Integrand size = 26, antiderivative size = 293 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {515 b^2 x}{128 a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b^2 x}{8 a^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {23 b^2 x}{48 a^4 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {259 b^2 x}{192 a^5 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{3 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 b \left (a+b x^2\right )}{a^6 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b^{3/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
515/128*b^2*x/a^6/((b*x^2+a)^2)^(1/2)+1/8*b^2*x/a^3/(b*x^2+a)^3/((b*x^2+a) ^2)^(1/2)+23/48*b^2*x/a^4/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)+259/192*b^2*x/a^ 5/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-1/3*(b*x^2+a)/a^5/x^3/((b*x^2+a)^2)^(1/2)+ 5*b*(b*x^2+a)/a^6/x/((b*x^2+a)^2)^(1/2)+1155/128*b^(3/2)*(b*x^2+a)*arctan( b^(1/2)*x/a^(1/2))/a^(13/2)/((b*x^2+a)^2)^(1/2)
Time = 1.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {\sqrt {a} \left (-128 a^5+1408 a^4 b x^2+9207 a^3 b^2 x^4+16863 a^2 b^3 x^6+12705 a b^4 x^8+3465 b^5 x^{10}\right )+3465 b^{3/2} x^3 \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{384 a^{13/2} x^3 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:
Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
Output:
(Sqrt[a]*(-128*a^5 + 1408*a^4*b*x^2 + 9207*a^3*b^2*x^4 + 16863*a^2*b^3*x^6 + 12705*a*b^4*x^8 + 3465*b^5*x^10) + 3465*b^(3/2)*x^3*(a + b*x^2)^4*ArcTa n[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(13/2)*x^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2 ])
Time = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.65, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1384, 27, 253, 253, 253, 253, 264, 264, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {1}{b^5 x^4 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {1}{x^4 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \int \frac {1}{x^4 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \int \frac {1}{x^4 \left (b x^2+a\right )^3}dx}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {1}{x^4 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
Output:
((a + b*x^2)*(1/(8*a*x^3*(a + b*x^2)^4) + (11*(1/(6*a*x^3*(a + b*x^2)^3) + (3*(1/(4*a*x^3*(a + b*x^2)^2) + (7*(1/(2*a*x^3*(a + b*x^2)) + (5*(-1/3*1/ (a*x^3) - (b*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))) /a))/(2*a)))/(4*a)))/(2*a)))/(8*a)))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
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_)^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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[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 - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 1.54 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {1155 b^{5} x^{10}}{128 a^{6}}+\frac {4235 b^{4} x^{8}}{128 a^{5}}+\frac {5621 b^{3} x^{6}}{128 a^{4}}+\frac {3069 b^{2} x^{4}}{128 a^{3}}+\frac {11 b \,x^{2}}{3 a^{2}}-\frac {1}{3 a}\right )}{\left (b \,x^{2}+a \right )^{5} x^{3}}+\frac {1155 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, b \ln \left (-b x -\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) a^{7}}-\frac {1155 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, b \ln \left (-b x +\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) a^{7}}\) | \(175\) |
| default | \(-\frac {\left (-3465 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) b^{6} x^{11}-3465 \sqrt {a b}\, b^{5} x^{10}-13860 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a \,b^{5} x^{9}-12705 \sqrt {a b}\, a \,b^{4} x^{8}-20790 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{2} b^{4} x^{7}-16863 \sqrt {a b}\, a^{2} b^{3} x^{6}-13860 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{3} b^{3} x^{5}-9207 \sqrt {a b}\, a^{3} b^{2} x^{4}-3465 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{4} b^{2} x^{3}-1408 \sqrt {a b}\, a^{4} b \,x^{2}+128 \sqrt {a b}\, a^{5}\right ) \left (b \,x^{2}+a \right )}{384 x^{3} \sqrt {a b}\, a^{6} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(211\) |
Input:
int(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)^2)^(1/2)/(b*x^2+a)^5*(1155/128*b^5/a^6*x^10+4235/128*b^4/a^5*x^ 8+5621/128*b^3/a^4*x^6+3069/128*b^2/a^3*x^4+11/3*b/a^2*x^2-1/3/a)/x^3+1155 /256*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/a^7*(-a*b)^(1/2)*b*ln(-b*x-(-a*b)^(1/2) )-1155/256*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/a^7*(-a*b)^(1/2)*b*ln(-b*x+(-a*b) ^(1/2))
Time = 0.08 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\left [\frac {6930 \, b^{5} x^{10} + 25410 \, a b^{4} x^{8} + 33726 \, a^{2} b^{3} x^{6} + 18414 \, a^{3} b^{2} x^{4} + 2816 \, a^{4} b x^{2} - 256 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{768 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}, \frac {3465 \, b^{5} x^{10} + 12705 \, a b^{4} x^{8} + 16863 \, a^{2} b^{3} x^{6} + 9207 \, a^{3} b^{2} x^{4} + 1408 \, a^{4} b x^{2} - 128 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{384 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}\right ] \] Input:
integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
Output:
[1/768*(6930*b^5*x^10 + 25410*a*b^4*x^8 + 33726*a^2*b^3*x^6 + 18414*a^3*b^ 2*x^4 + 2816*a^4*b*x^2 - 256*a^5 + 3465*(b^5*x^11 + 4*a*b^4*x^9 + 6*a^2*b^ 3*x^7 + 4*a^3*b^2*x^5 + a^4*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a ) - a)/(b*x^2 + a)))/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9 *b*x^5 + a^10*x^3), 1/384*(3465*b^5*x^10 + 12705*a*b^4*x^8 + 16863*a^2*b^3 *x^6 + 9207*a^3*b^2*x^4 + 1408*a^4*b*x^2 - 128*a^5 + 3465*(b^5*x^11 + 4*a* b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^3*b^2*x^5 + a^4*b*x^3)*sqrt(b/a)*arctan(x*sq rt(b/a)))/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^ 10*x^3)]
\[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{x^{4} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/x**4/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral(1/(x**4*((a + b*x**2)**2)**(5/2)), x)
Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {3465 \, b^{5} x^{10} + 12705 \, a b^{4} x^{8} + 16863 \, a^{2} b^{3} x^{6} + 9207 \, a^{3} b^{2} x^{4} + 1408 \, a^{4} b x^{2} - 128 \, a^{5}}{384 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} + \frac {1155 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} \] Input:
integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
Output:
1/384*(3465*b^5*x^10 + 12705*a*b^4*x^8 + 16863*a^2*b^3*x^6 + 9207*a^3*b^2* x^4 + 1408*a^4*b*x^2 - 128*a^5)/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2* x^7 + 4*a^9*b*x^5 + a^10*x^3) + 1155/128*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a *b)*a^6)
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1155 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {15 \, b x^{2} - a}{3 \, a^{6} x^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1545 \, b^{5} x^{7} + 5153 \, a b^{4} x^{5} + 5855 \, a^{2} b^{3} x^{3} + 2295 \, a^{3} b^{2} x}{384 \, {\left (b x^{2} + a\right )}^{4} a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} \] Input:
integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
1155/128*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6*sgn(b*x^2 + a)) + 1/3*(1 5*b*x^2 - a)/(a^6*x^3*sgn(b*x^2 + a)) + 1/384*(1545*b^5*x^7 + 5153*a*b^4*x ^5 + 5855*a^2*b^3*x^3 + 2295*a^3*b^2*x)/((b*x^2 + a)^4*a^6*sgn(b*x^2 + a))
Timed out. \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{x^4\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:
int(1/(x^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)
Output:
int(1/(x^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)
Time = 0.17 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{3}+13860 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{5}+20790 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{7}+13860 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{9}+3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{11}-128 a^{6}+1408 a^{5} b \,x^{2}+9207 a^{4} b^{2} x^{4}+16863 a^{3} b^{3} x^{6}+12705 a^{2} b^{4} x^{8}+3465 a \,b^{5} x^{10}}{384 a^{7} x^{3} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**3 + 13860*sq rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**5 + 20790*sqrt(b) *sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*x**7 + 13860*sqrt(b)*sqrt (a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*x**9 + 3465*sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*b**5*x**11 - 128*a**6 + 1408*a**5*b*x**2 + 9207*a **4*b**2*x**4 + 16863*a**3*b**3*x**6 + 12705*a**2*b**4*x**8 + 3465*a*b**5* x**10)/(384*a**7*x**3*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3* x**6 + b**4*x**8))