Integrand size = 24, antiderivative size = 145 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {6 a x}{b^7}+\frac {x^3}{3 b^6}-\frac {a^6 x}{10 b^7 \left (a+b x^2\right )^5}+\frac {61 a^5 x}{80 b^7 \left (a+b x^2\right )^4}-\frac {1253 a^4 x}{480 b^7 \left (a+b x^2\right )^3}+\frac {2107 a^3 x}{384 b^7 \left (a+b x^2\right )^2}-\frac {2373 a^2 x}{256 b^7 \left (a+b x^2\right )}+\frac {3003 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}} \] Output:
-6*a*x/b^7+1/3*x^3/b^6-1/10*a^6*x/b^7/(b*x^2+a)^5+61/80*a^5*x/b^7/(b*x^2+a )^4-1253/480*a^4*x/b^7/(b*x^2+a)^3+2107/384*a^3*x/b^7/(b*x^2+a)^2-2373/256 *a^2*x/b^7/(b*x^2+a)+3003/256*a^(3/2)*arctan(b^(1/2)*x/a^(1/2))/b^(15/2)
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {\sqrt {b} x \left (-45045 a^6-210210 a^5 b x^2-384384 a^4 b^2 x^4-338910 a^3 b^3 x^6-137995 a^2 b^4 x^8-16640 a b^5 x^{10}+1280 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}+45045 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3840 b^{15/2}} \] Input:
Integrate[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((Sqrt[b]*x*(-45045*a^6 - 210210*a^5*b*x^2 - 384384*a^4*b^2*x^4 - 338910*a ^3*b^3*x^6 - 137995*a^2*b^4*x^8 - 16640*a*b^5*x^10 + 1280*b^6*x^12))/(a + b*x^2)^5 + 45045*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*b^(15/2))
Time = 0.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1380, 27, 252, 252, 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.
| \(\displaystyle \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {x^{14}}{b^6 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^{14}}{\left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 \int \frac {x^{12}}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 \left (\frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \int \frac {x^8}{\left (b x^2+a\right )^3}dx}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {x^4}{b x^2+a}dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \left (\frac {a^2}{b^2 \left (b x^2+a\right )}-\frac {a}{b^2}+\frac {x^2}{b}\right )dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b}\right )}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*x^13/(b*(a + b*x^2)^5) + (13*(-1/8*x^11/(b*(a + b*x^2)^4) + (11*(-1/ 6*x^9/(b*(a + b*x^2)^3) + (3*(-1/4*x^7/(b*(a + b*x^2)^2) + (7*(-1/2*x^5/(b *(a + b*x^2)) + (5*(-((a*x)/b^2) + x^3/(3*b) + (a^(3/2)*ArcTan[(Sqrt[b]*x) /Sqrt[a]])/b^(5/2)))/(2*b)))/(4*b)))/(2*b)))/(8*b)))/(10*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 = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(-\frac {-\frac {1}{3} b \,x^{3}+6 a x}{b^{7}}+\frac {a^{2} \left (\frac {-\frac {2373}{256} b^{4} x^{9}-\frac {12131}{384} a \,b^{3} x^{7}-\frac {1253}{30} a^{2} b^{2} x^{5}-\frac {9629}{384} a^{3} b \,x^{3}-\frac {1467}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {3003 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{b^{7}}\) | \(96\) |
| risch | \(\frac {x^{3}}{3 b^{6}}-\frac {6 a x}{b^{7}}+\frac {-\frac {2373}{256} a^{2} b^{4} x^{9}-\frac {12131}{384} a^{3} b^{3} x^{7}-\frac {1253}{30} a^{4} b^{2} x^{5}-\frac {9629}{384} a^{5} b \,x^{3}-\frac {1467}{256} a^{6} x}{b^{7} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}+\frac {3003 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right )}{512 b^{8}}-\frac {3003 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right )}{512 b^{8}}\) | \(146\) |
Input:
int(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/b^7*(-1/3*b*x^3+6*a*x)+1/b^7*a^2*((-2373/256*b^4*x^9-12131/384*a*b^3*x^ 7-1253/30*a^2*b^2*x^5-9629/384*a^3*b*x^3-1467/256*a^4*x)/(b*x^2+a)^5+3003/ 256/(a*b)^(1/2)*arctan(b/(a*b)^(1/2)*x))
Time = 0.07 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.95 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {2560 \, b^{6} x^{13} - 33280 \, a b^{5} x^{11} - 275990 \, a^{2} b^{4} x^{9} - 677820 \, a^{3} b^{3} x^{7} - 768768 \, a^{4} b^{2} x^{5} - 420420 \, a^{5} b x^{3} - 90090 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{7680 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, \frac {1280 \, b^{6} x^{13} - 16640 \, a b^{5} x^{11} - 137995 \, a^{2} b^{4} x^{9} - 338910 \, a^{3} b^{3} x^{7} - 384384 \, a^{4} b^{2} x^{5} - 210210 \, a^{5} b x^{3} - 45045 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \] Input:
integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
[1/7680*(2560*b^6*x^13 - 33280*a*b^5*x^11 - 275990*a^2*b^4*x^9 - 677820*a^ 3*b^3*x^7 - 768768*a^4*b^2*x^5 - 420420*a^5*b*x^3 - 90090*a^6*x + 45045*(a *b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^12*x ^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^ 5*b^7), 1/3840*(1280*b^6*x^13 - 16640*a*b^5*x^11 - 137995*a^2*b^4*x^9 - 33 8910*a^3*b^3*x^7 - 384384*a^4*b^2*x^5 - 210210*a^5*b*x^3 - 45045*a^6*x + 4 5045*(a*b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5 *b*x^2 + a^6)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)]
Time = 0.45 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.41 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {6 a x}{b^{7}} - \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x - \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x + \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {- 22005 a^{6} x - 96290 a^{5} b x^{3} - 160384 a^{4} b^{2} x^{5} - 121310 a^{3} b^{3} x^{7} - 35595 a^{2} b^{4} x^{9}}{3840 a^{5} b^{7} + 19200 a^{4} b^{8} x^{2} + 38400 a^{3} b^{9} x^{4} + 38400 a^{2} b^{10} x^{6} + 19200 a b^{11} x^{8} + 3840 b^{12} x^{10}} + \frac {x^{3}}{3 b^{6}} \] Input:
integrate(x**14/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
-6*a*x/b**7 - 3003*sqrt(-a**3/b**15)*log(x - b**7*sqrt(-a**3/b**15)/a)/512 + 3003*sqrt(-a**3/b**15)*log(x + b**7*sqrt(-a**3/b**15)/a)/512 + (-22005* a**6*x - 96290*a**5*b*x**3 - 160384*a**4*b**2*x**5 - 121310*a**3*b**3*x**7 - 35595*a**2*b**4*x**9)/(3840*a**5*b**7 + 19200*a**4*b**8*x**2 + 38400*a* *3*b**9*x**4 + 38400*a**2*b**10*x**6 + 19200*a*b**11*x**8 + 3840*b**12*x** 10) + x**3/(3*b**6)
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} + \frac {b x^{3} - 18 \, a x}{3 \, b^{7}} \] Input:
integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
-1/3840*(35595*a^2*b^4*x^9 + 121310*a^3*b^3*x^7 + 160384*a^4*b^2*x^5 + 962 90*a^5*b*x^3 + 22005*a^6*x)/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7) + 3003/256*a^2*arctan(b*x/sqrt(a *b))/(sqrt(a*b)*b^7) + 1/3*(b*x^3 - 18*a*x)/b^7
Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.73 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} - \frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b x^{2} + a\right )}^{5} b^{7}} + \frac {b^{12} x^{3} - 18 \, a b^{11} x}{3 \, b^{18}} \] Input:
integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
3003/256*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/3840*(35595*a^2*b^4 *x^9 + 121310*a^3*b^3*x^7 + 160384*a^4*b^2*x^5 + 96290*a^5*b*x^3 + 22005*a ^6*x)/((b*x^2 + a)^5*b^7) + 1/3*(b^12*x^3 - 18*a*b^11*x)/b^18
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x^3}{3\,b^6}-\frac {\frac {1467\,a^6\,x}{256}+\frac {9629\,a^5\,b\,x^3}{384}+\frac {1253\,a^4\,b^2\,x^5}{30}+\frac {12131\,a^3\,b^3\,x^7}{384}+\frac {2373\,a^2\,b^4\,x^9}{256}}{a^5\,b^7+5\,a^4\,b^8\,x^2+10\,a^3\,b^9\,x^4+10\,a^2\,b^{10}\,x^6+5\,a\,b^{11}\,x^8+b^{12}\,x^{10}}+\frac {3003\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,b^{15/2}}-\frac {6\,a\,x}{b^7} \] Input:
int(x^14/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
x^3/(3*b^6) - ((1467*a^6*x)/256 + (9629*a^5*b*x^3)/384 + (1253*a^4*b^2*x^5 )/30 + (12131*a^3*b^3*x^7)/384 + (2373*a^2*b^4*x^9)/256)/(a^5*b^7 + b^12*x ^10 + 5*a*b^11*x^8 + 5*a^4*b^8*x^2 + 10*a^3*b^9*x^4 + 10*a^2*b^10*x^6) + ( 3003*a^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(256*b^(15/2)) - (6*a*x)/b^7
Time = 0.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.92 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6}+225225 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b \,x^{2}+450450 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{2} x^{4}+450450 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{3} x^{6}+225225 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{4} x^{8}+45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{5} x^{10}-45045 a^{6} b x -210210 a^{5} b^{2} x^{3}-384384 a^{4} b^{3} x^{5}-338910 a^{3} b^{4} x^{7}-137995 a^{2} b^{5} x^{9}-16640 a \,b^{6} x^{11}+1280 b^{7} x^{13}}{3840 b^{8} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:
int(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(45045*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6 + 225225*sqrt(b) *sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b*x**2 + 450450*sqrt(b)*sqrt(a )*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**2*x**4 + 450450*sqrt(b)*sqrt(a)*at an((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*x**6 + 225225*sqrt(b)*sqrt(a)*atan(( b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*x**8 + 45045*sqrt(b)*sqrt(a)*atan((b*x)/ (sqrt(b)*sqrt(a)))*a*b**5*x**10 - 45045*a**6*b*x - 210210*a**5*b**2*x**3 - 384384*a**4*b**3*x**5 - 338910*a**3*b**4*x**7 - 137995*a**2*b**5*x**9 - 1 6640*a*b**6*x**11 + 1280*b**7*x**13)/(3840*b**8*(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))