Integrand size = 30, antiderivative size = 115 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {A}{d^2 x}+\frac {D x}{e^2}+\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x}{2 d^2 e^2 \left (d+e x^2\right )}-\frac {\left (3 d^3 D-C d^2 e-B d e^2+3 A e^3\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} e^{5/2}} \] Output:
-A/d^2/x+D*x/e^2+1/2*(-A*e^3+B*d*e^2-C*d^2*e+D*d^3)*x/d^2/e^2/(e*x^2+d)-1/ 2*(3*A*e^3-B*d*e^2-C*d^2*e+3*D*d^3)*arctan(e^(1/2)*x/d^(1/2))/d^(5/2)/e^(5 /2)
Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {A}{d^2 x}+\frac {D x}{e^2}+\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x}{2 d^2 e^2 \left (d+e x^2\right )}-\frac {\left (3 d^3 D-C d^2 e-B d e^2+3 A e^3\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} e^{5/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^2*(d + e*x^2)^2),x]
Output:
-(A/(d^2*x)) + (D*x)/e^2 + ((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x)/(2*d^2* e^2*(d + e*x^2)) - ((3*d^3*D - C*d^2*e - B*d*e^2 + 3*A*e^3)*ArcTan[(Sqrt[e ]*x)/Sqrt[d]])/(2*d^(5/2)*e^(5/2))
Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle \frac {x \left (-\frac {A e}{d}+B-\frac {C d}{e}+\frac {d^2 D}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac {\int -\frac {\frac {2 d D x^4}{e}+\left (-\frac {D d^2}{e^2}+\frac {C d}{e}+B-\frac {A e}{d}\right ) x^2+2 A}{x^2 \left (e x^2+d\right )}dx}{2 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {2 d D x^4}{e}+\left (-\frac {D d^2}{e^2}+\frac {C d}{e}+B-\frac {A e}{d}\right ) x^2+2 A}{x^2 \left (e x^2+d\right )}dx}{2 d}+\frac {x \left (-\frac {A e}{d}+B-\frac {C d}{e}+\frac {d^2 D}{e^2}\right )}{2 d \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {\int \left (\frac {2 A}{d x^2}+\frac {-3 D d^3+C e d^2+B e^2 d-3 A e^3}{d e^2 \left (e x^2+d\right )}+\frac {2 d D}{e^2}\right )dx}{2 d}+\frac {x \left (-\frac {A e}{d}+B-\frac {C d}{e}+\frac {d^2 D}{e^2}\right )}{2 d \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 A e^3-B d e^2-C d^2 e+3 d^3 D\right )}{d^{3/2} e^{5/2}}-\frac {2 A}{d x}+\frac {2 d D x}{e^2}}{2 d}+\frac {x \left (-\frac {A e}{d}+B-\frac {C d}{e}+\frac {d^2 D}{e^2}\right )}{2 d \left (d+e x^2\right )}\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^2*(d + e*x^2)^2),x]
Output:
((B + (d^2*D)/e^2 - (C*d)/e - (A*e)/d)*x)/(2*d*(d + e*x^2)) + ((-2*A)/(d*x ) + (2*d*D*x)/e^2 - ((3*d^3*D - C*d^2*e - B*d*e^2 + 3*A*e^3)*ArcTan[(Sqrt[ e]*x)/Sqrt[d]])/(d^(3/2)*e^(5/2)))/(2*d)
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 = 0.47 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {D x}{e^{2}}-\frac {\frac {\left (\frac {1}{2} A \,e^{3}-\frac {1}{2} B d \,e^{2}+\frac {1}{2} C e \,d^{2}-\frac {1}{2} D d^{3}\right ) x}{e \,x^{2}+d}+\frac {\left (3 A \,e^{3}-B d \,e^{2}-C e \,d^{2}+3 D d^{3}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}}{d^{2} e^{2}}-\frac {A}{d^{2} x}\) | \(107\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/x^2/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
D*x/e^2-1/d^2/e^2*((1/2*A*e^3-1/2*B*d*e^2+1/2*C*e*d^2-1/2*D*d^3)*x/(e*x^2+ d)+1/2*(3*A*e^3-B*d*e^2-C*d^2*e+3*D*d^3)/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2 )))-A/d^2/x
Time = 0.08 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.07 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\left [\frac {4 \, D d^{3} e^{2} x^{4} - 4 \, A d^{2} e^{3} + 2 \, {\left (3 \, D d^{4} e - C d^{3} e^{2} + B d^{2} e^{3} - 3 \, A d e^{4}\right )} x^{2} - {\left ({\left (3 \, D d^{3} e - C d^{2} e^{2} - B d e^{3} + 3 \, A e^{4}\right )} x^{3} + {\left (3 \, D d^{4} - C d^{3} e - B d^{2} e^{2} + 3 \, A d e^{3}\right )} x\right )} \sqrt {-d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{4 \, {\left (d^{3} e^{4} x^{3} + d^{4} e^{3} x\right )}}, \frac {2 \, D d^{3} e^{2} x^{4} - 2 \, A d^{2} e^{3} + {\left (3 \, D d^{4} e - C d^{3} e^{2} + B d^{2} e^{3} - 3 \, A d e^{4}\right )} x^{2} - {\left ({\left (3 \, D d^{3} e - C d^{2} e^{2} - B d e^{3} + 3 \, A e^{4}\right )} x^{3} + {\left (3 \, D d^{4} - C d^{3} e - B d^{2} e^{2} + 3 \, A d e^{3}\right )} x\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right )}{2 \, {\left (d^{3} e^{4} x^{3} + d^{4} e^{3} x\right )}}\right ] \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^2/(e*x^2+d)^2,x, algorithm="fricas")
Output:
[1/4*(4*D*d^3*e^2*x^4 - 4*A*d^2*e^3 + 2*(3*D*d^4*e - C*d^3*e^2 + B*d^2*e^3 - 3*A*d*e^4)*x^2 - ((3*D*d^3*e - C*d^2*e^2 - B*d*e^3 + 3*A*e^4)*x^3 + (3* D*d^4 - C*d^3*e - B*d^2*e^2 + 3*A*d*e^3)*x)*sqrt(-d*e)*log((e*x^2 + 2*sqrt (-d*e)*x - d)/(e*x^2 + d)))/(d^3*e^4*x^3 + d^4*e^3*x), 1/2*(2*D*d^3*e^2*x^ 4 - 2*A*d^2*e^3 + (3*D*d^4*e - C*d^3*e^2 + B*d^2*e^3 - 3*A*d*e^4)*x^2 - (( 3*D*d^3*e - C*d^2*e^2 - B*d*e^3 + 3*A*e^4)*x^3 + (3*D*d^4 - C*d^3*e - B*d^ 2*e^2 + 3*A*d*e^3)*x)*sqrt(d*e)*arctan(sqrt(d*e)*x/d))/(d^3*e^4*x^3 + d^4* e^3*x)]
Time = 2.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {D x}{e^{2}} + \frac {\sqrt {- \frac {1}{d^{5} e^{5}}} \cdot \left (3 A e^{3} - B d e^{2} - C d^{2} e + 3 D d^{3}\right ) \log {\left (- d^{3} e^{2} \sqrt {- \frac {1}{d^{5} e^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{d^{5} e^{5}}} \cdot \left (3 A e^{3} - B d e^{2} - C d^{2} e + 3 D d^{3}\right ) \log {\left (d^{3} e^{2} \sqrt {- \frac {1}{d^{5} e^{5}}} + x \right )}}{4} + \frac {- 2 A d e^{2} + x^{2} \left (- 3 A e^{3} + B d e^{2} - C d^{2} e + D d^{3}\right )}{2 d^{3} e^{2} x + 2 d^{2} e^{3} x^{3}} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/x**2/(e*x**2+d)**2,x)
Output:
D*x/e**2 + sqrt(-1/(d**5*e**5))*(3*A*e**3 - B*d*e**2 - C*d**2*e + 3*D*d**3 )*log(-d**3*e**2*sqrt(-1/(d**5*e**5)) + x)/4 - sqrt(-1/(d**5*e**5))*(3*A*e **3 - B*d*e**2 - C*d**2*e + 3*D*d**3)*log(d**3*e**2*sqrt(-1/(d**5*e**5)) + x)/4 + (-2*A*d*e**2 + x**2*(-3*A*e**3 + B*d*e**2 - C*d**2*e + D*d**3))/(2 *d**3*e**2*x + 2*d**2*e**3*x**3)
Exception generated. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^2/(e*x^2+d)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {D x}{e^{2}} - \frac {{\left (3 \, D d^{3} - C d^{2} e - B d e^{2} + 3 \, A e^{3}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{2} e^{2}} + \frac {D d^{3} x^{2} - C d^{2} e x^{2} + B d e^{2} x^{2} - 3 \, A e^{3} x^{2} - 2 \, A d e^{2}}{2 \, {\left (e x^{3} + d x\right )} d^{2} e^{2}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^2/(e*x^2+d)^2,x, algorithm="giac")
Output:
D*x/e^2 - 1/2*(3*D*d^3 - C*d^2*e - B*d*e^2 + 3*A*e^3)*arctan(e*x/sqrt(d*e) )/(sqrt(d*e)*d^2*e^2) + 1/2*(D*d^3*x^2 - C*d^2*e*x^2 + B*d*e^2*x^2 - 3*A*e ^3*x^2 - 2*A*d*e^2)/((e*x^3 + d*x)*d^2*e^2)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^2*(d + e*x^2)^2),x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^2*(d + e*x^2)^2), x)
Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.43 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {-3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d \,e^{3} x -3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{4} x^{3}+\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{2} e^{2} x +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b d \,e^{3} x^{3}+\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c \,d^{3} e x +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c \,d^{2} e^{2} x^{3}-3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d^{5} x -3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d^{4} e \,x^{3}-2 a \,d^{2} e^{3}-3 a d \,e^{4} x^{2}+b \,d^{2} e^{3} x^{2}-c \,d^{3} e^{2} x^{2}+3 d^{5} e \,x^{2}+2 d^{4} e^{2} x^{4}}{2 d^{3} e^{3} x \left (e \,x^{2}+d \right )} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/x^2/(e*x^2+d)^2,x)
Output:
( - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e**3*x - 3*sqrt(e) *sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e**4*x**3 + sqrt(e)*sqrt(d)*atan( (e*x)/(sqrt(e)*sqrt(d)))*b*d**2*e**2*x + sqrt(e)*sqrt(d)*atan((e*x)/(sqrt( e)*sqrt(d)))*b*d*e**3*x**3 + sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d))) *c*d**3*e*x + sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*c*d**2*e**2*x* *3 - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d**5*x - 3*sqrt(e)*sq rt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d**4*e*x**3 - 2*a*d**2*e**3 - 3*a*d*e* *4*x**2 + b*d**2*e**3*x**2 - c*d**3*e**2*x**2 + 3*d**5*e*x**2 + 2*d**4*e** 2*x**4)/(2*d**3*e**3*x*(d + e*x**2))