Integrand size = 30, antiderivative size = 137 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\sqrt {b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \]
-1/7*c/a/x^7+1/5*(-a*d+b*c)/a^2/x^5+1/3*(-a^2*e+a*b*d-b^2*c)/a^3/x^3+(-a^3 *f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x+(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*arctan(x*b^ (1/2)/a^(1/2))*b^(1/2)/a^(9/2)
Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}+\frac {-b^2 c+a b d-a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac {\sqrt {b} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \]
-1/7*c/(a*x^7) + (b*c - a*d)/(5*a^2*x^5) + (-(b^2*c) + a*b*d - a^2*e)/(3*a ^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a^4*x) - (Sqrt[b]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)
Time = 0.35 (sec) , antiderivative size = 137, 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.
\(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \int \left (\frac {a d-b c}{a^2 x^6}+\frac {a^2 e-a b d+b^2 c}{a^3 x^4}-\frac {b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^4 \left (a+b x^2\right )}+\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a^4 x^2}+\frac {c}{a x^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c-a d}{5 a^2 x^5}-\frac {a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{9/2}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{7 a x^7}\) |
-1/7*c/(a*x^7) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(3*a^3* x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a^4*x) + (Sqrt[b]*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)
3.2.21.3.1 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 = 3.48 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {c}{7 a \,x^{7}}-\frac {a d -b c}{5 a^{2} x^{5}}-\frac {a^{2} e -a b d +b^{2} c}{3 a^{3} x^{3}}-\frac {f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c}{a^{4} x}-\frac {b \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}\) | \(129\) |
risch | \(\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{6}}{a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{3 a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{5 a^{2}}-\frac {c}{7 a}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{2}+a^{6} b \,f^{2}-2 a^{5} b^{2} e f +2 a^{4} b^{3} d f +a^{4} b^{3} e^{2}-2 a^{3} b^{4} c f -2 a^{3} b^{4} d e +2 a^{2} b^{5} c e +a^{2} b^{5} d^{2}-2 a \,b^{6} c d +b^{7} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9}+2 a^{6} b \,f^{2}-4 a^{5} b^{2} e f +4 a^{4} b^{3} d f +2 a^{4} b^{3} e^{2}-4 a^{3} b^{4} c f -4 a^{3} b^{4} d e +4 a^{2} b^{5} c e +2 a^{2} b^{5} d^{2}-4 a \,b^{6} c d +2 b^{7} c^{2}\right ) x +\left (a^{8} f -a^{7} b e +a^{6} b^{2} d -a^{5} b^{3} c \right ) \textit {\_R} \right )\right )}{2}\) | \(337\) |
-1/7*c/a/x^7-1/5*(a*d-b*c)/a^2/x^5-1/3*(a^2*e-a*b*d+b^2*c)/a^3/x^3-(a^3*f- a^2*b*e+a*b^2*d-b^3*c)/a^4/x-b*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^4/(a*b)^(1/ 2)*arctan(b*x/(a*b)^(1/2))
Time = 0.28 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.13 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=\left [-\frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 210 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} + 70 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} + 30 \, a^{3} c - 42 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{210 \, a^{4} x^{7}}, \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \]
[-1/210*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 210*(b^3*c - a*b^2*d + a^2*b*e - a ^3*f)*x^6 + 70*(a*b^2*c - a^2*b*d + a^3*e)*x^4 + 30*a^3*c - 42*(a^2*b*c - a^3*d)*x^2)/(a^4*x^7), 1/105*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7* sqrt(b/a)*arctan(x*sqrt(b/a)) + 105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^ 6 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^4 - 15*a^3*c + 21*(a^2*b*c - a^3*d)*x ^2)/(a^4*x^7)]
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (128) = 256\).
Time = 5.86 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.20 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=\frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} + \frac {- 15 a^{3} c + x^{6} \left (- 105 a^{3} f + 105 a^{2} b e - 105 a b^{2} d + 105 b^{3} c\right ) + x^{4} \left (- 35 a^{3} e + 35 a^{2} b d - 35 a b^{2} c\right ) + x^{2} \left (- 21 a^{3} d + 21 a^{2} b c\right )}{105 a^{4} x^{7}} \]
sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**5*sqrt(-b/a* *9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b*f - a**2*b**2*e + a*b* *3*d - b**4*c) + x)/2 - sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3 *c)*log(a**5*sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b *f - a**2*b**2*e + a*b**3*d - b**4*c) + x)/2 + (-15*a**3*c + x**6*(-105*a* *3*f + 105*a**2*b*e - 105*a*b**2*d + 105*b**3*c) + x**4*(-35*a**3*e + 35*a **2*b*d - 35*a*b**2*c) + x**2*(-21*a**3*d + 21*a**2*b*c))/(105*a**4*x**7)
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}} \]
(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a ^4) + 1/105*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6 - 35*(a*b^2*c - a ^2*b*d + a^3*e)*x^4 - 15*a^3*c + 21*(a^2*b*c - a^3*d)*x^2)/(a^4*x^7)
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} c x^{6} - 105 \, a b^{2} d x^{6} + 105 \, a^{2} b e x^{6} - 105 \, a^{3} f x^{6} - 35 \, a b^{2} c x^{4} + 35 \, a^{2} b d x^{4} - 35 \, a^{3} e x^{4} + 21 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{4} x^{7}} \]
(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a ^4) + 1/105*(105*b^3*c*x^6 - 105*a*b^2*d*x^6 + 105*a^2*b*e*x^6 - 105*a^3*f *x^6 - 35*a*b^2*c*x^4 + 35*a^2*b*d*x^4 - 35*a^3*e*x^4 + 21*a^2*b*c*x^2 - 2 1*a^3*d*x^2 - 15*a^3*c)/(a^4*x^7)
Time = 6.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{9/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{5\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,a^3}}{x^7} \]