Integrand size = 25, antiderivative size = 136 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {a}{5 d^2 x^5}-\frac {b d-2 a e}{3 d^3 x^3}-\frac {c d^2-e (2 b d-3 a e)}{d^4 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (3 c d^2-e (5 b d-7 a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{9/2}} \]
-1/5*a/d^2/x^5+1/3*(2*a*e-b*d)/d^3/x^3+(-c*d^2+e*(-3*a*e+2*b*d))/d^4/x-1/2 *e*(a*e^2-b*d*e+c*d^2)*x/d^4/(e*x^2+d)-1/2*(3*c*d^2-e*(-7*a*e+5*b*d))*arct an(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(9/2)
Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {a}{5 d^2 x^5}+\frac {-b d+2 a e}{3 d^3 x^3}+\frac {-c d^2+2 b d e-3 a e^2}{d^4 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (3 c d^2-5 b d e+7 a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{9/2}} \]
-1/5*a/(d^2*x^5) + (-(b*d) + 2*a*e)/(3*d^3*x^3) + (-(c*d^2) + 2*b*d*e - 3* a*e^2)/(d^4*x) - (e*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^4*(d + e*x^2)) - (Sqrt [e]*(3*c*d^2 - 5*b*d*e + 7*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(9/2))
Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1582, 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.
\(\displaystyle \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1582 |
\(\displaystyle -\frac {\int -\frac {-e^3 \left (c d^2-b e d+a e^2\right ) x^6+2 d e^2 \left (c d^2-b e d+a e^2\right ) x^4+2 d^2 e^2 (b d-a e) x^2+2 a d^3 e^2}{x^6 \left (e x^2+d\right )}dx}{2 d^4 e^2}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {-e^3 \left (c d^2-b e d+a e^2\right ) x^6+2 d e^2 \left (c d^2-b e d+a e^2\right ) x^4+2 d^2 e^2 (b d-a e) x^2+2 a d^3 e^2}{x^6 \left (e x^2+d\right )}dx}{2 d^4 e^2}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {\int \left (\frac {\left (e (5 b d-7 a e)-3 c d^2\right ) e^3}{e x^2+d}+\frac {2 \left (c d^2-e (2 b d-3 a e)\right ) e^2}{x^2}+\frac {2 d (b d-2 a e) e^2}{x^4}+\frac {2 a d^2 e^2}{x^6}\right )dx}{2 d^4 e^2}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {e^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{\sqrt {d}}-\frac {2 e^2 \left (c d^2-e (2 b d-3 a e)\right )}{x}-\frac {2 d e^2 (b d-2 a e)}{3 x^3}-\frac {2 a d^2 e^2}{5 x^5}}{2 d^4 e^2}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}\) |
-1/2*(e*(c*d^2 - b*d*e + a*e^2)*x)/(d^4*(d + e*x^2)) + ((-2*a*d^2*e^2)/(5* x^5) - (2*d*e^2*(b*d - 2*a*e))/(3*x^3) - (2*e^2*(c*d^2 - e*(2*b*d - 3*a*e) ))/x - (e^(5/2)*(3*c*d^2 - e*(5*b*d - 7*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]]) /Sqrt[d])/(2*d^4*e^2)
3.3.86.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
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 = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a}{5 d^{2} x^{5}}-\frac {-2 a e +b d}{3 d^{3} x^{3}}-\frac {3 a \,e^{2}-2 b d e +c \,d^{2}}{d^{4} x}-\frac {e \left (\frac {\left (\frac {1}{2} a \,e^{2}-\frac {1}{2} b d e +\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (7 a \,e^{2}-5 b d e +3 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 \sqrt {e d}}\right )}{d^{4}}\) | \(122\) |
risch | \(\frac {-\frac {e \left (7 a \,e^{2}-5 b d e +3 c \,d^{2}\right ) x^{6}}{2 d^{4}}-\frac {\left (7 a \,e^{2}-5 b d e +3 c \,d^{2}\right ) x^{4}}{3 d^{3}}+\frac {\left (7 a e -5 b d \right ) x^{2}}{15 d^{2}}-\frac {a}{5 d}}{x^{5} \left (e \,x^{2}+d \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{9} \textit {\_Z}^{2}+49 a^{2} e^{5}-70 a b d \,e^{4}+42 a c \,d^{2} e^{3}+25 b^{2} d^{2} e^{3}-30 b c \,d^{3} e^{2}+9 c^{2} d^{4} e \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} d^{9}+98 a^{2} e^{5}-140 a b d \,e^{4}+84 a c \,d^{2} e^{3}+50 b^{2} d^{2} e^{3}-60 b c \,d^{3} e^{2}+18 c^{2} d^{4} e \right ) x +\left (7 a \,d^{5} e^{2}-5 b \,d^{6} e +3 c \,d^{7}\right ) \textit {\_R} \right )\right )}{4}\) | \(258\) |
-1/5*a/d^2/x^5-1/3*(-2*a*e+b*d)/d^3/x^3-(3*a*e^2-2*b*d*e+c*d^2)/d^4/x-e/d^ 4*((1/2*a*e^2-1/2*b*d*e+1/2*c*d^2)*x/(e*x^2+d)+1/2*(7*a*e^2-5*b*d*e+3*c*d^ 2)/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.65 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=\left [-\frac {30 \, {\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 20 \, {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 12 \, a d^{3} + 4 \, {\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} - 15 \, {\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} + {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} - 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right )}{60 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac {15 \, {\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 10 \, {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 6 \, a d^{3} + 2 \, {\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} + 15 \, {\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} + {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right )}{30 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \]
[-1/60*(30*(3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*x^6 + 20*(3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^4 + 12*a*d^3 + 4*(5*b*d^3 - 7*a*d^2*e)*x^2 - 15*((3*c*d^2* e - 5*b*d*e^2 + 7*a*e^3)*x^7 + (3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^5)*sqrt (-e/d)*log((e*x^2 - 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)))/(d^4*e*x^7 + d^5*x ^5), -1/30*(15*(3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*x^6 + 10*(3*c*d^3 - 5*b*d ^2*e + 7*a*d*e^2)*x^4 + 6*a*d^3 + 2*(5*b*d^3 - 7*a*d^2*e)*x^2 + 15*((3*c*d ^2*e - 5*b*d*e^2 + 7*a*e^3)*x^7 + (3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^5)*s qrt(e/d)*arctan(x*sqrt(e/d)))/(d^4*e*x^7 + d^5*x^5)]
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (126) = 252\).
Time = 1.05 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.09 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=\frac {\sqrt {- \frac {e}{d^{9}}} \cdot \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log {\left (- \frac {d^{5} \sqrt {- \frac {e}{d^{9}}} \cdot \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac {\sqrt {- \frac {e}{d^{9}}} \cdot \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log {\left (\frac {d^{5} \sqrt {- \frac {e}{d^{9}}} \cdot \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} + \frac {- 6 a d^{3} + x^{6} \left (- 105 a e^{3} + 75 b d e^{2} - 45 c d^{2} e\right ) + x^{4} \left (- 70 a d e^{2} + 50 b d^{2} e - 30 c d^{3}\right ) + x^{2} \cdot \left (14 a d^{2} e - 10 b d^{3}\right )}{30 d^{5} x^{5} + 30 d^{4} e x^{7}} \]
sqrt(-e/d**9)*(7*a*e**2 - 5*b*d*e + 3*c*d**2)*log(-d**5*sqrt(-e/d**9)*(7*a *e**2 - 5*b*d*e + 3*c*d**2)/(7*a*e**3 - 5*b*d*e**2 + 3*c*d**2*e) + x)/4 - sqrt(-e/d**9)*(7*a*e**2 - 5*b*d*e + 3*c*d**2)*log(d**5*sqrt(-e/d**9)*(7*a* e**2 - 5*b*d*e + 3*c*d**2)/(7*a*e**3 - 5*b*d*e**2 + 3*c*d**2*e) + x)/4 + ( -6*a*d**3 + x**6*(-105*a*e**3 + 75*b*d*e**2 - 45*c*d**2*e) + x**4*(-70*a*d *e**2 + 50*b*d**2*e - 30*c*d**3) + x**2*(14*a*d**2*e - 10*b*d**3))/(30*d** 5*x**5 + 30*d**4*e*x**7)
Exception generated. \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{4}} - \frac {c d^{2} e x - b d e^{2} x + a e^{3} x}{2 \, {\left (e x^{2} + d\right )} d^{4}} - \frac {15 \, c d^{2} x^{4} - 30 \, b d e x^{4} + 45 \, a e^{2} x^{4} + 5 \, b d^{2} x^{2} - 10 \, a d e x^{2} + 3 \, a d^{2}}{15 \, d^{4} x^{5}} \]
-1/2*(3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^ 4) - 1/2*(c*d^2*e*x - b*d*e^2*x + a*e^3*x)/((e*x^2 + d)*d^4) - 1/15*(15*c* d^2*x^4 - 30*b*d*e*x^4 + 45*a*e^2*x^4 + 5*b*d^2*x^2 - 10*a*d*e*x^2 + 3*a*d ^2)/(d^4*x^5)
Time = 7.70 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {\frac {a}{5\,d}-\frac {x^2\,\left (7\,a\,e-5\,b\,d\right )}{15\,d^2}+\frac {x^4\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{3\,d^3}+\frac {e\,x^6\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{2\,d^4}}{e\,x^7+d\,x^5}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{2\,d^{9/2}} \]