Integrand size = 30, antiderivative size = 112 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {c}{a^2 x}+\frac {f x}{b^2}-\frac {\left (\frac {b c}{a}-d+\frac {a e}{b}-\frac {a^2 f}{b^2}\right ) x}{2 a \left (a+b x^2\right )}-\frac {\left (3 b^3 c-a b^2 d-a^2 b e+3 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{5/2}} \]
-c/a^2/x+f*x/b^2-1/2*(b*c/a-d+a*e/b-a^2*f/b^2)*x/a/(b*x^2+a)-1/2*(3*a^3*f- a^2*b*e-a*b^2*d+3*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(5/2)/b^(5/2)
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {c}{a^2 x}+\frac {f x}{b^2}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {\left (3 b^3 c-a b^2 d-a^2 b e+3 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{5/2}} \]
-(c/(a^2*x)) + (f*x)/b^2 + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a ^2*b^2*(a + b*x^2)) - ((3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*ArcTan[(Sqr t[b]*x)/Sqrt[a]])/(2*a^(5/2)*b^(5/2))
Time = 0.37 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2336, 25, 1584, 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.
\(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle -\frac {\int -\frac {\frac {2 a f x^4}{b}-\left (\frac {f a^2}{b^2}-\frac {e a}{b}-d+\frac {b c}{a}\right ) x^2+2 c}{x^2 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (-\frac {a^2 f}{b^2}+\frac {b c}{a}+\frac {a e}{b}-d\right )}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {2 a f x^4}{b}-\left (\frac {f a^2}{b^2}-\frac {e a}{b}-d+\frac {b c}{a}\right ) x^2+2 c}{x^2 \left (b x^2+a\right )}dx}{2 a}-\frac {x \left (-\frac {a^2 f}{b^2}+\frac {b c}{a}+\frac {a e}{b}-d\right )}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {\int \left (\frac {2 c}{a x^2}+\frac {2 a f}{b^2}+\frac {-3 f a^3+b e a^2+b^2 d a-3 b^3 c}{a b^2 \left (b x^2+a\right )}\right )dx}{2 a}-\frac {x \left (-\frac {a^2 f}{b^2}+\frac {b c}{a}+\frac {a e}{b}-d\right )}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{a^{3/2} b^{5/2}}+\frac {2 a f x}{b^2}-\frac {2 c}{a x}}{2 a}-\frac {x \left (-\frac {a^2 f}{b^2}+\frac {b c}{a}+\frac {a e}{b}-d\right )}{2 a \left (a+b x^2\right )}\) |
-1/2*(((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(a*(a + b*x^2)) + ((-2*c)/( a*x) + (2*a*f*x)/b^2 - ((3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*ArcTan[(Sq rt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2)))/(2*a)
3.2.28.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -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 = 3.47 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {f x}{b^{2}}-\frac {c}{a^{2} x}-\frac {\frac {\left (-\frac {1}{2} f \,a^{3}+\frac {1}{2} a^{2} b e -\frac {1}{2} a \,b^{2} d +\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (3 f \,a^{3}-a^{2} b e -a \,b^{2} d +3 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{2} b^{2}}\) | \(107\) |
risch | \(\frac {f x}{b^{2}}+\frac {\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -3 b^{3} c \right ) x^{2}}{2 a^{2}}-\frac {b^{2} c}{a}}{b^{2} x \left (b \,x^{2}+a \right )}-\frac {3 a \ln \left (-\sqrt {-a b}\, x -a \right ) f}{4 b^{2} \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) e}{4 b \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) d}{4 \sqrt {-a b}\, a}-\frac {3 b \ln \left (-\sqrt {-a b}\, x -a \right ) c}{4 \sqrt {-a b}\, a^{2}}+\frac {3 a \ln \left (-\sqrt {-a b}\, x +a \right ) f}{4 b^{2} \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) e}{4 b \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) d}{4 \sqrt {-a b}\, a}+\frac {3 b \ln \left (-\sqrt {-a b}\, x +a \right ) c}{4 \sqrt {-a b}\, a^{2}}\) | \(271\) |
f*x/b^2-c/a^2/x-1/a^2/b^2*((-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*(3*a^3*f-a^2*b*e-a*b^2*d+3*b^3*c)/(a*b)^(1/2)*arctan(b*x/( a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.16 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a^{3} b^{2} f x^{4} - 4 \, a^{2} b^{3} c - 2 \, {\left (3 \, a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{2} - {\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} + {\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}, \frac {2 \, a^{3} b^{2} f x^{4} - 2 \, a^{2} b^{3} c - {\left (3 \, a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{2} - {\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} + {\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}\right ] \]
[1/4*(4*a^3*b^2*f*x^4 - 4*a^2*b^3*c - 2*(3*a*b^4*c - a^2*b^3*d + a^3*b^2*e - 3*a^4*b*f)*x^2 - ((3*b^4*c - a*b^3*d - a^2*b^2*e + 3*a^3*b*f)*x^3 + (3* a*b^3*c - a^2*b^2*d - a^3*b*e + 3*a^4*f)*x)*sqrt(-a*b)*log((b*x^2 + 2*sqrt (-a*b)*x - a)/(b*x^2 + a)))/(a^3*b^4*x^3 + a^4*b^3*x), 1/2*(2*a^3*b^2*f*x^ 4 - 2*a^2*b^3*c - (3*a*b^4*c - a^2*b^3*d + a^3*b^2*e - 3*a^4*b*f)*x^2 - (( 3*b^4*c - a*b^3*d - a^2*b^2*e + 3*a^3*b*f)*x^3 + (3*a*b^3*c - a^2*b^2*d - a^3*b*e + 3*a^4*f)*x)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^3*b^4*x^3 + a^4* b^3*x)]
Time = 2.03 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.76 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {- \frac {1}{a^{5} b^{5}}} \cdot \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log {\left (- a^{3} b^{2} \sqrt {- \frac {1}{a^{5} b^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{5} b^{5}}} \cdot \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log {\left (a^{3} b^{2} \sqrt {- \frac {1}{a^{5} b^{5}}} + x \right )}}{4} + \frac {- 2 a b^{2} c + x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - 3 b^{3} c\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac {f x}{b^{2}} \]
sqrt(-1/(a**5*b**5))*(3*a**3*f - a**2*b*e - a*b**2*d + 3*b**3*c)*log(-a**3 *b**2*sqrt(-1/(a**5*b**5)) + x)/4 - sqrt(-1/(a**5*b**5))*(3*a**3*f - a**2* b*e - a*b**2*d + 3*b**3*c)*log(a**3*b**2*sqrt(-1/(a**5*b**5)) + x)/4 + (-2 *a*b**2*c + x**2*(a**3*f - a**2*b*e + a*b**2*d - 3*b**3*c))/(2*a**3*b**2*x + 2*a**2*b**3*x**3) + f*x/b**2
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {2 \, a b^{2} c + {\left (3 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )}} + \frac {f x}{b^{2}} - \frac {{\left (3 \, b^{3} c - a b^{2} d - a^{2} b e + 3 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2} b^{2}} \]
-1/2*(2*a*b^2*c + (3*b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a^2*b^3*x^3 + a^3*b^2*x) + f*x/b^2 - 1/2*(3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*arcta n(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^2)
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {f x}{b^{2}} - \frac {{\left (3 \, b^{3} c - a b^{2} d - a^{2} b e + 3 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2} b^{2}} - \frac {3 \, b^{3} c x^{2} - a b^{2} d x^{2} + a^{2} b e x^{2} - a^{3} f x^{2} + 2 \, a b^{2} c}{2 \, {\left (b x^{3} + a x\right )} a^{2} b^{2}} \]
f*x/b^2 - 1/2*(3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*arctan(b*x/sqrt(a*b) )/(sqrt(a*b)*a^2*b^2) - 1/2*(3*b^3*c*x^2 - a*b^2*d*x^2 + a^2*b*e*x^2 - a^3 *f*x^2 + 2*a*b^2*c)/((b*x^3 + a*x)*a^2*b^2)
Time = 6.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {f\,x}{b^2}-\frac {\frac {x^2\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+3\,c\,b^3\right )}{2\,a^2}+\frac {b^2\,c}{a}}{b^3\,x^3+a\,b^2\,x}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,f\,a^3-e\,a^2\,b-d\,a\,b^2+3\,c\,b^3\right )}{2\,a^{5/2}\,b^{5/2}} \]