Integrand size = 25, antiderivative size = 168 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac {(2 c d-b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {d^{3/2} \left (9 c d^2-e (7 b d-5 a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{11/2}} \]
-d*(4*c*d^2-e*(-2*a*e+3*b*d))*x/e^5+1/3*(3*c*d^2-e*(-a*e+2*b*d))*x^3/e^4-1 /5*(-b*e+2*c*d)*x^5/e^3+1/7*c*x^7/e^2-1/2*d^2*(a*e^2-b*d*e+c*d^2)*x/e^5/(e *x^2+d)+1/2*d^(3/2)*(9*c*d^2-e*(-5*a*e+7*b*d))*arctan(x*e^(1/2)/d^(1/2))/e ^(11/2)
Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {d \left (4 c d^2-3 b d e+2 a e^2\right ) x}{e^5}+\frac {\left (3 c d^2-2 b d e+a e^2\right ) x^3}{3 e^4}+\frac {(-2 c d+b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {\left (c d^4-b d^3 e+a d^2 e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {d^{3/2} \left (9 c d^2-7 b d e+5 a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{11/2}} \]
-((d*(4*c*d^2 - 3*b*d*e + 2*a*e^2)*x)/e^5) + ((3*c*d^2 - 2*b*d*e + a*e^2)* x^3)/(3*e^4) + ((-2*c*d + b*e)*x^5)/(5*e^3) + (c*x^7)/(7*e^2) - ((c*d^4 - b*d^3*e + a*d^2*e^2)*x)/(2*e^5*(d + e*x^2)) + (d^(3/2)*(9*c*d^2 - 7*b*d*e + 5*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(11/2))
Time = 0.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1580, 25, 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.
\(\displaystyle \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1580 |
\(\displaystyle -\frac {\int -\frac {2 c e^4 x^8-2 e^3 (c d-b e) x^6+2 e^2 \left (c d^2-b e d+a e^2\right ) x^4-2 d e \left (c d^2-b e d+a e^2\right ) x^2+d^2 \left (c d^2-b e d+a e^2\right )}{e x^2+d}dx}{2 e^5}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 c e^4 x^8-2 e^3 (c d-b e) x^6+2 e^2 \left (c d^2-b e d+a e^2\right ) x^4-2 d e \left (c d^2-b e d+a e^2\right ) x^2+d^2 \left (c d^2-b e d+a e^2\right )}{e x^2+d}dx}{2 e^5}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {\int \left (2 c e^3 x^6-2 e^2 (2 c d-b e) x^4+2 e \left (3 c d^2-e (2 b d-a e)\right ) x^2-2 d \left (4 c d^2-e (3 b d-2 a e)\right )+\frac {9 c d^4-7 b e d^3+5 a e^2 d^2}{e x^2+d}\right )dx}{2 e^5}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{\sqrt {e}}+\frac {2}{3} e x^3 \left (3 c d^2-e (2 b d-a e)\right )-2 d x \left (4 c d^2-e (3 b d-2 a e)\right )-\frac {2}{5} e^2 x^5 (2 c d-b e)+\frac {2}{7} c e^3 x^7}{2 e^5}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}\) |
-1/2*(d^2*(c*d^2 - b*d*e + a*e^2)*x)/(e^5*(d + e*x^2)) + (-2*d*(4*c*d^2 - e*(3*b*d - 2*a*e))*x + (2*e*(3*c*d^2 - e*(2*b*d - a*e))*x^3)/3 - (2*e^2*(2 *c*d - b*e)*x^5)/5 + (2*c*e^3*x^7)/7 + (d^(3/2)*(9*c*d^2 - e*(7*b*d - 5*a* e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e])/(2*e^5)
3.3.80.3.1 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 = 159, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {-\frac {1}{7} c \,x^{7} e^{3}-\frac {1}{5} b \,e^{3} x^{5}+\frac {2}{5} c d \,x^{5} e^{2}-\frac {1}{3} a \,e^{3} x^{3}+\frac {2}{3} b d \,e^{2} x^{3}-c \,d^{2} e \,x^{3}+2 d \,e^{2} a x -3 b \,d^{2} e x +4 d^{3} c x}{e^{5}}+\frac {d^{2} \left (\frac {\left (-\frac {1}{2} a \,e^{2}+\frac {1}{2} b d e -\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (5 a \,e^{2}-7 b d e +9 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 \sqrt {e d}}\right )}{e^{5}}\) | \(159\) |
risch | \(\frac {c \,x^{7}}{7 e^{2}}+\frac {b \,x^{5}}{5 e^{2}}-\frac {2 c d \,x^{5}}{5 e^{3}}+\frac {a \,x^{3}}{3 e^{2}}-\frac {2 b d \,x^{3}}{3 e^{3}}+\frac {c \,d^{2} x^{3}}{e^{4}}-\frac {2 d a x}{e^{3}}+\frac {3 b \,d^{2} x}{e^{4}}-\frac {4 d^{3} c x}{e^{5}}+\frac {\left (-\frac {1}{2} e^{2} d^{2} a +\frac {1}{2} d^{3} e b -\frac {1}{2} d^{4} c \right ) x}{e^{5} \left (e \,x^{2}+d \right )}+\frac {5 \sqrt {-e d}\, d \ln \left (-\sqrt {-e d}\, x +d \right ) a}{4 e^{4}}-\frac {7 \sqrt {-e d}\, d^{2} \ln \left (-\sqrt {-e d}\, x +d \right ) b}{4 e^{5}}+\frac {9 \sqrt {-e d}\, d^{3} \ln \left (-\sqrt {-e d}\, x +d \right ) c}{4 e^{6}}-\frac {5 \sqrt {-e d}\, d \ln \left (\sqrt {-e d}\, x +d \right ) a}{4 e^{4}}+\frac {7 \sqrt {-e d}\, d^{2} \ln \left (\sqrt {-e d}\, x +d \right ) b}{4 e^{5}}-\frac {9 \sqrt {-e d}\, d^{3} \ln \left (\sqrt {-e d}\, x +d \right ) c}{4 e^{6}}\) | \(280\) |
-1/e^5*(-1/7*c*x^7*e^3-1/5*b*e^3*x^5+2/5*c*d*x^5*e^2-1/3*a*e^3*x^3+2/3*b*d *e^2*x^3-c*d^2*e*x^3+2*d*e^2*a*x-3*b*d^2*e*x+4*d^3*c*x)+d^2/e^5*((-1/2*a*e ^2+1/2*b*d*e-1/2*c*d^2)*x/(e*x^2+d)+1/2*(5*a*e^2-7*b*d*e+9*c*d^2)/(e*d)^(1 /2)*arctan(e*x/(e*d)^(1/2)))
Time = 0.33 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.54 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\left [\frac {60 \, c e^{4} x^{9} - 12 \, {\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 28 \, {\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 140 \, {\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \, {\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} + {\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) - 210 \, {\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{420 \, {\left (e^{6} x^{2} + d e^{5}\right )}}, \frac {30 \, c e^{4} x^{9} - 6 \, {\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 14 \, {\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 70 \, {\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \, {\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} + {\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) - 105 \, {\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{210 \, {\left (e^{6} x^{2} + d e^{5}\right )}}\right ] \]
[1/420*(60*c*e^4*x^9 - 12*(9*c*d*e^3 - 7*b*e^4)*x^7 + 28*(9*c*d^2*e^2 - 7* b*d*e^3 + 5*a*e^4)*x^5 - 140*(9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e^3)*x^3 + 1 05*(9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2 + (9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e ^3)*x^2)*sqrt(-d/e)*log((e*x^2 + 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) - 210* (9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2)*x)/(e^6*x^2 + d*e^5), 1/210*(30*c*e^4* x^9 - 6*(9*c*d*e^3 - 7*b*e^4)*x^7 + 14*(9*c*d^2*e^2 - 7*b*d*e^3 + 5*a*e^4) *x^5 - 70*(9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e^3)*x^3 + 105*(9*c*d^4 - 7*b*d ^3*e + 5*a*d^2*e^2 + (9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e^3)*x^2)*sqrt(d/e)* arctan(e*x*sqrt(d/e)/d) - 105*(9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2)*x)/(e^6* x^2 + d*e^5)]
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (153) = 306\).
Time = 0.65 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.90 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {c x^{7}}{7 e^{2}} + x^{5} \left (\frac {b}{5 e^{2}} - \frac {2 c d}{5 e^{3}}\right ) + x^{3} \left (\frac {a}{3 e^{2}} - \frac {2 b d}{3 e^{3}} + \frac {c d^{2}}{e^{4}}\right ) + x \left (- \frac {2 a d}{e^{3}} + \frac {3 b d^{2}}{e^{4}} - \frac {4 c d^{3}}{e^{5}}\right ) + \frac {x \left (- a d^{2} e^{2} + b d^{3} e - c d^{4}\right )}{2 d e^{5} + 2 e^{6} x^{2}} - \frac {\sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log {\left (- \frac {e^{5} \sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log {\left (\frac {e^{5} \sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} \]
c*x**7/(7*e**2) + x**5*(b/(5*e**2) - 2*c*d/(5*e**3)) + x**3*(a/(3*e**2) - 2*b*d/(3*e**3) + c*d**2/e**4) + x*(-2*a*d/e**3 + 3*b*d**2/e**4 - 4*c*d**3/ e**5) + x*(-a*d**2*e**2 + b*d**3*e - c*d**4)/(2*d*e**5 + 2*e**6*x**2) - sq rt(-d**3/e**11)*(5*a*e**2 - 7*b*d*e + 9*c*d**2)*log(-e**5*sqrt(-d**3/e**11 )*(5*a*e**2 - 7*b*d*e + 9*c*d**2)/(5*a*d*e**2 - 7*b*d**2*e + 9*c*d**3) + x )/4 + sqrt(-d**3/e**11)*(5*a*e**2 - 7*b*d*e + 9*c*d**2)*log(e**5*sqrt(-d** 3/e**11)*(5*a*e**2 - 7*b*d*e + 9*c*d**2)/(5*a*d*e**2 - 7*b*d**2*e + 9*c*d* *3) + x)/4
Exception generated. \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.42 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.04 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} e^{5}} - \frac {c d^{4} x - b d^{3} e x + a d^{2} e^{2} x}{2 \, {\left (e x^{2} + d\right )} e^{5}} + \frac {15 \, c e^{12} x^{7} - 42 \, c d e^{11} x^{5} + 21 \, b e^{12} x^{5} + 105 \, c d^{2} e^{10} x^{3} - 70 \, b d e^{11} x^{3} + 35 \, a e^{12} x^{3} - 420 \, c d^{3} e^{9} x + 315 \, b d^{2} e^{10} x - 210 \, a d e^{11} x}{105 \, e^{14}} \]
1/2*(9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e ^5) - 1/2*(c*d^4*x - b*d^3*e*x + a*d^2*e^2*x)/((e*x^2 + d)*e^5) + 1/105*(1 5*c*e^12*x^7 - 42*c*d*e^11*x^5 + 21*b*e^12*x^5 + 105*c*d^2*e^10*x^3 - 70*b *d*e^11*x^3 + 35*a*e^12*x^3 - 420*c*d^3*e^9*x + 315*b*d^2*e^10*x - 210*a*d *e^11*x)/e^14
Time = 7.70 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.49 \[ \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx=x^5\,\left (\frac {b}{5\,e^2}-\frac {2\,c\,d}{5\,e^3}\right )-x^3\,\left (\frac {c\,d^2}{3\,e^4}-\frac {a}{3\,e^2}+\frac {2\,d\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{3\,e}\right )+x\,\left (\frac {2\,d\,\left (\frac {c\,d^2}{e^4}-\frac {a}{e^2}+\frac {2\,d\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{e^2}\right )-\frac {x\,\left (\frac {c\,d^4}{2}-\frac {b\,d^3\,e}{2}+\frac {a\,d^2\,e^2}{2}\right )}{e^6\,x^2+d\,e^5}+\frac {c\,x^7}{7\,e^2}+\frac {d^{3/2}\,\mathrm {atan}\left (\frac {d^{3/2}\,\sqrt {e}\,x\,\left (9\,c\,d^2-7\,b\,d\,e+5\,a\,e^2\right )}{9\,c\,d^4-7\,b\,d^3\,e+5\,a\,d^2\,e^2}\right )\,\left (9\,c\,d^2-7\,b\,d\,e+5\,a\,e^2\right )}{2\,e^{11/2}} \]
x^5*(b/(5*e^2) - (2*c*d)/(5*e^3)) - x^3*((c*d^2)/(3*e^4) - a/(3*e^2) + (2* d*(b/e^2 - (2*c*d)/e^3))/(3*e)) + x*((2*d*((c*d^2)/e^4 - a/e^2 + (2*d*(b/e ^2 - (2*c*d)/e^3))/e))/e - (d^2*(b/e^2 - (2*c*d)/e^3))/e^2) - (x*((c*d^4)/ 2 + (a*d^2*e^2)/2 - (b*d^3*e)/2))/(d*e^5 + e^6*x^2) + (c*x^7)/(7*e^2) + (d ^(3/2)*atan((d^(3/2)*e^(1/2)*x*(5*a*e^2 + 9*c*d^2 - 7*b*d*e))/(9*c*d^4 + 5 *a*d^2*e^2 - 7*b*d^3*e))*(5*a*e^2 + 9*c*d^2 - 7*b*d*e))/(2*e^(11/2))