Integrand size = 30, antiderivative size = 152 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}-\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} \sqrt {b}} \]
-1/5*c/a^2/x^5+1/3*(-a*d+2*b*c)/a^3/x^3+(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x-1/2 *(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^4/(b*x^2+a)-1/2*(-a^3*f+3*a^2*b*e-5*a* b^2*d+7*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)/b^(1/2)
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}+\frac {-3 b^2 c+2 a b d-a^2 e}{a^4 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}+\frac {\left (-7 b^3 c+5 a b^2 d-3 a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} \sqrt {b}} \]
-1/5*c/(a^2*x^5) + (2*b*c - a*d)/(3*a^3*x^3) + (-3*b^2*c + 2*a*b*d - a^2*e )/(a^4*x) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^4*(a + b*x^2)) + ((-7*b^3*c + 5*a*b^2*d - 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] )/(2*a^(9/2)*Sqrt[b])
Time = 0.57 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-\frac {\left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {2 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-2 \left (\frac {b c}{a}-d\right ) x^2+2 c}{x^6 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {\left (-f a^3+b e a^2-b^2 d a+b^3 c\right ) x^6}{a^3}+\frac {2 \left (e a^2-b d a+b^2 c\right ) x^4}{a^2}-2 \left (\frac {b c}{a}-d\right ) x^2+2 c}{x^6 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\int \left (\frac {2 c}{a x^6}+\frac {f a^3-3 b e a^2+5 b^2 d a-7 b^3 c}{a^3 \left (b x^2+a\right )}+\frac {2 \left (e a^2-2 b d a+3 b^2 c\right )}{a^3 x^2}+\frac {2 (a d-2 b c)}{a^2 x^4}\right )dx}{2 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 (2 b c-a d)}{3 a^2 x^3}-\frac {2 \left (a^2 e-2 a b d+3 b^2 c\right )}{a^3 x}-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+3 a^2 b e-5 a b^2 d+7 b^3 c\right )}{a^{7/2} \sqrt {b}}-\frac {2 c}{5 a x^5}}{2 a}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}\) |
-1/2*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a^4*(a + b*x^2)) + ((-2*c)/( 5*a*x^5) + (2*(2*b*c - a*d))/(3*a^2*x^3) - (2*(3*b^2*c - 2*a*b*d + a^2*e)) /(a^3*x) - ((7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/S qrt[a]])/(a^(7/2)*Sqrt[b]))/(2*a)
3.2.30.3.1 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 = 3.46 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {c}{5 a^{2} x^{5}}-\frac {a d -2 b c}{3 a^{3} x^{3}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{a^{4} x}+\frac {\frac {\left (\frac {1}{2} f \,a^{3}-\frac {1}{2} a^{2} b e +\frac {1}{2} a \,b^{2} d -\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (f \,a^{3}-3 a^{2} b e +5 a \,b^{2} d -7 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{4}}\) | \(137\) |
| risch | \(\frac {\frac {\left (f \,a^{3}-3 a^{2} b e +5 a \,b^{2} d -7 b^{3} c \right ) x^{6}}{2 a^{4}}-\frac {\left (3 a^{2} e -5 a b d +7 b^{2} c \right ) x^{4}}{3 a^{3}}-\frac {\left (5 a d -7 b c \right ) x^{2}}{15 a^{2}}-\frac {c}{5 a}}{x^{5} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{2} b +a^{6} f^{2}-6 a^{5} b e f +10 a^{4} b^{2} d f +9 a^{4} b^{2} e^{2}-14 a^{3} b^{3} c f -30 a^{3} b^{3} d e +42 a^{2} b^{4} c e +25 a^{2} b^{4} d^{2}-70 a c d \,b^{5}+49 c^{2} b^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9} b +2 a^{6} f^{2}-12 a^{5} b e f +20 a^{4} b^{2} d f +18 a^{4} b^{2} e^{2}-28 a^{3} b^{3} c f -60 a^{3} b^{3} d e +84 a^{2} b^{4} c e +50 a^{2} b^{4} d^{2}-140 a c d \,b^{5}+98 c^{2} b^{6}\right ) x +\left (-a^{8} f +3 a^{7} b e -5 a^{6} b^{2} d +7 a^{5} b^{3} c \right ) \textit {\_R} \right )\right )}{4}\) | \(351\) |
-1/5*c/a^2/x^5-1/3*(a*d-2*b*c)/a^3/x^3-(a^2*e-2*a*b*d+3*b^2*c)/a^4/x+1/a^4 *((1/2*f*a^3-1/2*a^2*b*e+1/2*a*b^2*d-1/2*b^3*c)*x/(b*x^2+a)+1/2*(a^3*f-3*a ^2*b*e+5*a*b^2*d-7*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.88 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=\left [-\frac {30 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 12 \, a^{4} b c + 20 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 4 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} - 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{60 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}, -\frac {15 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 6 \, a^{4} b c + 10 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 2 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{30 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}\right ] \]
[-1/60*(30*(7*a*b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 12*a^4* b*c + 20*(7*a^2*b^3*c - 5*a^3*b^2*d + 3*a^4*b*e)*x^4 - 4*(7*a^3*b^2*c - 5* a^4*b*d)*x^2 - 15*((7*b^4*c - 5*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^7 + (7* a*b^3*c - 5*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(-a*b)*log((b*x^2 - 2* sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^5*b^2*x^7 + a^6*b*x^5), -1/30*(15*(7*a* b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 6*a^4*b*c + 10*(7*a^2*b ^3*c - 5*a^3*b^2*d + 3*a^4*b*e)*x^4 - 2*(7*a^3*b^2*c - 5*a^4*b*d)*x^2 + 15 *((7*b^4*c - 5*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^7 + (7*a*b^3*c - 5*a^2*b ^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5*b^2*x ^7 + a^6*b*x^5)]
Time = 19.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.49 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (- a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {- 6 a^{3} c + x^{6} \cdot \left (15 a^{3} f - 45 a^{2} b e + 75 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (- 30 a^{3} e + 50 a^{2} b d - 70 a b^{2} c\right ) + x^{2} \left (- 10 a^{3} d + 14 a^{2} b c\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]
-sqrt(-1/(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b**2*d - 7*b**3*c)*log(-a**5 *sqrt(-1/(a**9*b)) + x)/4 + sqrt(-1/(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b **2*d - 7*b**3*c)*log(a**5*sqrt(-1/(a**9*b)) + x)/4 + (-6*a**3*c + x**6*(1 5*a**3*f - 45*a**2*b*e + 75*a*b**2*d - 105*b**3*c) + x**4*(-30*a**3*e + 50 *a**2*b*d - 70*a*b**2*c) + x**2*(-10*a**3*d + 14*a**2*b*c))/(30*a**5*x**5 + 30*a**4*b*x**7)
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=-\frac {15 \, {\left (7 \, b^{3} c - 5 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} x^{6} + 10 \, {\left (7 \, a b^{2} c - 5 \, a^{2} b d + 3 \, a^{3} e\right )} x^{4} + 6 \, a^{3} c - 2 \, {\left (7 \, a^{2} b c - 5 \, a^{3} d\right )} x^{2}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} - \frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \]
-1/30*(15*(7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*x^6 + 10*(7*a*b^2*c - 5*a^2*b*d + 3*a^3*e)*x^4 + 6*a^3*c - 2*(7*a^2*b*c - 5*a^3*d)*x^2)/(a^4*b*x ^7 + a^5*x^5) - 1/2*(7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*arctan(b*x/s qrt(a*b))/(sqrt(a*b)*a^4)
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} - \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {45 \, b^{2} c x^{4} - 30 \, a b d x^{4} + 15 \, a^{2} e x^{4} - 10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{4} x^{5}} \]
-1/2*(7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*arctan(b*x/sqrt(a*b))/(sqrt (a*b)*a^4) - 1/2*(b^3*c*x - a*b^2*d*x + a^2*b*e*x - a^3*f*x)/((b*x^2 + a)* a^4) - 1/15*(45*b^2*c*x^4 - 30*a*b*d*x^4 + 15*a^2*e*x^4 - 10*a*b*c*x^2 + 5 *a^2*d*x^2 + 3*a^2*c)/(a^4*x^5)
Time = 6.01 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx=-\frac {\frac {c}{5\,a}+\frac {x^6\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^4}+\frac {x^2\,\left (5\,a\,d-7\,b\,c\right )}{15\,a^2}+\frac {x^4\,\left (3\,e\,a^2-5\,d\,a\,b+7\,c\,b^2\right )}{3\,a^3}}{b\,x^7+a\,x^5}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^{9/2}\,\sqrt {b}} \]