Integrand size = 30, antiderivative size = 196 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{3 a^4 x^3}-\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}-\frac {\left (63 b^3 c-35 a b^2 d+15 a^2 b e-3 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2} \sqrt {b}} \]
-1/5*c/a^3/x^5+1/3*(-a*d+3*b*c)/a^4/x^3+(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x-1/4 *(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^4/(b*x^2+a)^2-1/8*(-3*a^3*f+7*a^2*b*e- 11*a*b^2*d+15*b^3*c)*x/a^5/(b*x^2+a)-1/8*(-3*a^3*f+15*a^2*b*e-35*a*b^2*d+6 3*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(11/2)/b^(1/2)
Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{3 a^4 x^3}+\frac {-6 b^2 c+3 a b d-a^2 e}{a^5 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}+\frac {\left (-15 b^3 c+11 a b^2 d-7 a^2 b e+3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}+\frac {\left (-63 b^3 c+35 a b^2 d-15 a^2 b e+3 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2} \sqrt {b}} \]
-1/5*c/(a^3*x^5) + (3*b*c - a*d)/(3*a^4*x^3) + (-6*b^2*c + 3*a*b*d - a^2*e )/(a^5*x) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^4*(a + b*x^2)^ 2) + ((-15*b^3*c + 11*a*b^2*d - 7*a^2*b*e + 3*a^3*f)*x)/(8*a^5*(a + b*x^2) ) + ((-63*b^3*c + 35*a*b^2*d - 15*a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sq rt[a]])/(8*a^(11/2)*Sqrt[b])
Time = 0.84 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2336, 25, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-\frac {3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {4 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^6 \left (b x^2+a\right )^2}dx}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {3 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {4 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^6 \left (b x^2+a\right )^2}dx}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\frac {\left (-3 f a^3+7 b e a^2-11 b^2 d a+15 b^3 c\right ) x^6}{a^3}+\frac {8 \left (e a^2-2 b d a+3 b^2 c\right ) x^4}{a^2}-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 c}{x^6 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{2 a^4 \left (a+b x^2\right )}}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {\left (-3 f a^3+7 b e a^2-11 b^2 d a+15 b^3 c\right ) x^6}{a^3}+\frac {8 \left (e a^2-2 b d a+3 b^2 c\right ) x^4}{a^2}-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 c}{x^6 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{2 a^4 \left (a+b x^2\right )}}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\frac {\int \left (\frac {8 c}{a x^6}+\frac {3 f a^3-15 b e a^2+35 b^2 d a-63 b^3 c}{a^3 \left (b x^2+a\right )}+\frac {8 \left (e a^2-3 b d a+6 b^2 c\right )}{a^3 x^2}+\frac {8 (a d-3 b c)}{a^2 x^4}\right )dx}{2 a}-\frac {x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{2 a^4 \left (a+b x^2\right )}}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {8 (3 b c-a d)}{3 a^2 x^3}-\frac {8 \left (a^2 e-3 a b d+6 b^2 c\right )}{a^3 x}-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^3 f+15 a^2 b e-35 a b^2 d+63 b^3 c\right )}{a^{7/2} \sqrt {b}}-\frac {8 c}{5 a x^5}}{2 a}-\frac {x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{2 a^4 \left (a+b x^2\right )}}{4 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}\) |
-1/4*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a^4*(a + b*x^2)^2) + (-1/2*( (15*b^3*c - 11*a*b^2*d + 7*a^2*b*e - 3*a^3*f)*x)/(a^4*(a + b*x^2)) + ((-8* c)/(5*a*x^5) + (8*(3*b*c - a*d))/(3*a^2*x^3) - (8*(6*b^2*c - 3*a*b*d + a^2 *e))/(a^3*x) - ((63*b^3*c - 35*a*b^2*d + 15*a^2*b*e - 3*a^3*f)*ArcTan[(Sqr t[b]*x)/Sqrt[a]])/(a^(7/2)*Sqrt[b]))/(2*a))/(4*a)
3.2.40.3.1 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 = 3.52 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {c}{5 a^{3} x^{5}}-\frac {a d -3 b c}{3 a^{4} x^{3}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{a^{5} x}+\frac {\frac {\left (\frac {3}{8} a^{3} b f -\frac {7}{8} a^{2} e \,b^{2}+\frac {11}{8} a \,b^{3} d -\frac {15}{8} b^{4} c \right ) x^{3}+\frac {a \left (5 f \,a^{3}-9 a^{2} b e +13 a \,b^{2} d -17 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 f \,a^{3}-15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{a^{5}}\) | \(176\) |
| risch | \(\frac {\frac {b \left (3 f \,a^{3}-15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) x^{8}}{8 a^{5}}+\frac {5 \left (3 f \,a^{3}-15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) x^{6}}{24 a^{4}}-\frac {\left (15 a^{2} e -35 a b d +63 b^{2} c \right ) x^{4}}{15 a^{3}}-\frac {\left (5 a d -9 b c \right ) x^{2}}{15 a^{2}}-\frac {c}{5 a}}{x^{5} \left (b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (-\sqrt {-a b}\, x +a \right ) f}{16 \sqrt {-a b}\, a^{2}}+\frac {15 \ln \left (-\sqrt {-a b}\, x +a \right ) b e}{16 \sqrt {-a b}\, a^{3}}-\frac {35 \ln \left (-\sqrt {-a b}\, x +a \right ) b^{2} d}{16 \sqrt {-a b}\, a^{4}}+\frac {63 \ln \left (-\sqrt {-a b}\, x +a \right ) b^{3} c}{16 \sqrt {-a b}\, a^{5}}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) f}{16 \sqrt {-a b}\, a^{2}}-\frac {15 \ln \left (-\sqrt {-a b}\, x -a \right ) b e}{16 \sqrt {-a b}\, a^{3}}+\frac {35 \ln \left (-\sqrt {-a b}\, x -a \right ) b^{2} d}{16 \sqrt {-a b}\, a^{4}}-\frac {63 \ln \left (-\sqrt {-a b}\, x -a \right ) b^{3} c}{16 \sqrt {-a b}\, a^{5}}\) | \(350\) |
-1/5*c/a^3/x^5-1/3*(a*d-3*b*c)/a^4/x^3-(a^2*e-3*a*b*d+6*b^2*c)/a^5/x+1/a^5 *(((3/8*a^3*b*f-7/8*a^2*e*b^2+11/8*a*b^3*d-15/8*b^4*c)*x^3+1/8*a*(5*a^3*f- 9*a^2*b*e+13*a*b^2*d-17*b^3*c)*x)/(b*x^2+a)^2+1/8*(3*a^3*f-15*a^2*b*e+35*a *b^2*d-63*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 628, normalized size of antiderivative = 3.20 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=\left [-\frac {30 \, {\left (63 \, a b^{5} c - 35 \, a^{2} b^{4} d + 15 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{8} + 48 \, a^{5} b c + 50 \, {\left (63 \, a^{2} b^{4} c - 35 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{6} + 16 \, {\left (63 \, a^{3} b^{3} c - 35 \, a^{4} b^{2} d + 15 \, a^{5} b e\right )} x^{4} - 16 \, {\left (9 \, a^{4} b^{2} c - 5 \, a^{5} b d\right )} x^{2} - 15 \, {\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{240 \, {\left (a^{6} b^{3} x^{9} + 2 \, a^{7} b^{2} x^{7} + a^{8} b x^{5}\right )}}, -\frac {15 \, {\left (63 \, a b^{5} c - 35 \, a^{2} b^{4} d + 15 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{8} + 24 \, a^{5} b c + 25 \, {\left (63 \, a^{2} b^{4} c - 35 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{6} + 8 \, {\left (63 \, a^{3} b^{3} c - 35 \, a^{4} b^{2} d + 15 \, a^{5} b e\right )} x^{4} - 8 \, {\left (9 \, a^{4} b^{2} c - 5 \, a^{5} b d\right )} x^{2} + 15 \, {\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{120 \, {\left (a^{6} b^{3} x^{9} + 2 \, a^{7} b^{2} x^{7} + a^{8} b x^{5}\right )}}\right ] \]
[-1/240*(30*(63*a*b^5*c - 35*a^2*b^4*d + 15*a^3*b^3*e - 3*a^4*b^2*f)*x^8 + 48*a^5*b*c + 50*(63*a^2*b^4*c - 35*a^3*b^3*d + 15*a^4*b^2*e - 3*a^5*b*f)* x^6 + 16*(63*a^3*b^3*c - 35*a^4*b^2*d + 15*a^5*b*e)*x^4 - 16*(9*a^4*b^2*c - 5*a^5*b*d)*x^2 - 15*((63*b^5*c - 35*a*b^4*d + 15*a^2*b^3*e - 3*a^3*b^2*f )*x^9 + 2*(63*a*b^4*c - 35*a^2*b^3*d + 15*a^3*b^2*e - 3*a^4*b*f)*x^7 + (63 *a^2*b^3*c - 35*a^3*b^2*d + 15*a^4*b*e - 3*a^5*f)*x^5)*sqrt(-a*b)*log((b*x ^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^6*b^3*x^9 + 2*a^7*b^2*x^7 + a^8* b*x^5), -1/120*(15*(63*a*b^5*c - 35*a^2*b^4*d + 15*a^3*b^3*e - 3*a^4*b^2*f )*x^8 + 24*a^5*b*c + 25*(63*a^2*b^4*c - 35*a^3*b^3*d + 15*a^4*b^2*e - 3*a^ 5*b*f)*x^6 + 8*(63*a^3*b^3*c - 35*a^4*b^2*d + 15*a^5*b*e)*x^4 - 8*(9*a^4*b ^2*c - 5*a^5*b*d)*x^2 + 15*((63*b^5*c - 35*a*b^4*d + 15*a^2*b^3*e - 3*a^3* b^2*f)*x^9 + 2*(63*a*b^4*c - 35*a^2*b^3*d + 15*a^3*b^2*e - 3*a^4*b*f)*x^7 + (63*a^2*b^3*c - 35*a^3*b^2*d + 15*a^4*b*e - 3*a^5*f)*x^5)*sqrt(a*b)*arct an(sqrt(a*b)*x/a))/(a^6*b^3*x^9 + 2*a^7*b^2*x^7 + a^8*b*x^5)]
Timed out. \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {15 \, {\left (63 \, b^{4} c - 35 \, a b^{3} d + 15 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 25 \, {\left (63 \, a b^{3} c - 35 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} + 24 \, a^{4} c + 8 \, {\left (63 \, a^{2} b^{2} c - 35 \, a^{3} b d + 15 \, a^{4} e\right )} x^{4} - 8 \, {\left (9 \, a^{3} b c - 5 \, a^{4} d\right )} x^{2}}{120 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} - \frac {{\left (63 \, b^{3} c - 35 \, a b^{2} d + 15 \, a^{2} b e - 3 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} \]
-1/120*(15*(63*b^4*c - 35*a*b^3*d + 15*a^2*b^2*e - 3*a^3*b*f)*x^8 + 25*(63 *a*b^3*c - 35*a^2*b^2*d + 15*a^3*b*e - 3*a^4*f)*x^6 + 24*a^4*c + 8*(63*a^2 *b^2*c - 35*a^3*b*d + 15*a^4*e)*x^4 - 8*(9*a^3*b*c - 5*a^4*d)*x^2)/(a^5*b^ 2*x^9 + 2*a^6*b*x^7 + a^7*x^5) - 1/8*(63*b^3*c - 35*a*b^2*d + 15*a^2*b*e - 3*a^3*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {{\left (63 \, b^{3} c - 35 \, a b^{2} d + 15 \, a^{2} b e - 3 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} - \frac {15 \, b^{4} c x^{3} - 11 \, a b^{3} d x^{3} + 7 \, a^{2} b^{2} e x^{3} - 3 \, a^{3} b f x^{3} + 17 \, a b^{3} c x - 13 \, a^{2} b^{2} d x + 9 \, a^{3} b e x - 5 \, a^{4} f x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{5}} - \frac {90 \, b^{2} c x^{4} - 45 \, a b d x^{4} + 15 \, a^{2} e x^{4} - 15 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{5} x^{5}} \]
-1/8*(63*b^3*c - 35*a*b^2*d + 15*a^2*b*e - 3*a^3*f)*arctan(b*x/sqrt(a*b))/ (sqrt(a*b)*a^5) - 1/8*(15*b^4*c*x^3 - 11*a*b^3*d*x^3 + 7*a^2*b^2*e*x^3 - 3 *a^3*b*f*x^3 + 17*a*b^3*c*x - 13*a^2*b^2*d*x + 9*a^3*b*e*x - 5*a^4*f*x)/(( b*x^2 + a)^2*a^5) - 1/15*(90*b^2*c*x^4 - 45*a*b*d*x^4 + 15*a^2*e*x^4 - 15* a*b*c*x^2 + 5*a^2*d*x^2 + 3*a^2*c)/(a^5*x^5)
Time = 5.60 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {\frac {c}{5\,a}+\frac {5\,x^6\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{24\,a^4}+\frac {x^2\,\left (5\,a\,d-9\,b\,c\right )}{15\,a^2}+\frac {x^4\,\left (15\,e\,a^2-35\,d\,a\,b+63\,c\,b^2\right )}{15\,a^3}+\frac {b\,x^8\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{8\,a^5}}{a^2\,x^5+2\,a\,b\,x^7+b^2\,x^9}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{8\,a^{11/2}\,\sqrt {b}} \]
- (c/(5*a) + (5*x^6*(63*b^3*c - 3*a^3*f - 35*a*b^2*d + 15*a^2*b*e))/(24*a^ 4) + (x^2*(5*a*d - 9*b*c))/(15*a^2) + (x^4*(63*b^2*c + 15*a^2*e - 35*a*b*d ))/(15*a^3) + (b*x^8*(63*b^3*c - 3*a^3*f - 35*a*b^2*d + 15*a^2*b*e))/(8*a^ 5))/(a^2*x^5 + b^2*x^9 + 2*a*b*x^7) - (atan((b^(1/2)*x)/a^(1/2))*(63*b^3*c - 3*a^3*f - 35*a*b^2*d + 15*a^2*b*e))/(8*a^(11/2)*b^(1/2))