x^8))/(a + b*x^2)^5 + (315*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b]) /(1280*a^(11/2))
Time = 0.42 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.56, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1379, 215, 215, 215, 215, 215, 218}
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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1379 |
| \(\displaystyle b^6 \int \frac {1}{\left (b^2 x^2+a b\right )^6}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b^6 \left (\frac {9 \int \frac {1}{\left (b^2 x^2+a b\right )^5}dx}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b^6 \left (\frac {9 \left (\frac {7 \int \frac {1}{\left (b^2 x^2+a b\right )^4}dx}{8 a b}+\frac {x}{8 a b^5 \left (a+b x^2\right )^4}\right )}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b^6 \left (\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{\left (b^2 x^2+a b\right )^3}dx}{6 a b}+\frac {x}{6 a b^4 \left (a+b x^2\right )^3}\right )}{8 a b}+\frac {x}{8 a b^5 \left (a+b x^2\right )^4}\right )}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b^6 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\left (b^2 x^2+a b\right )^2}dx}{4 a b}+\frac {x}{4 a b^3 \left (a+b x^2\right )^2}\right )}{6 a b}+\frac {x}{6 a b^4 \left (a+b x^2\right )^3}\right )}{8 a b}+\frac {x}{8 a b^5 \left (a+b x^2\right )^4}\right )}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle b^6 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{b^2 x^2+a b}dx}{2 a b}+\frac {x}{2 a b^2 \left (a+b x^2\right )}\right )}{4 a b}+\frac {x}{4 a b^3 \left (a+b x^2\right )^2}\right )}{6 a b}+\frac {x}{6 a b^4 \left (a+b x^2\right )^3}\right )}{8 a b}+\frac {x}{8 a b^5 \left (a+b x^2\right )^4}\right )}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle b^6 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {x}{2 a b^2 \left (a+b x^2\right )}\right )}{4 a b}+\frac {x}{4 a b^3 \left (a+b x^2\right )^2}\right )}{6 a b}+\frac {x}{6 a b^4 \left (a+b x^2\right )^3}\right )}{8 a b}+\frac {x}{8 a b^5 \left (a+b x^2\right )^4}\right )}{10 a b}+\frac {x}{10 a b^6 \left (a+b x^2\right )^5}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3),x]
Output:
b^6*(x/(10*a*b^6*(a + b*x^2)^5) + (9*(x/(8*a*b^5*(a + b*x^2)^4) + (7*(x/(6 *a*b^4*(a + b*x^2)^3) + (5*(x/(4*a*b^3*(a + b*x^2)^2) + (3*(x/(2*a*b^2*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*b^(5/2))))/(4*a*b)))/(6 *a*b)))/(8*a*b)))/(10*a*b))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/ c^p Int[(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n 2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && NeQ[p, 1]
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {x}{10 a \left (b \,x^{2}+a \right )^{5}}+\frac {\frac {9 x}{80 a \left (b \,x^{2}+a \right )^{4}}+\frac {9 \left (\frac {7 x}{48 a \left (b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (b \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}\right )}{10 a}}{a}\) | \(120\) |
| risch | \(\frac {\frac {63 b^{4} x^{9}}{256 a^{5}}+\frac {147 b^{3} x^{7}}{128 a^{4}}+\frac {21 b^{2} x^{5}}{10 a^{3}}+\frac {237 b \,x^{3}}{128 a^{2}}+\frac {193 x}{256 a}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}-\frac {63 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, a^{5}}+\frac {63 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, a^{5}}\) | \(126\) |
Input:
int(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/10*x/a/(b*x^2+a)^5+9/10/a*(1/8*x/a/(b*x^2+a)^4+7/8/a*(1/6*x/a/(b*x^2+a)^ 3+5/6/a*(1/4*x/a/(b*x^2+a)^2+3/4/a*(1/2*x/a/(b*x^2+a)+1/2/a/(a*b)^(1/2)*ar ctan(b/(a*b)^(1/2)*x)))))
Time = 0.10 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.42 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {630 \, a b^{5} x^{9} + 2940 \, a^{2} b^{4} x^{7} + 5376 \, a^{3} b^{3} x^{5} + 4740 \, a^{4} b^{2} x^{3} + 1930 \, a^{5} b x - 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2560 \, {\left (a^{6} b^{6} x^{10} + 5 \, a^{7} b^{5} x^{8} + 10 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{4} + 5 \, a^{10} b^{2} x^{2} + a^{11} b\right )}}, \frac {315 \, a b^{5} x^{9} + 1470 \, a^{2} b^{4} x^{7} + 2688 \, a^{3} b^{3} x^{5} + 2370 \, a^{4} b^{2} x^{3} + 965 \, a^{5} b x + 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{1280 \, {\left (a^{6} b^{6} x^{10} + 5 \, a^{7} b^{5} x^{8} + 10 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{4} + 5 \, a^{10} b^{2} x^{2} + a^{11} b\right )}}\right ] \] Input:
integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
[1/2560*(630*a*b^5*x^9 + 2940*a^2*b^4*x^7 + 5376*a^3*b^3*x^5 + 4740*a^4*b^ 2*x^3 + 1930*a^5*b*x - 315*(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)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a )/(b*x^2 + a)))/(a^6*b^6*x^10 + 5*a^7*b^5*x^8 + 10*a^8*b^4*x^6 + 10*a^9*b^ 3*x^4 + 5*a^10*b^2*x^2 + a^11*b), 1/1280*(315*a*b^5*x^9 + 1470*a^2*b^4*x^7 + 2688*a^3*b^3*x^5 + 2370*a^4*b^2*x^3 + 965*a^5*b*x + 315*(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)*sqrt(a*b)* arctan(sqrt(a*b)*x/a))/(a^6*b^6*x^10 + 5*a^7*b^5*x^8 + 10*a^8*b^4*x^6 + 10 *a^9*b^3*x^4 + 5*a^10*b^2*x^2 + a^11*b)]
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {63 \sqrt {- \frac {1}{a^{11} b}} \log {\left (- a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{512} + \frac {63 \sqrt {- \frac {1}{a^{11} b}} \log {\left (a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{512} + \frac {965 a^{4} x + 2370 a^{3} b x^{3} + 2688 a^{2} b^{2} x^{5} + 1470 a b^{3} x^{7} + 315 b^{4} x^{9}}{1280 a^{10} + 6400 a^{9} b x^{2} + 12800 a^{8} b^{2} x^{4} + 12800 a^{7} b^{3} x^{6} + 6400 a^{6} b^{4} x^{8} + 1280 a^{5} b^{5} x^{10}} \] Input:
integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
-63*sqrt(-1/(a**11*b))*log(-a**6*sqrt(-1/(a**11*b)) + x)/512 + 63*sqrt(-1/ (a**11*b))*log(a**6*sqrt(-1/(a**11*b)) + x)/512 + (965*a**4*x + 2370*a**3* b*x**3 + 2688*a**2*b**2*x**5 + 1470*a*b**3*x**7 + 315*b**4*x**9)/(1280*a** 10 + 6400*a**9*b*x**2 + 12800*a**8*b**2*x**4 + 12800*a**7*b**3*x**6 + 6400 *a**6*b**4*x**8 + 1280*a**5*b**5*x**10)
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {315 \, b^{4} x^{9} + 1470 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 2370 \, a^{3} b x^{3} + 965 \, a^{4} x}{1280 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} + \frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{5}} \] Input:
integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/1280*(315*b^4*x^9 + 1470*a*b^3*x^7 + 2688*a^2*b^2*x^5 + 2370*a^3*b*x^3 + 965*a^4*x)/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^ 4 + 5*a^9*b*x^2 + a^10) + 63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{5}} + \frac {315 \, b^{4} x^{9} + 1470 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 2370 \, a^{3} b x^{3} + 965 \, a^{4} x}{1280 \, {\left (b x^{2} + a\right )}^{5} a^{5}} \] Input:
integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/1280*(315*b^4*x^9 + 1470* a*b^3*x^7 + 2688*a^2*b^2*x^5 + 2370*a^3*b*x^3 + 965*a^4*x)/((b*x^2 + a)^5* a^5)
Time = 17.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {193\,x}{256\,a}+\frac {237\,b\,x^3}{128\,a^2}+\frac {21\,b^2\,x^5}{10\,a^3}+\frac {147\,b^3\,x^7}{128\,a^4}+\frac {63\,b^4\,x^9}{256\,a^5}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}}+\frac {63\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{11/2}\,\sqrt {b}} \] Input:
int(1/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((193*x)/(256*a) + (237*b*x^3)/(128*a^2) + (21*b^2*x^5)/(10*a^3) + (147*b^ 3*x^7)/(128*a^4) + (63*b^4*x^9)/(256*a^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) + (63*atan((b^(1/2)*x)/a^( 1/2)))/(256*a^(11/2)*b^(1/2))
Time = 0.17 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5}+1575 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{2}+3150 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{4}+3150 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{6}+1575 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{8}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{10}+965 a^{5} b x +2370 a^{4} b^{2} x^{3}+2688 a^{3} b^{3} x^{5}+1470 a^{2} b^{4} x^{7}+315 a \,b^{5} x^{9}}{1280 a^{6} b \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:
int(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5 + 1575*sqrt(b)*sqr t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**2 + 3150*sqrt(b)*sqrt(a)*atan ((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**4 + 3150*sqrt(b)*sqrt(a)*atan((b*x) /(sqrt(b)*sqrt(a)))*a**2*b**3*x**6 + 1575*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt (b)*sqrt(a)))*a*b**4*x**8 + 315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a )))*b**5*x**10 + 965*a**5*b*x + 2370*a**4*b**2*x**3 + 2688*a**3*b**3*x**5 + 1470*a**2*b**4*x**7 + 315*a*b**5*x**9)/(1280*a**6*b*(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))