Integrand size = 30, antiderivative size = 230 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=-\frac {c}{9 a^2 x^9}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac {b^{3/2} \left (11 b^3 c-9 a b^2 d+7 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{13/2}} \]
-1/9*c/a^2/x^9+1/7*(-a*d+2*b*c)/a^3/x^7+1/5*(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x ^5+1/3*(-a^3*f+2*a^2*b*e-3*a*b^2*d+4*b^3*c)/a^5/x^3-b*(-2*a^3*f+3*a^2*b*e- 4*a*b^2*d+5*b^3*c)/a^6/x-1/2*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^6/(b*x ^2+a)-1/2*b^(3/2)*(-5*a^3*f+7*a^2*b*e-9*a*b^2*d+11*b^3*c)*arctan(x*b^(1/2) /a^(1/2))/a^(13/2)
Time = 0.08 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=-\frac {c}{9 a^2 x^9}+\frac {2 b c-a d}{7 a^3 x^7}+\frac {-3 b^2 c+2 a b d-a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}+\frac {b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}+\frac {b^{3/2} \left (-11 b^3 c+9 a b^2 d-7 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{13/2}} \]
-1/9*c/(a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) + (-3*b^2*c + 2*a*b*d - a^2*e )/(5*a^4*x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*x^3) + (b *(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^6*(a + b*x^2)) + (b^(3/2)*(-11*b^3*c + 9*a*b^2*d - 7*a^2*b*e + 5*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))
Time = 0.76 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2336, 25, 2333, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-\frac {b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{10}}{a^5}+\frac {2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^8}{a^4}-\frac {2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {2 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-2 \left (\frac {b c}{a}-d\right ) x^2+2 c}{x^{10} \left (b x^2+a\right )}dx}{2 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{10}}{a^5}+\frac {2 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^8}{a^4}-\frac {2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {2 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-2 \left (\frac {b c}{a}-d\right ) x^2+2 c}{x^{10} \left (b x^2+a\right )}dx}{2 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\int \left (\frac {\left (5 f a^3-7 b e a^2+9 b^2 d a-11 b^3 c\right ) b^2}{a^5 \left (b x^2+a\right )}-\frac {2 \left (2 f a^3-3 b e a^2+4 b^2 d a-5 b^3 c\right ) b}{a^5 x^2}+\frac {2 \left (f a^3-2 b e a^2+3 b^2 d a-4 b^3 c\right )}{a^4 x^4}+\frac {2 \left (e a^2-2 b d a+3 b^2 c\right )}{a^3 x^6}+\frac {2 (a d-2 b c)}{a^2 x^8}+\frac {2 c}{a x^{10}}\right )dx}{2 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 (2 b c-a d)}{7 a^2 x^7}-\frac {2 \left (a^2 e-2 a b d+3 b^2 c\right )}{5 a^3 x^5}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-5 a^3 f+7 a^2 b e-9 a b^2 d+11 b^3 c\right )}{a^{11/2}}-\frac {2 b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^5 x}+\frac {2 \left (a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right )}{3 a^4 x^3}-\frac {2 c}{9 a x^9}}{2 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}\) |
-1/2*(b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a^6*(a + b*x^2)) + ((-2* c)/(9*a*x^9) + (2*(2*b*c - a*d))/(7*a^2*x^7) - (2*(3*b^2*c - 2*a*b*d + a^2 *e))/(5*a^3*x^5) + (2*(4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f))/(3*a^4*x^ 3) - (2*b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^5*x) - (b^(3/2)* (11*b^3*c - 9*a*b^2*d + 7*a^2*b*e - 5*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/ a^(11/2))/(2*a)
3.2.32.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Time = 3.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(-\frac {c}{9 a^{2} x^{9}}-\frac {a d -2 b c}{7 a^{3} x^{7}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{5 a^{4} x^{5}}-\frac {f \,a^{3}-2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{3 a^{5} x^{3}}+\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right )}{a^{6} x}+\frac {b^{2} \left (\frac {\left (\frac {1}{2} f \,a^{3}-\frac {1}{2} a^{2} b e +\frac {1}{2} a \,b^{2} d -\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (5 f \,a^{3}-7 a^{2} b e +9 a \,b^{2} d -11 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{6}}\) | \(210\) |
| risch | \(\frac {\frac {b^{2} \left (5 f \,a^{3}-7 a^{2} b e +9 a \,b^{2} d -11 b^{3} c \right ) x^{10}}{2 a^{6}}+\frac {b \left (5 f \,a^{3}-7 a^{2} b e +9 a \,b^{2} d -11 b^{3} c \right ) x^{8}}{3 a^{5}}-\frac {\left (5 f \,a^{3}-7 a^{2} b e +9 a \,b^{2} d -11 b^{3} c \right ) x^{6}}{15 a^{4}}-\frac {\left (7 a^{2} e -9 a b d +11 b^{2} c \right ) x^{4}}{35 a^{3}}-\frac {\left (9 a d -11 b c \right ) x^{2}}{63 a^{2}}-\frac {c}{9 a}}{x^{9} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{2}+25 a^{6} b^{3} f^{2}-70 a^{5} b^{4} e f +90 a^{4} b^{5} d f +49 a^{4} b^{5} e^{2}-110 a^{3} b^{6} c f -126 a^{3} b^{6} d e +154 a^{2} b^{7} c e +81 a^{2} b^{7} d^{2}-198 a \,b^{8} c d +121 b^{9} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{13}+50 a^{6} b^{3} f^{2}-140 a^{5} b^{4} e f +180 a^{4} b^{5} d f +98 a^{4} b^{5} e^{2}-220 a^{3} b^{6} c f -252 a^{3} b^{6} d e +308 a^{2} b^{7} c e +162 a^{2} b^{7} d^{2}-396 a \,b^{8} c d +242 b^{9} c^{2}\right ) x +\left (-5 a^{10} b f +7 a^{9} b^{2} e -9 a^{8} b^{3} d +11 a^{7} b^{4} c \right ) \textit {\_R} \right )\right )}{4}\) | \(438\) |
-1/9*c/a^2/x^9-1/7*(a*d-2*b*c)/a^3/x^7-1/5*(a^2*e-2*a*b*d+3*b^2*c)/a^4/x^5 -1/3*(a^3*f-2*a^2*b*e+3*a*b^2*d-4*b^3*c)/a^5/x^3+b*(2*a^3*f-3*a^2*b*e+4*a* b^2*d-5*b^3*c)/a^6/x+b^2/a^6*((1/2*f*a^3-1/2*a^2*b*e+1/2*a*b^2*d-1/2*b^3*c )*x/(b*x^2+a)+1/2*(5*a^3*f-7*a^2*b*e+9*a*b^2*d-11*b^3*c)/(a*b)^(1/2)*arcta n(b*x/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.53 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=\left [-\frac {630 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 420 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 84 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 140 \, a^{5} c + 36 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 20 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \, {\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} + {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{1260 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}, -\frac {315 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 210 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 42 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 70 \, a^{5} c + 18 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 10 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \, {\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} + {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{630 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}\right ] \]
[-1/1260*(630*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 42 0*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 84*(11*a^2*b^ 3*c - 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 140*a^5*c + 36*(11*a^3*b^2* c - 9*a^4*b*d + 7*a^5*e)*x^4 - 20*(11*a^4*b*c - 9*a^5*d)*x^2 + 315*((11*b^ 5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^ 3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^9)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/ a) - a)/(b*x^2 + a)))/(a^6*b*x^11 + a^7*x^9), -1/630*(315*(11*b^5*c - 9*a* b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 210*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 42*(11*a^2*b^3*c - 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 70*a^5*c + 18*(11*a^3*b^2*c - 9*a^4*b*d + 7*a^5*e)*x^4 - 10*(11*a^4*b*c - 9*a^5*d)*x^2 + 315*((11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)* x^9)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b*x^11 + a^7*x^9)]
Timed out. \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=-\frac {315 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 210 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 42 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 70 \, a^{5} c + 18 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 10 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2}}{630 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}} - \frac {{\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{6}} \]
-1/630*(315*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 210* (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 42*(11*a^2*b^3* c - 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 70*a^5*c + 18*(11*a^3*b^2*c - 9*a^4*b*d + 7*a^5*e)*x^4 - 10*(11*a^4*b*c - 9*a^5*d)*x^2)/(a^6*b*x^11 + a ^7*x^9) - 1/2*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*arctan(b* x/sqrt(a*b))/(sqrt(a*b)*a^6)
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=-\frac {{\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{6}} - \frac {b^{5} c x - a b^{4} d x + a^{2} b^{3} e x - a^{3} b^{2} f x}{2 \, {\left (b x^{2} + a\right )} a^{6}} - \frac {1575 \, b^{4} c x^{8} - 1260 \, a b^{3} d x^{8} + 945 \, a^{2} b^{2} e x^{8} - 630 \, a^{3} b f x^{8} - 420 \, a b^{3} c x^{6} + 315 \, a^{2} b^{2} d x^{6} - 210 \, a^{3} b e x^{6} + 105 \, a^{4} f x^{6} + 189 \, a^{2} b^{2} c x^{4} - 126 \, a^{3} b d x^{4} + 63 \, a^{4} e x^{4} - 90 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{6} x^{9}} \]
-1/2*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*arctan(b*x/sqrt(a* b))/(sqrt(a*b)*a^6) - 1/2*(b^5*c*x - a*b^4*d*x + a^2*b^3*e*x - a^3*b^2*f*x )/((b*x^2 + a)*a^6) - 1/315*(1575*b^4*c*x^8 - 1260*a*b^3*d*x^8 + 945*a^2*b ^2*e*x^8 - 630*a^3*b*f*x^8 - 420*a*b^3*c*x^6 + 315*a^2*b^2*d*x^6 - 210*a^3 *b*e*x^6 + 105*a^4*f*x^6 + 189*a^2*b^2*c*x^4 - 126*a^3*b*d*x^4 + 63*a^4*e* x^4 - 90*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^6*x^9)
Time = 5.61 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx=-\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{15\,a^4}+\frac {x^2\,\left (9\,a\,d-11\,b\,c\right )}{63\,a^2}+\frac {x^4\,\left (7\,e\,a^2-9\,d\,a\,b+11\,c\,b^2\right )}{35\,a^3}+\frac {b\,x^8\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{3\,a^5}+\frac {b^2\,x^{10}\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{2\,a^6}}{b\,x^{11}+a\,x^9}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{2\,a^{13/2}} \]
- (c/(9*a) - (x^6*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(15*a^4) + (x^2*(9*a*d - 11*b*c))/(63*a^2) + (x^4*(11*b^2*c + 7*a^2*e - 9*a*b*d))/(3 5*a^3) + (b*x^8*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(3*a^5) + (b ^2*x^10*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(2*a^6))/(a*x^9 + b* x^11) - (b^(3/2)*atan((b^(1/2)*x)/a^(1/2))*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(2*a^(13/2))