Integrand size = 165, antiderivative size = 24 \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x \left (-25+e^2+e^{e^5}+x\right ) \left (x+x^2\right )} \end {dmath*}
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^2 (1+x) \left (-25+e^2+e^{e^5}+x\right )} \end {dmath*}
Integrate[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 + E^4*(x^3 + 2*x^4 + x^5) + E^(2*E^5) *(x^3 + 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(-5 0*x^3 - 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(24)=48\).
Time = 0.70 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.50, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 6, 2026, 2462, 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.
\(\displaystyle \int \frac {-4 x^2+72 x+e^{e^5} (-3 x-2)+e^2 (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+e^{2 e^5} \left (x^5+2 x^4+x^3\right )+e^4 \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+e^{2 e^5} \left (x^5+2 x^4+x^3\right )+e^4 \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^7-48 x^6+526 x^5+1200 x^4+625 x^3+\left (e^4+e^{2 e^5}\right ) \left (x^5+2 x^4+x^3\right )+e^2 \left (2 x^6-46 x^5-98 x^4-50 x^3\right )+e^{e^5} \left (2 x^6-46 x^5-98 x^4-50 x^3+e^2 \left (2 x^5+4 x^4+2 x^3\right )\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-4 x^2+72 x+\left (e^2+e^{e^5}\right ) (-3 x-2)+50}{x^3 \left (x^4-2 \left (24-e^2-e^{e^5}\right ) x^3+\left (526-46 e^2+e^4-46 e^{e^5}+e^{2 e^5}+2 e^{2+e^5}\right ) x^2+2 \left (24-e^2-e^{e^5}\right ) \left (25-e^2-e^{e^5}\right ) x+\left (-25+e^2+e^{e^5}\right )^2\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (-\frac {2}{\left (-25+e^2+e^{e^5}\right ) x^3}+\frac {-24+e^2+e^{e^5}}{\left (-25+e^2+e^{e^5}\right )^2 x^2}-\frac {1}{\left (-26+e^2+e^{e^5}\right ) (x+1)^2}+\frac {1}{\left (-26+e^2+e^{e^5}\right ) \left (-25+e^2+e^{e^5}\right )^2 \left (x+e^{e^5}+e^2-25\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{\left (25-e^2-e^{e^5}\right ) x^2}+\frac {24-e^2-e^{e^5}}{\left (25-e^2-e^{e^5}\right )^2 x}-\frac {1}{\left (26-e^2-e^{e^5}\right ) (x+1)}-\frac {1}{\left (25-e^2-e^{e^5}\right )^2 \left (26-e^2-e^{e^5}\right ) \left (-x-e^{e^5}-e^2+25\right )}\) |
Int[(50 + E^2*(-2 - 3*x) + E^E^5*(-2 - 3*x) + 72*x - 4*x^2)/(625*x^3 + 120 0*x^4 + 526*x^5 - 48*x^6 + x^7 + E^4*(x^3 + 2*x^4 + x^5) + E^(2*E^5)*(x^3 + 2*x^4 + x^5) + E^2*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6) + E^E^5*(-50*x^3 - 98*x^4 - 46*x^5 + 2*x^6 + E^2*(2*x^3 + 4*x^4 + 2*x^5))),x]
-(1/((25 - E^2 - E^E^5)^2*(26 - E^2 - E^E^5)*(25 - E^2 - E^E^5 - x))) - 1/ ((25 - E^2 - E^E^5)*x^2) + (24 - E^2 - E^E^5)/((25 - E^2 - E^E^5)^2*x) - 1 /((26 - E^2 - E^E^5)*(1 + x))
3.3.44.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 4.64 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
norman | \(\frac {1}{x^{2} \left (1+x \right ) \left ({\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}+x -25\right )}\) | \(20\) |
gosper | \(\frac {1}{x^{2} \left ({\mathrm e}^{2} x +x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+{\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}-24 x -25\right )}\) | \(29\) |
risch | \(\frac {1}{x^{2} \left ({\mathrm e}^{2} x +x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+{\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}-24 x -25\right )}\) | \(29\) |
parallelrisch | \(\frac {1}{x^{2} \left ({\mathrm e}^{2} x +x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+{\mathrm e}^{2}+{\mathrm e}^{{\mathrm e}^{5}}-24 x -25\right )}\) | \(29\) |
int(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4+x^3)* exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3)*exp( exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2)+x^7-4 8*x^6+526*x^5+1200*x^4+625*x^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} - 24 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + x^{2}\right )} e^{2} + {\left (x^{3} + x^{2}\right )} e^{\left (e^{5}\right )}} \end {dmath*}
integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 +x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 )*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) +x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm=\
Time = 3.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} + x^{3} \left (-24 + e^{2} + e^{e^{5}}\right ) + x^{2} \left (-25 + e^{2} + e^{e^{5}}\right )} \end {dmath*}
integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x**2+72*x+50)/((x**5+2*x **4+x**3)*exp(exp(5))**2+((2*x**5+4*x**4+2*x**3)*exp(2)+2*x**6-46*x**5-98* x**4-50*x**3)*exp(exp(5))+(x**5+2*x**4+x**3)*exp(2)**2+(2*x**6-46*x**5-98* x**4-50*x**3)*exp(2)+x**7-48*x**6+526*x**5+1200*x**4+625*x**3),x)
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\frac {1}{x^{4} + x^{3} {\left (e^{2} + e^{\left (e^{5}\right )} - 24\right )} + x^{2} {\left (e^{2} + e^{\left (e^{5}\right )} - 25\right )}} \end {dmath*}
integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 +x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 )*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) +x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm=\
\begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\int { -\frac {4 \, x^{2} + {\left (3 \, x + 2\right )} e^{2} + {\left (3 \, x + 2\right )} e^{\left (e^{5}\right )} - 72 \, x - 50}{x^{7} - 48 \, x^{6} + 526 \, x^{5} + 1200 \, x^{4} + 625 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{4} + 2 \, {\left (x^{6} - 23 \, x^{5} - 49 \, x^{4} - 25 \, x^{3}\right )} e^{2} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{\left (2 \, e^{5}\right )} + 2 \, {\left (x^{6} - 23 \, x^{5} - 49 \, x^{4} - 25 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} e^{2}\right )} e^{\left (e^{5}\right )}} \,d x } \end {dmath*}
integrate(((-2-3*x)*exp(exp(5))+(-2-3*x)*exp(2)-4*x^2+72*x+50)/((x^5+2*x^4 +x^3)*exp(exp(5))^2+((2*x^5+4*x^4+2*x^3)*exp(2)+2*x^6-46*x^5-98*x^4-50*x^3 )*exp(exp(5))+(x^5+2*x^4+x^3)*exp(2)^2+(2*x^6-46*x^5-98*x^4-50*x^3)*exp(2) +x^7-48*x^6+526*x^5+1200*x^4+625*x^3),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {50+e^2 (-2-3 x)+e^{e^5} (-2-3 x)+72 x-4 x^2}{625 x^3+1200 x^4+526 x^5-48 x^6+x^7+e^4 \left (x^3+2 x^4+x^5\right )+e^{2 e^5} \left (x^3+2 x^4+x^5\right )+e^2 \left (-50 x^3-98 x^4-46 x^5+2 x^6\right )+e^{e^5} \left (-50 x^3-98 x^4-46 x^5+2 x^6+e^2 \left (2 x^3+4 x^4+2 x^5\right )\right )} \, dx=\text {Hanged} \end {dmath*}
int(-(exp(exp(5))*(3*x + 2) - 72*x + 4*x^2 + exp(2)*(3*x + 2) - 50)/(exp(4 )*(x^3 + 2*x^4 + x^5) - exp(exp(5))*(50*x^3 - exp(2)*(2*x^3 + 4*x^4 + 2*x^ 5) + 98*x^4 + 46*x^5 - 2*x^6) + exp(2*exp(5))*(x^3 + 2*x^4 + x^5) + 625*x^ 3 + 1200*x^4 + 526*x^5 - 48*x^6 + x^7 - exp(2)*(50*x^3 + 98*x^4 + 46*x^5 - 2*x^6)),x)