Integrand size = 30, antiderivative size = 211 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{13/2}} \]
-1/11*c/a/x^11+1/9*(-a*d+b*c)/a^2/x^9+1/7*(-a^2*e+a*b*d-b^2*c)/a^3/x^7+1/5 *(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x^5-1/3*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3* c)/a^5/x^3+b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^6/x+b^(5/2)*(-a^3*f+a^2*b* e-a*b^2*d+b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(13/2)
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}+\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{13/2}} \]
-1/11*c/(a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^ 3*x^7) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(S qrt[b]*x)/Sqrt[a]])/a^(13/2)
Time = 0.42 (sec) , antiderivative size = 211, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (\frac {a d-b c}{a^2 x^{10}}+\frac {a^2 e-a b d+b^2 c}{a^3 x^8}-\frac {b^3 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^6 \left (a+b x^2\right )}+\frac {b^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^6 x^2}-\frac {b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 x^4}+\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a^4 x^6}+\frac {c}{a x^{12}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c-a d}{9 a^2 x^9}-\frac {a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{13/2}}+\frac {b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac {c}{11 a x^{11}}\) |
-1/11*c/(a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^ 3*x^7) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a ^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt [b]*x)/Sqrt[a]])/a^(13/2)
3.2.23.3.1 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 = 3.50 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {c}{11 a \,x^{11}}-\frac {a d -b c}{9 a^{2} x^{9}}-\frac {a^{2} e -a b d +b^{2} c}{7 a^{3} x^{7}}-\frac {f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c}{5 a^{4} x^{5}}-\frac {b^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{6} x}+\frac {b \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{3 a^{5} x^{3}}-\frac {b^{3} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{6} \sqrt {a b}}\) | \(201\) |
| risch | \(\frac {-\frac {b^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{10}}{a^{6}}+\frac {b \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{8}}{3 a^{5}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{6}}{5 a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{7 a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{9 a^{2}}-\frac {c}{11 a}}{x^{11}}+\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) f}{2 a^{4}}-\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) e}{2 a^{5}}+\frac {\sqrt {-a b}\, b^{4} \ln \left (-b x +\sqrt {-a b}\right ) d}{2 a^{6}}-\frac {\sqrt {-a b}\, b^{5} \ln \left (-b x +\sqrt {-a b}\right ) c}{2 a^{7}}-\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) f}{2 a^{4}}+\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) e}{2 a^{5}}-\frac {\sqrt {-a b}\, b^{4} \ln \left (-b x -\sqrt {-a b}\right ) d}{2 a^{6}}+\frac {\sqrt {-a b}\, b^{5} \ln \left (-b x -\sqrt {-a b}\right ) c}{2 a^{7}}\) | \(380\) |
-1/11*c/a/x^11-1/9*(a*d-b*c)/a^2/x^9-1/7*(a^2*e-a*b*d+b^2*c)/a^3/x^7-1/5*( a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^4/x^5-b^2*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^6 /x+1/3*b*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5/x^3-b^3*(a^3*f-a^2*b*e+a*b^2*d- b^3*c)/a^6/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.28 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.17 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=\left [-\frac {3465 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 6930 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} + 2310 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} - 1386 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} + 630 \, a^{5} c + 990 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} - 770 \, {\left (a^{4} b c - a^{5} d\right )} x^{2}}{6930 \, a^{6} x^{11}}, \frac {3465 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3465 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} + 693 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} - 315 \, a^{5} c - 495 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} + 385 \, {\left (a^{4} b c - a^{5} d\right )} x^{2}}{3465 \, a^{6} x^{11}}\right ] \]
[-1/6930*(3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^11*sqrt(-b/a)*l og((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 6930*(b^5*c - a*b^4*d + a ^2*b^3*e - a^3*b^2*f)*x^10 + 2310*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b *f)*x^8 - 1386*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^6 + 630*a^5*c + 990*(a^3*b^2*c - a^4*b*d + a^5*e)*x^4 - 770*(a^4*b*c - a^5*d)*x^2)/(a^6*x ^11), 1/3465*(3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^11*sqrt(b/a )*arctan(x*sqrt(b/a)) + 3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^1 0 - 1155*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^8 + 693*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x^6 - 315*a^5*c - 495*(a^3*b^2*c - a^4*b*d + a^5*e)*x^4 + 385*(a^4*b*c - a^5*d)*x^2)/(a^6*x^11)]
Time = 38.92 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.89 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=\frac {\sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} + \frac {- 315 a^{5} c + x^{10} \left (- 3465 a^{3} b^{2} f + 3465 a^{2} b^{3} e - 3465 a b^{4} d + 3465 b^{5} c\right ) + x^{8} \cdot \left (1155 a^{4} b f - 1155 a^{3} b^{2} e + 1155 a^{2} b^{3} d - 1155 a b^{4} c\right ) + x^{6} \left (- 693 a^{5} f + 693 a^{4} b e - 693 a^{3} b^{2} d + 693 a^{2} b^{3} c\right ) + x^{4} \left (- 495 a^{5} e + 495 a^{4} b d - 495 a^{3} b^{2} c\right ) + x^{2} \left (- 385 a^{5} d + 385 a^{4} b c\right )}{3465 a^{6} x^{11}} \]
sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**7*sqrt(- b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**3*f - a**2*b* *4*e + a*b**5*d - b**6*c) + x)/2 - sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**7*sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2* d - b**3*c)/(a**3*b**3*f - a**2*b**4*e + a*b**5*d - b**6*c) + x)/2 + (-315 *a**5*c + x**10*(-3465*a**3*b**2*f + 3465*a**2*b**3*e - 3465*a*b**4*d + 34 65*b**5*c) + x**8*(1155*a**4*b*f - 1155*a**3*b**2*e + 1155*a**2*b**3*d - 1 155*a*b**4*c) + x**6*(-693*a**5*f + 693*a**4*b*e - 693*a**3*b**2*d + 693*a **2*b**3*c) + x**4*(-495*a**5*e + 495*a**4*b*d - 495*a**3*b**2*c) + x**2*( -385*a**5*d + 385*a**4*b*c))/(3465*a**6*x**11)
Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=\frac {{\left (b^{6} c - a b^{5} d + a^{2} b^{4} e - a^{3} b^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{6}} + \frac {3465 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} + 693 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} - 315 \, a^{5} c - 495 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} + 385 \, {\left (a^{4} b c - a^{5} d\right )} x^{2}}{3465 \, a^{6} x^{11}} \]
(b^6*c - a*b^5*d + a^2*b^4*e - a^3*b^3*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b) *a^6) + 1/3465*(3465*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^10 - 1155 *(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^8 + 693*(a^2*b^3*c - a^3*b^ 2*d + a^4*b*e - a^5*f)*x^6 - 315*a^5*c - 495*(a^3*b^2*c - a^4*b*d + a^5*e) *x^4 + 385*(a^4*b*c - a^5*d)*x^2)/(a^6*x^11)
Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=\frac {{\left (b^{6} c - a b^{5} d + a^{2} b^{4} e - a^{3} b^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{6}} + \frac {3465 \, b^{5} c x^{10} - 3465 \, a b^{4} d x^{10} + 3465 \, a^{2} b^{3} e x^{10} - 3465 \, a^{3} b^{2} f x^{10} - 1155 \, a b^{4} c x^{8} + 1155 \, a^{2} b^{3} d x^{8} - 1155 \, a^{3} b^{2} e x^{8} + 1155 \, a^{4} b f x^{8} + 693 \, a^{2} b^{3} c x^{6} - 693 \, a^{3} b^{2} d x^{6} + 693 \, a^{4} b e x^{6} - 693 \, a^{5} f x^{6} - 495 \, a^{3} b^{2} c x^{4} + 495 \, a^{4} b d x^{4} - 495 \, a^{5} e x^{4} + 385 \, a^{4} b c x^{2} - 385 \, a^{5} d x^{2} - 315 \, a^{5} c}{3465 \, a^{6} x^{11}} \]
(b^6*c - a*b^5*d + a^2*b^4*e - a^3*b^3*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b) *a^6) + 1/3465*(3465*b^5*c*x^10 - 3465*a*b^4*d*x^10 + 3465*a^2*b^3*e*x^10 - 3465*a^3*b^2*f*x^10 - 1155*a*b^4*c*x^8 + 1155*a^2*b^3*d*x^8 - 1155*a^3*b ^2*e*x^8 + 1155*a^4*b*f*x^8 + 693*a^2*b^3*c*x^6 - 693*a^3*b^2*d*x^6 + 693* a^4*b*e*x^6 - 693*a^5*f*x^6 - 495*a^3*b^2*c*x^4 + 495*a^4*b*d*x^4 - 495*a^ 5*e*x^4 + 385*a^4*b*c*x^2 - 385*a^5*d*x^2 - 315*a^5*c)/(a^6*x^11)
Time = 5.86 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx=\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{13/2}}-\frac {\frac {c}{11\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{5\,a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{9\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{7\,a^3}+\frac {b\,x^8\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^5}-\frac {b^2\,x^{10}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^6}}{x^{11}} \]
(b^(5/2)*atan((b^(1/2)*x)/a^(1/2))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/a^ (13/2) - (c/(11*a) - (x^6*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(5*a^4) + ( x^2*(a*d - b*c))/(9*a^2) + (x^4*(b^2*c + a^2*e - a*b*d))/(7*a^3) + (b*x^8* (b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^5) - (b^2*x^10*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/a^6)/x^11