\(\int \frac {1}{x^2 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [416]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1}{a^6 x}-\frac {b x}{10 a^2 \left (a+b x^2\right )^5}-\frac {19 b x}{80 a^3 \left (a+b x^2\right )^4}-\frac {71 b x}{160 a^4 \left (a+b x^2\right )^3}-\frac {103 b x}{128 a^5 \left (a+b x^2\right )^2}-\frac {437 b x}{256 a^6 \left (a+b x^2\right )}-\frac {693 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}} \] Output:

-1/a^6/x-1/10*b*x/a^2/(b*x^2+a)^5-19/80*b*x/a^3/(b*x^2+a)^4-71/160*b*x/a^4 
/(b*x^2+a)^3-103/128*b*x/a^5/(b*x^2+a)^2-437/256*b*x/a^6/(b*x^2+a)-693/256 
*b^(1/2)*arctan(b^(1/2)*x/a^(1/2))/a^(13/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1280 a^5+10615 a^4 b x^2+26070 a^3 b^2 x^4+29568 a^2 b^3 x^6+16170 a b^4 x^8+3465 b^5 x^{10}}{1280 a^6 x \left (a+b x^2\right )^5}-\frac {693 \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{13/2}} \] Input:

Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
 

Output:

-1/1280*(1280*a^5 + 10615*a^4*b*x^2 + 26070*a^3*b^2*x^4 + 29568*a^2*b^3*x^ 
6 + 16170*a*b^4*x^8 + 3465*b^5*x^10)/(a^6*x*(a + b*x^2)^5) - (693*Sqrt[b]* 
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(13/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.34, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1380, 27, 253, 253, 253, 253, 253, 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.

\(\displaystyle \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^6 \int \frac {1}{b^6 x^2 \left (b x^2+a\right )^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^6}dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \int \frac {1}{x^2 \left (b x^2+a\right )^5}dx}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \int \frac {1}{x^2 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{x^2 \left (b x^2+a\right )^3}dx}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^2 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x \left (a+b x^2\right )^5}\)

Input:

Int[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
 

Output:

1/(10*a*x*(a + b*x^2)^5) + (11*(1/(8*a*x*(a + b*x^2)^4) + (9*(1/(6*a*x*(a 
+ b*x^2)^3) + (7*(1/(4*a*x*(a + b*x^2)^2) + (5*(1/(2*a*x*(a + b*x^2)) + (3 
*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)))/(2*a)))/(4* 
a)))/(6*a)))/(8*a)))/(10*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 253

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]
 

rule 264

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]
 

rule 1380

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]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {b \left (\frac {\frac {437}{256} b^{4} x^{9}+\frac {977}{128} a \,b^{3} x^{7}+\frac {131}{10} a^{2} b^{2} x^{5}+\frac {1327}{128} a^{3} b \,x^{3}+\frac {843}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {693 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{6}}-\frac {1}{a^{6} x}\)	\(87\)
risch	\(\frac {-\frac {693 b^{5} x^{10}}{256 a^{6}}-\frac {1617 b^{4} x^{8}}{128 a^{5}}-\frac {231 b^{3} x^{6}}{10 a^{4}}-\frac {2607 b^{2} x^{4}}{128 a^{3}}-\frac {2123 b \,x^{2}}{256 a^{2}}-\frac {1}{a}}{x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {693 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{13}+2 b \right ) x +a^{7} \textit {\_R} \right )\right )}{512}\)	\(132\)

Input:

int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-b/a^6*((437/256*b^4*x^9+977/128*a*b^3*x^7+131/10*a^2*b^2*x^5+1327/128*a^3 
*b*x^3+843/256*a^4*x)/(b*x^2+a)^5+693/256/(a*b)^(1/2)*arctan(b/(a*b)^(1/2) 
*x))-1/a^6/x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.17 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [-\frac {6930 \, b^{5} x^{10} + 32340 \, a b^{4} x^{8} + 59136 \, a^{2} b^{3} x^{6} + 52140 \, a^{3} b^{2} x^{4} + 21230 \, a^{4} b x^{2} + 2560 \, a^{5} - 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2560 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}, -\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 5 \, a b^{4} x^{9} + 10 \, a^{2} b^{3} x^{7} + 10 \, a^{3} b^{2} x^{5} + 5 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}}\right ] \] Input:

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

[-1/2560*(6930*b^5*x^10 + 32340*a*b^4*x^8 + 59136*a^2*b^3*x^6 + 52140*a^3* 
b^2*x^4 + 21230*a^4*b*x^2 + 2560*a^5 - 3465*(b^5*x^11 + 5*a*b^4*x^9 + 10*a 
^2*b^3*x^7 + 10*a^3*b^2*x^5 + 5*a^4*b*x^3 + a^5*x)*sqrt(-b/a)*log((b*x^2 - 
 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^6*b^5*x^11 + 5*a^7*b^4*x^9 + 10*a^ 
8*b^3*x^7 + 10*a^9*b^2*x^5 + 5*a^10*b*x^3 + a^11*x), -1/1280*(3465*b^5*x^1 
0 + 16170*a*b^4*x^8 + 29568*a^2*b^3*x^6 + 26070*a^3*b^2*x^4 + 10615*a^4*b* 
x^2 + 1280*a^5 + 3465*(b^5*x^11 + 5*a*b^4*x^9 + 10*a^2*b^3*x^7 + 10*a^3*b^ 
2*x^5 + 5*a^4*b*x^3 + a^5*x)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b^5*x^11 
+ 5*a^7*b^4*x^9 + 10*a^8*b^3*x^7 + 10*a^9*b^2*x^5 + 5*a^10*b*x^3 + a^11*x) 
]
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (- \frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} - \frac {693 \sqrt {- \frac {b}{a^{13}}} \log {\left (\frac {a^{7} \sqrt {- \frac {b}{a^{13}}}}{b} + x \right )}}{512} + \frac {- 1280 a^{5} - 10615 a^{4} b x^{2} - 26070 a^{3} b^{2} x^{4} - 29568 a^{2} b^{3} x^{6} - 16170 a b^{4} x^{8} - 3465 b^{5} x^{10}}{1280 a^{11} x + 6400 a^{10} b x^{3} + 12800 a^{9} b^{2} x^{5} + 12800 a^{8} b^{3} x^{7} + 6400 a^{7} b^{4} x^{9} + 1280 a^{6} b^{5} x^{11}} \] Input:

integrate(1/x**2/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

693*sqrt(-b/a**13)*log(-a**7*sqrt(-b/a**13)/b + x)/512 - 693*sqrt(-b/a**13 
)*log(a**7*sqrt(-b/a**13)/b + x)/512 + (-1280*a**5 - 10615*a**4*b*x**2 - 2 
6070*a**3*b**2*x**4 - 29568*a**2*b**3*x**6 - 16170*a*b**4*x**8 - 3465*b**5 
*x**10)/(1280*a**11*x + 6400*a**10*b*x**3 + 12800*a**9*b**2*x**5 + 12800*a 
**8*b**3*x**7 + 6400*a**7*b**4*x**9 + 1280*a**6*b**5*x**11)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {3465 \, b^{5} x^{10} + 16170 \, a b^{4} x^{8} + 29568 \, a^{2} b^{3} x^{6} + 26070 \, a^{3} b^{2} x^{4} + 10615 \, a^{4} b x^{2} + 1280 \, a^{5}}{1280 \, {\left (a^{6} b^{5} x^{11} + 5 \, a^{7} b^{4} x^{9} + 10 \, a^{8} b^{3} x^{7} + 10 \, a^{9} b^{2} x^{5} + 5 \, a^{10} b x^{3} + a^{11} x\right )}} - \frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} \] Input:

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

-1/1280*(3465*b^5*x^10 + 16170*a*b^4*x^8 + 29568*a^2*b^3*x^6 + 26070*a^3*b 
^2*x^4 + 10615*a^4*b*x^2 + 1280*a^5)/(a^6*b^5*x^11 + 5*a^7*b^4*x^9 + 10*a^ 
8*b^3*x^7 + 10*a^9*b^2*x^5 + 5*a^10*b*x^3 + a^11*x) - 693/256*b*arctan(b*x 
/sqrt(a*b))/(sqrt(a*b)*a^6)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {693 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{6}} - \frac {1}{a^{6} x} - \frac {2185 \, b^{5} x^{9} + 9770 \, a b^{4} x^{7} + 16768 \, a^{2} b^{3} x^{5} + 13270 \, a^{3} b^{2} x^{3} + 4215 \, a^{4} b x}{1280 \, {\left (b x^{2} + a\right )}^{5} a^{6}} \] Input:

integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

-693/256*b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) - 1/(a^6*x) - 1/1280*(218 
5*b^5*x^9 + 9770*a*b^4*x^7 + 16768*a^2*b^3*x^5 + 13270*a^3*b^2*x^3 + 4215* 
a^4*b*x)/((b*x^2 + a)^5*a^6)
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {1}{a}+\frac {2123\,b\,x^2}{256\,a^2}+\frac {2607\,b^2\,x^4}{128\,a^3}+\frac {231\,b^3\,x^6}{10\,a^4}+\frac {1617\,b^4\,x^8}{128\,a^5}+\frac {693\,b^5\,x^{10}}{256\,a^6}}{a^5\,x+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^5+10\,a^2\,b^3\,x^7+5\,a\,b^4\,x^9+b^5\,x^{11}}-\frac {693\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{13/2}} \] Input:

int(1/(x^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)
 

Output:

- (1/a + (2123*b*x^2)/(256*a^2) + (2607*b^2*x^4)/(128*a^3) + (231*b^3*x^6) 
/(10*a^4) + (1617*b^4*x^8)/(128*a^5) + (693*b^5*x^10)/(256*a^6))/(a^5*x + 
b^5*x^11 + 5*a^4*b*x^3 + 5*a*b^4*x^9 + 10*a^3*b^2*x^5 + 10*a^2*b^3*x^7) - 
(693*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(256*a^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} x -17325 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{3}-34650 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{5}-34650 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{7}-17325 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{9}-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{11}-1280 a^{6}-10615 a^{5} b \,x^{2}-26070 a^{4} b^{2} x^{4}-29568 a^{3} b^{3} x^{6}-16170 a^{2} b^{4} x^{8}-3465 a \,b^{5} x^{10}}{1280 a^{7} x \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:

int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

( - 3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*x - 17325*sqrt 
(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**3 - 34650*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**5 - 34650*sqrt(b)*sqrt(a)*a 
tan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*x**7 - 17325*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a*b**4*x**9 - 3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*b**5*x**11 - 1280*a**6 - 10615*a**5*b*x**2 - 26070*a**4*b** 
2*x**4 - 29568*a**3*b**3*x**6 - 16170*a**2*b**4*x**8 - 3465*a*b**5*x**10)/ 
(1280*a**7*x*(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))