Integrand size = 30, antiderivative size = 175 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {b^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{11/2}} \]
-1/9*c/a/x^9+1/7*(-a*d+b*c)/a^2/x^7+1/5*(-a^2*e+a*b*d-b^2*c)/a^3/x^5+1/3*( -a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x^3-b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^5 /x-b^(3/2)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(11/ 2)
Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}+\frac {-b^2 c+a b d-a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}+\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x}+\frac {b^{3/2} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{11/2}} \]
-1/9*c/(a*x^9) + (b*c - a*d)/(7*a^2*x^7) + (-(b^2*c) + a*b*d - a^2*e)/(5*a ^3*x^5) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(3*a^4*x^3) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a^5*x) + (b^(3/2)*(-(b^3*c) + a*b^2*d - a^2* b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)
Time = 0.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
\(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \int \left (\frac {a d-b c}{a^2 x^8}+\frac {a^2 e-a b d+b^2 c}{a^3 x^6}+\frac {b^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 \left (a+b x^2\right )}-\frac {b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 x^2}+\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a^4 x^4}+\frac {c}{a x^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c-a d}{7 a^2 x^7}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{11/2}}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5 x}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {c}{9 a x^9}\) |
-1/9*c/(a*x^9) + (b*c - a*d)/(7*a^2*x^7) - (b^2*c - a*b*d + a^2*e)/(5*a^3* x^5) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(3*a^4*x^3) - (b*(b^3*c - a*b^2 *d + a^2*b*e - a^3*f))/(a^5*x) - (b^(3/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3 *f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)
3.2.22.3.1 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]
Time = 3.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {c}{9 a \,x^{9}}-\frac {a d -b c}{7 a^{2} x^{7}}-\frac {a^{2} e -a b d +b^{2} c}{5 a^{3} x^{5}}-\frac {f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c}{3 a^{4} x^{3}}+\frac {b \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{5} x}+\frac {b^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{5} \sqrt {a b}}\) | \(163\) |
risch | \(\frac {\frac {b \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{8}}{a^{5}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{6}}{3 a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{5 a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{7 a^{2}}-\frac {c}{9 a}}{x^{9}}+\frac {\sqrt {-a b}\, b \ln \left (-b x -\sqrt {-a b}\right ) f}{2 a^{3}}-\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) e}{2 a^{4}}+\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) d}{2 a^{5}}-\frac {\sqrt {-a b}\, b^{4} \ln \left (-b x -\sqrt {-a b}\right ) c}{2 a^{6}}-\frac {\sqrt {-a b}\, b \ln \left (-b x +\sqrt {-a b}\right ) f}{2 a^{3}}+\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) e}{2 a^{4}}-\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) d}{2 a^{5}}+\frac {\sqrt {-a b}\, b^{4} \ln \left (-b x +\sqrt {-a b}\right ) c}{2 a^{6}}\) | \(339\) |
-1/9*c/a/x^9-1/7*(a*d-b*c)/a^2/x^7-1/5*(a^2*e-a*b*d+b^2*c)/a^3/x^5-1/3*(a^ 3*f-a^2*b*e+a*b^2*d-b^3*c)/a^4/x^3+b*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5/x+b ^2*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.30 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.14 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=\left [-\frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 630 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 210 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 70 \, a^{4} c + 126 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 90 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{630 \, a^{5} x^{9}}, -\frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}}\right ] \]
[-1/630*(315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^9*sqrt(-b/a)*log((b *x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 630*(b^4*c - a*b^3*d + a^2*b^2 *e - a^3*b*f)*x^8 - 210*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^6 + 70*a ^4*c + 126*(a^2*b^2*c - a^3*b*d + a^4*e)*x^4 - 90*(a^3*b*c - a^4*d)*x^2)/( a^5*x^9), -1/315*(315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^9*sqrt(b/a )*arctan(x*sqrt(b/a)) + 315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^8 - 105*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^6 + 35*a^4*c + 63*(a^2*b^2*c - a^3*b*d + a^4*e)*x^4 - 45*(a^3*b*c - a^4*d)*x^2)/(a^5*x^9)]
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (167) = 334\).
Time = 24.35 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.02 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=- \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {- 35 a^{4} c + x^{8} \cdot \left (315 a^{3} b f - 315 a^{2} b^{2} e + 315 a b^{3} d - 315 b^{4} c\right ) + x^{6} \left (- 105 a^{4} f + 105 a^{3} b e - 105 a^{2} b^{2} d + 105 a b^{3} c\right ) + x^{4} \left (- 63 a^{4} e + 63 a^{3} b d - 63 a^{2} b^{2} c\right ) + x^{2} \left (- 45 a^{4} d + 45 a^{3} b c\right )}{315 a^{5} x^{9}} \]
-sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**6*sqrt( -b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**2*f - a**2*b **3*e + a*b**4*d - b**5*c) + x)/2 + sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**6*sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2 *d - b**3*c)/(a**3*b**2*f - a**2*b**3*e + a*b**4*d - b**5*c) + x)/2 + (-35 *a**4*c + x**8*(315*a**3*b*f - 315*a**2*b**2*e + 315*a*b**3*d - 315*b**4*c ) + x**6*(-105*a**4*f + 105*a**3*b*e - 105*a**2*b**2*d + 105*a*b**3*c) + x **4*(-63*a**4*e + 63*a**3*b*d - 63*a**2*b**2*c) + x**2*(-45*a**4*d + 45*a* *3*b*c))/(315*a**5*x**9)
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}} \]
-(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b )*a^5) - 1/315*(315*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^8 - 105*(a*b ^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^6 + 35*a^4*c + 63*(a^2*b^2*c - a^3*b *d + a^4*e)*x^4 - 45*(a^3*b*c - a^4*d)*x^2)/(a^5*x^9)
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.13 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, b^{4} c x^{8} - 315 \, a b^{3} d x^{8} + 315 \, a^{2} b^{2} e x^{8} - 315 \, a^{3} b f x^{8} - 105 \, a b^{3} c x^{6} + 105 \, a^{2} b^{2} d x^{6} - 105 \, a^{3} b e x^{6} + 105 \, a^{4} f x^{6} + 63 \, a^{2} b^{2} c x^{4} - 63 \, a^{3} b d x^{4} + 63 \, a^{4} e x^{4} - 45 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{5} x^{9}} \]
-(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b )*a^5) - 1/315*(315*b^4*c*x^8 - 315*a*b^3*d*x^8 + 315*a^2*b^2*e*x^8 - 315* a^3*b*f*x^8 - 105*a*b^3*c*x^6 + 105*a^2*b^2*d*x^6 - 105*a^3*b*e*x^6 + 105* a^4*f*x^6 + 63*a^2*b^2*c*x^4 - 63*a^3*b*d*x^4 + 63*a^4*e*x^4 - 45*a^3*b*c* x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^5*x^9)
Time = 5.96 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx=-\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{7\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{5\,a^3}+\frac {b\,x^8\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^5}}{x^9}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{11/2}} \]