Integrand size = 25, antiderivative size = 135 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\left (3 c d^2-e (2 b d-a e)\right ) x}{e^4}-\frac {(2 c d-b e) x^3}{3 e^3}+\frac {c x^5}{5 e^2}+\frac {d \left (c d^2-b d e+a e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac {\sqrt {d} \left (7 c d^2-e (5 b d-3 a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{9/2}} \]
(3*c*d^2-e*(-a*e+2*b*d))*x/e^4-1/3*(-b*e+2*c*d)*x^3/e^3+1/5*c*x^5/e^2+1/2* d*(a*e^2-b*d*e+c*d^2)*x/e^4/(e*x^2+d)-1/2*(7*c*d^2-e*(-3*a*e+5*b*d))*arcta n(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(9/2)
Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\left (3 c d^2-2 b d e+a e^2\right ) x}{e^4}+\frac {(-2 c d+b e) x^3}{3 e^3}+\frac {c x^5}{5 e^2}+\frac {\left (c d^3-b d^2 e+a d e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac {\sqrt {d} \left (7 c d^2-5 b d e+3 a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{9/2}} \]
((3*c*d^2 - 2*b*d*e + a*e^2)*x)/e^4 + ((-2*c*d + b*e)*x^3)/(3*e^3) + (c*x^ 5)/(5*e^2) + ((c*d^3 - b*d^2*e + a*d*e^2)*x)/(2*e^4*(d + e*x^2)) - (Sqrt[d ]*(7*c*d^2 - 5*b*d*e + 3*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(9/2))
Time = 0.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1580, 2341, 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.
\(\displaystyle \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1580 |
\(\displaystyle \frac {d x \left (a e^2-b d e+c d^2\right )}{2 e^4 \left (d+e x^2\right )}-\frac {\int \frac {-2 c e^3 x^6+2 e^2 (c d-b e) x^4-2 e \left (c d^2-b e d+a e^2\right ) x^2+d \left (c d^2-b e d+a e^2\right )}{e x^2+d}dx}{2 e^4}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {d x \left (a e^2-b d e+c d^2\right )}{2 e^4 \left (d+e x^2\right )}-\frac {\int \left (-2 c e^2 x^4+2 e (2 c d-b e) x^2-2 \left (3 c d^2-2 b e d+a e^2\right )+\frac {7 c d^3-5 b e d^2+3 a e^2 d}{e x^2+d}\right )dx}{2 e^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d x \left (a e^2-b d e+c d^2\right )}{2 e^4 \left (d+e x^2\right )}-\frac {\frac {\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (7 c d^2-e (5 b d-3 a e)\right )}{\sqrt {e}}-2 x \left (3 c d^2-e (2 b d-a e)\right )+\frac {2}{3} e x^3 (2 c d-b e)-\frac {2}{5} c e^2 x^5}{2 e^4}\) |
(d*(c*d^2 - b*d*e + a*e^2)*x)/(2*e^4*(d + e*x^2)) - (-2*(3*c*d^2 - e*(2*b* d - a*e))*x + (2*e*(2*c*d - b*e)*x^3)/3 - (2*c*e^2*x^5)/5 + (Sqrt[d]*(7*c* d^2 - e*(5*b*d - 3*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e])/(2*e^4)
3.3.81.3.1 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[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\frac {1}{5} c \,x^{5} e^{2}+\frac {1}{3} b \,e^{2} x^{3}-\frac {2}{3} d c \,x^{3} e +a \,e^{2} x -2 b d e x +3 c \,d^{2} x}{e^{4}}-\frac {d \left (\frac {\left (-\frac {1}{2} a \,e^{2}+\frac {1}{2} b d e -\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (3 a \,e^{2}-5 b d e +7 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 \sqrt {e d}}\right )}{e^{4}}\) | \(122\) |
risch | \(\frac {c \,x^{5}}{5 e^{2}}+\frac {b \,x^{3}}{3 e^{2}}-\frac {2 d c \,x^{3}}{3 e^{3}}+\frac {a x}{e^{2}}-\frac {2 b d x}{e^{3}}+\frac {3 c \,d^{2} x}{e^{4}}+\frac {\left (\frac {1}{2} d \,e^{2} a -\frac {1}{2} b \,d^{2} e +\frac {1}{2} d^{3} c \right ) x}{e^{4} \left (e \,x^{2}+d \right )}+\frac {3 \sqrt {-e d}\, \ln \left (-\sqrt {-e d}\, x -d \right ) a}{4 e^{3}}-\frac {5 \sqrt {-e d}\, \ln \left (-\sqrt {-e d}\, x -d \right ) b d}{4 e^{4}}+\frac {7 \sqrt {-e d}\, \ln \left (-\sqrt {-e d}\, x -d \right ) c \,d^{2}}{4 e^{5}}-\frac {3 \sqrt {-e d}\, \ln \left (\sqrt {-e d}\, x -d \right ) a}{4 e^{3}}+\frac {5 \sqrt {-e d}\, \ln \left (\sqrt {-e d}\, x -d \right ) b d}{4 e^{4}}-\frac {7 \sqrt {-e d}\, \ln \left (\sqrt {-e d}\, x -d \right ) c \,d^{2}}{4 e^{5}}\) | \(250\) |
1/e^4*(1/5*c*x^5*e^2+1/3*b*e^2*x^3-2/3*d*c*x^3*e+a*e^2*x-2*b*d*e*x+3*c*d^2 *x)-d/e^4*((-1/2*a*e^2+1/2*b*d*e-1/2*c*d^2)*x/(e*x^2+d)+1/2*(3*a*e^2-5*b*d *e+7*c*d^2)/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.59 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\left [\frac {12 \, c e^{3} x^{7} - 4 \, {\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 20 \, {\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} + 15 \, {\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} + {\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 30 \, {\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{60 \, {\left (e^{5} x^{2} + d e^{4}\right )}}, \frac {6 \, c e^{3} x^{7} - 2 \, {\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 10 \, {\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} - 15 \, {\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} + {\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 15 \, {\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{30 \, {\left (e^{5} x^{2} + d e^{4}\right )}}\right ] \]
[1/60*(12*c*e^3*x^7 - 4*(7*c*d*e^2 - 5*b*e^3)*x^5 + 20*(7*c*d^2*e - 5*b*d* e^2 + 3*a*e^3)*x^3 + 15*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2 + (7*c*d^2*e - 5* b*d*e^2 + 3*a*e^3)*x^2)*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x ^2 + d)) + 30*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2)*x)/(e^5*x^2 + d*e^4), 1/30 *(6*c*e^3*x^7 - 2*(7*c*d*e^2 - 5*b*e^3)*x^5 + 10*(7*c*d^2*e - 5*b*d*e^2 + 3*a*e^3)*x^3 - 15*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2 + (7*c*d^2*e - 5*b*d*e^ 2 + 3*a*e^3)*x^2)*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) + 15*(7*c*d^3 - 5*b*d^ 2*e + 3*a*d*e^2)*x)/(e^5*x^2 + d*e^4)]
Time = 0.62 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.40 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {c x^{5}}{5 e^{2}} + x^{3} \left (\frac {b}{3 e^{2}} - \frac {2 c d}{3 e^{3}}\right ) + x \left (\frac {a}{e^{2}} - \frac {2 b d}{e^{3}} + \frac {3 c d^{2}}{e^{4}}\right ) + \frac {x \left (a d e^{2} - b d^{2} e + c d^{3}\right )}{2 d e^{4} + 2 e^{5} x^{2}} + \frac {\sqrt {- \frac {d}{e^{9}}} \cdot \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log {\left (- e^{4} \sqrt {- \frac {d}{e^{9}}} + x \right )}}{4} - \frac {\sqrt {- \frac {d}{e^{9}}} \cdot \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log {\left (e^{4} \sqrt {- \frac {d}{e^{9}}} + x \right )}}{4} \]
c*x**5/(5*e**2) + x**3*(b/(3*e**2) - 2*c*d/(3*e**3)) + x*(a/e**2 - 2*b*d/e **3 + 3*c*d**2/e**4) + x*(a*d*e**2 - b*d**2*e + c*d**3)/(2*d*e**4 + 2*e**5 *x**2) + sqrt(-d/e**9)*(3*a*e**2 - 5*b*d*e + 7*c*d**2)*log(-e**4*sqrt(-d/e **9) + x)/4 - sqrt(-d/e**9)*(3*a*e**2 - 5*b*d*e + 7*c*d**2)*log(e**4*sqrt( -d/e**9) + x)/4
Exception generated. \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} e^{4}} + \frac {c d^{3} x - b d^{2} e x + a d e^{2} x}{2 \, {\left (e x^{2} + d\right )} e^{4}} + \frac {3 \, c e^{8} x^{5} - 10 \, c d e^{7} x^{3} + 5 \, b e^{8} x^{3} + 45 \, c d^{2} e^{6} x - 30 \, b d e^{7} x + 15 \, a e^{8} x}{15 \, e^{10}} \]
-1/2*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e^ 4) + 1/2*(c*d^3*x - b*d^2*e*x + a*d*e^2*x)/((e*x^2 + d)*e^4) + 1/15*(3*c*e ^8*x^5 - 10*c*d*e^7*x^3 + 5*b*e^8*x^3 + 45*c*d^2*e^6*x - 30*b*d*e^7*x + 15 *a*e^8*x)/e^10
Time = 7.68 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.33 \[ \int \frac {x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=x^3\,\left (\frac {b}{3\,e^2}-\frac {2\,c\,d}{3\,e^3}\right )-x\,\left (\frac {c\,d^2}{e^4}-\frac {a}{e^2}+\frac {2\,d\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{e}\right )+\frac {c\,x^5}{5\,e^2}+\frac {x\,\left (\frac {c\,d^3}{2}-\frac {b\,d^2\,e}{2}+\frac {a\,d\,e^2}{2}\right )}{e^5\,x^2+d\,e^4}-\frac {\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e}\,x\,\left (7\,c\,d^2-5\,b\,d\,e+3\,a\,e^2\right )}{7\,c\,d^3-5\,b\,d^2\,e+3\,a\,d\,e^2}\right )\,\left (7\,c\,d^2-5\,b\,d\,e+3\,a\,e^2\right )}{2\,e^{9/2}} \]
x^3*(b/(3*e^2) - (2*c*d)/(3*e^3)) - x*((c*d^2)/e^4 - a/e^2 + (2*d*(b/e^2 - (2*c*d)/e^3))/e) + (c*x^5)/(5*e^2) + (x*((c*d^3)/2 + (a*d*e^2)/2 - (b*d^2 *e)/2))/(d*e^4 + e^5*x^2) - (d^(1/2)*atan((d^(1/2)*e^(1/2)*x*(3*a*e^2 + 7* c*d^2 - 5*b*d*e))/(7*c*d^3 + 3*a*d*e^2 - 5*b*d^2*e))*(3*a*e^2 + 7*c*d^2 - 5*b*d*e))/(2*e^(9/2))