Integrand size = 30, antiderivative size = 277 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=-\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac {b^{3/2} \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{15/2}} \]
-1/9*c/a^3/x^9+1/7*(-a*d+3*b*c)/a^4/x^7+1/5*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x ^5+1/3*(-a^3*f+3*a^2*b*e-6*a*b^2*d+10*b^3*c)/a^6/x^3-b*(-3*a^3*f+6*a^2*b*e -10*a*b^2*d+15*b^3*c)/a^7/x-1/4*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^6/( b*x^2+a)^2-1/8*b^2*(-11*a^3*f+15*a^2*b*e-19*a*b^2*d+23*b^3*c)*x/a^7/(b*x^2 +a)-1/8*b^(3/2)*(-35*a^3*f+63*a^2*b*e-99*a*b^2*d+143*b^3*c)*arctan(x*b^(1/ 2)/a^(1/2))/a^(15/2)
Time = 0.11 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=-\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}+\frac {b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^7 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}+\frac {b^2 \left (-23 b^3 c+19 a b^2 d-15 a^2 b e+11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}+\frac {b^{3/2} \left (-143 b^3 c+99 a b^2 d-63 a^2 b e+35 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{15/2}} \]
-1/9*c/(a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e) /(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b *(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^6*(a + b*x^2)^2) + (b^2*(-23*b^3*c + 19*a*b^2*d - 15*a^2*b*e + 11*a^3*f)*x)/(8*a^7*(a + b*x^2)) + (b^(3/2)*(-14 3*b^3*c + 99*a*b^2*d - 63*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]) /(8*a^(15/2))
Time = 1.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-\frac {3 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{10}}{a^5}+\frac {4 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^8}{a^4}-\frac {4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {4 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^{10} \left (b x^2+a\right )^2}dx}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {3 b^2 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^{10}}{a^5}+\frac {4 b \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^8}{a^4}-\frac {4 \left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {4 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^{10} \left (b x^2+a\right )^2}dx}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\frac {b^2 \left (-11 f a^3+15 b e a^2-19 b^2 d a+23 b^3 c\right ) x^{10}}{a^5}+\frac {8 b \left (-2 f a^3+3 b e a^2-4 b^2 d a+5 b^3 c\right ) x^8}{a^4}-\frac {8 \left (-f a^3+2 b e a^2-3 b^2 d a+4 b^3 c\right ) x^6}{a^3}+\frac {8 \left (e a^2-2 b d a+3 b^2 c\right ) x^4}{a^2}-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 c}{x^{10} \left (b x^2+a\right )}dx}{2 a}-\frac {b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{2 a^6 \left (a+b x^2\right )}}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {b^2 \left (-11 f a^3+15 b e a^2-19 b^2 d a+23 b^3 c\right ) x^{10}}{a^5}+\frac {8 b \left (-2 f a^3+3 b e a^2-4 b^2 d a+5 b^3 c\right ) x^8}{a^4}-\frac {8 \left (-f a^3+2 b e a^2-3 b^2 d a+4 b^3 c\right ) x^6}{a^3}+\frac {8 \left (e a^2-2 b d a+3 b^2 c\right ) x^4}{a^2}-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 c}{x^{10} \left (b x^2+a\right )}dx}{2 a}-\frac {b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{2 a^6 \left (a+b x^2\right )}}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\frac {\int \left (\frac {\left (35 f a^3-63 b e a^2+99 b^2 d a-143 b^3 c\right ) b^2}{a^5 \left (b x^2+a\right )}-\frac {8 \left (3 f a^3-6 b e a^2+10 b^2 d a-15 b^3 c\right ) b}{a^5 x^2}+\frac {8 \left (f a^3-3 b e a^2+6 b^2 d a-10 b^3 c\right )}{a^4 x^4}+\frac {8 \left (e a^2-3 b d a+6 b^2 c\right )}{a^3 x^6}+\frac {8 (a d-3 b c)}{a^2 x^8}+\frac {8 c}{a x^{10}}\right )dx}{2 a}-\frac {b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{2 a^6 \left (a+b x^2\right )}}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {8 (3 b c-a d)}{7 a^2 x^7}-\frac {8 \left (a^2 e-3 a b d+6 b^2 c\right )}{5 a^3 x^5}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-35 a^3 f+63 a^2 b e-99 a b^2 d+143 b^3 c\right )}{a^{11/2}}-\frac {8 b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^5 x}+\frac {8 \left (a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right )}{3 a^4 x^3}-\frac {8 c}{9 a x^9}}{2 a}-\frac {b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{2 a^6 \left (a+b x^2\right )}}{4 a}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2}\) |
-1/4*(b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a^6*(a + b*x^2)^2) + (-1 /2*(b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^3*f)*x)/(a^6*(a + b*x^2 )) + ((-8*c)/(9*a*x^9) + (8*(3*b*c - a*d))/(7*a^2*x^7) - (8*(6*b^2*c - 3*a *b*d + a^2*e))/(5*a^3*x^5) + (8*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f) )/(3*a^4*x^3) - (8*b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^5*x ) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35*a^3*f)*ArcTan[(Sqrt [b]*x)/Sqrt[a]])/a^(11/2))/(2*a))/(4*a)
3.2.42.3.1 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 = 3.50 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {c}{9 a^{3} x^{9}}-\frac {a d -3 b c}{7 a^{4} x^{7}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{5 a^{5} x^{5}}-\frac {f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{3 a^{6} x^{3}}+\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right )}{a^{7} x}+\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{3} b f -\frac {15}{8} a^{2} e \,b^{2}+\frac {19}{8} a \,b^{3} d -\frac {23}{8} b^{4} c \right ) x^{3}+\frac {a \left (13 f \,a^{3}-17 a^{2} b e +21 a \,b^{2} d -25 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (35 f \,a^{3}-63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{7}}\) | \(248\) |
| risch | \(\frac {\frac {b^{3} \left (35 f \,a^{3}-63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{12}}{8 a^{7}}+\frac {5 b^{2} \left (35 f \,a^{3}-63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{10}}{24 a^{6}}+\frac {b \left (35 f \,a^{3}-63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{8}}{15 a^{5}}-\frac {\left (35 f \,a^{3}-63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{6}}{105 a^{4}}-\frac {\left (63 a^{2} e -99 a b d +143 b^{2} c \right ) x^{4}}{315 a^{3}}-\frac {\left (9 a d -13 b c \right ) x^{2}}{63 a^{2}}-\frac {c}{9 a}}{x^{9} \left (b \,x^{2}+a \right )^{2}}+\frac {35 \sqrt {-a b}\, b \ln \left (-b x -\sqrt {-a b}\right ) f}{16 a^{5}}-\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) e}{16 a^{6}}+\frac {99 \sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) d}{16 a^{7}}-\frac {143 \sqrt {-a b}\, b^{4} \ln \left (-b x -\sqrt {-a b}\right ) c}{16 a^{8}}-\frac {35 \sqrt {-a b}\, b \ln \left (-b x +\sqrt {-a b}\right ) f}{16 a^{5}}+\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) e}{16 a^{6}}-\frac {99 \sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) d}{16 a^{7}}+\frac {143 \sqrt {-a b}\, b^{4} \ln \left (-b x +\sqrt {-a b}\right ) c}{16 a^{8}}\) | \(432\) |
-1/9*c/a^3/x^9-1/7*(a*d-3*b*c)/a^4/x^7-1/5*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^5 -1/3*(a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3*c)/a^6/x^3+b*(3*a^3*f-6*a^2*b*e+10* a*b^2*d-15*b^3*c)/a^7/x+b^2/a^7*(((11/8*a^3*b*f-15/8*a^2*e*b^2+19/8*a*b^3* d-23/8*b^4*c)*x^3+1/8*a*(13*a^3*f-17*a^2*b*e+21*a*b^2*d-25*b^3*c)*x)/(b*x^ 2+a)^2+1/8*(35*a^3*f-63*a^2*b*e+99*a*b^2*d-143*b^3*c)/(a*b)^(1/2)*arctan(b *x/(a*b)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.79 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=\left [-\frac {630 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d + 63 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{12} + 1050 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d + 63 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{10} + 336 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d + 63 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{8} + 560 \, a^{6} c - 48 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d + 63 \, a^{5} b e - 35 \, a^{6} f\right )} x^{6} + 16 \, {\left (143 \, a^{4} b^{2} c - 99 \, a^{5} b d + 63 \, a^{6} e\right )} x^{4} - 80 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2} + 315 \, {\left ({\left (143 \, b^{6} c - 99 \, a b^{5} d + 63 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{13} + 2 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d + 63 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{11} + {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d + 63 \, a^{4} b^{2} e - 35 \, a^{5} 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 )}{5040 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\right )}}, -\frac {315 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d + 63 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{12} + 525 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d + 63 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{10} + 168 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d + 63 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{8} + 280 \, a^{6} c - 24 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d + 63 \, a^{5} b e - 35 \, a^{6} f\right )} x^{6} + 8 \, {\left (143 \, a^{4} b^{2} c - 99 \, a^{5} b d + 63 \, a^{6} e\right )} x^{4} - 40 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2} + 315 \, {\left ({\left (143 \, b^{6} c - 99 \, a b^{5} d + 63 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{13} + 2 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d + 63 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{11} + {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d + 63 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{9}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{2520 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\right )}}\right ] \]
[-1/5040*(630*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^12 + 1050*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^10 + 3 36*(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^8 + 560*a^ 6*c - 48*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 16*( 143*a^4*b^2*c - 99*a^5*b*d + 63*a^6*e)*x^4 - 80*(13*a^5*b*c - 9*a^6*d)*x^2 + 315*((143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^13 + 2*(1 43*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^11 + (143*a^2*b ^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^9)*sqrt(-b/a)*log((b*x^ 2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9 *x^9), -1/2520*(315*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f) *x^12 + 525*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^1 0 + 168*(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^8 + 2 80*a^6*c - 24*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 8*(143*a^4*b^2*c - 99*a^5*b*d + 63*a^6*e)*x^4 - 40*(13*a^5*b*c - 9*a^6*d) *x^2 + 315*((143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^13 + 2*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^11 + (143*a ^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^9)*sqrt(b/a)*arctan (x*sqrt(b/a)))/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9)]
Timed out. \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=-\frac {315 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d + 63 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{12} + 525 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d + 63 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{10} + 168 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d + 63 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{8} + 280 \, a^{6} c - 24 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d + 63 \, a^{5} b e - 35 \, a^{6} f\right )} x^{6} + 8 \, {\left (143 \, a^{4} b^{2} c - 99 \, a^{5} b d + 63 \, a^{6} e\right )} x^{4} - 40 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2}}{2520 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\right )}} - \frac {{\left (143 \, b^{5} c - 99 \, a b^{4} d + 63 \, a^{2} b^{3} e - 35 \, a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{7}} \]
-1/2520*(315*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^12 + 525*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^10 + 168 *(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^8 + 280*a^6* c - 24*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 8*(143 *a^4*b^2*c - 99*a^5*b*d + 63*a^6*e)*x^4 - 40*(13*a^5*b*c - 9*a^6*d)*x^2)/( a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9) - 1/8*(143*b^5*c - 99*a*b^4*d + 63* a^2*b^3*e - 35*a^3*b^2*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7)
Time = 0.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=-\frac {{\left (143 \, b^{5} c - 99 \, a b^{4} d + 63 \, a^{2} b^{3} e - 35 \, a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{7}} - \frac {23 \, b^{6} c x^{3} - 19 \, a b^{5} d x^{3} + 15 \, a^{2} b^{4} e x^{3} - 11 \, a^{3} b^{3} f x^{3} + 25 \, a b^{5} c x - 21 \, a^{2} b^{4} d x + 17 \, a^{3} b^{3} e x - 13 \, a^{4} b^{2} f x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{7}} - \frac {4725 \, b^{4} c x^{8} - 3150 \, a b^{3} d x^{8} + 1890 \, a^{2} b^{2} e x^{8} - 945 \, a^{3} b f x^{8} - 1050 \, a b^{3} c x^{6} + 630 \, a^{2} b^{2} d x^{6} - 315 \, a^{3} b e x^{6} + 105 \, a^{4} f x^{6} + 378 \, a^{2} b^{2} c x^{4} - 189 \, a^{3} b d x^{4} + 63 \, a^{4} e x^{4} - 135 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{7} x^{9}} \]
-1/8*(143*b^5*c - 99*a*b^4*d + 63*a^2*b^3*e - 35*a^3*b^2*f)*arctan(b*x/sqr t(a*b))/(sqrt(a*b)*a^7) - 1/8*(23*b^6*c*x^3 - 19*a*b^5*d*x^3 + 15*a^2*b^4* e*x^3 - 11*a^3*b^3*f*x^3 + 25*a*b^5*c*x - 21*a^2*b^4*d*x + 17*a^3*b^3*e*x - 13*a^4*b^2*f*x)/((b*x^2 + a)^2*a^7) - 1/315*(4725*b^4*c*x^8 - 3150*a*b^3 *d*x^8 + 1890*a^2*b^2*e*x^8 - 945*a^3*b*f*x^8 - 1050*a*b^3*c*x^6 + 630*a^2 *b^2*d*x^6 - 315*a^3*b*e*x^6 + 105*a^4*f*x^6 + 378*a^2*b^2*c*x^4 - 189*a^3 *b*d*x^4 + 63*a^4*e*x^4 - 135*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^7* x^9)
Time = 5.65 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx=-\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{105\,a^4}+\frac {x^2\,\left (9\,a\,d-13\,b\,c\right )}{63\,a^2}+\frac {x^4\,\left (63\,e\,a^2-99\,d\,a\,b+143\,c\,b^2\right )}{315\,a^3}+\frac {b\,x^8\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{15\,a^5}+\frac {5\,b^2\,x^{10}\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{24\,a^6}+\frac {b^3\,x^{12}\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{8\,a^7}}{a^2\,x^9+2\,a\,b\,x^{11}+b^2\,x^{13}}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{8\,a^{15/2}} \]
- (c/(9*a) - (x^6*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(105*a ^4) + (x^2*(9*a*d - 13*b*c))/(63*a^2) + (x^4*(143*b^2*c + 63*a^2*e - 99*a* b*d))/(315*a^3) + (b*x^8*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e)) /(15*a^5) + (5*b^2*x^10*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/ (24*a^6) + (b^3*x^12*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(8* a^7))/(a^2*x^9 + b^2*x^13 + 2*a*b*x^11) - (b^(3/2)*atan((b^(1/2)*x)/a^(1/2 ))*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(8*a^(15/2))