Integrand size = 200, antiderivative size = 36 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\frac {e^x}{x}-\frac {4-x}{-x+\left (5 \sqrt [5]{e}-x\right )^2 x} \] Output:
exp(x)/x-(4-x)/(x*(5*exp(1/5)-x)^2-x)
Time = 10.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\frac {-4+25 e^{\frac {2}{5}+x}+x-10 e^{\frac {1}{5}+x} x+e^x \left (-1+x^2\right )}{x \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )} \] Input:
Integrate[(-4 + 100*E^(2/5) + 12*x^2 - 2*x^3 + E^(1/5)*(-80*x + 10*x^2) + E^x*(-1 + x + 2*x^2 - 2*x^3 - x^4 + x^5 + E^(4/5)*(-625 + 625*x) + E^(3/5) *(500*x - 500*x^2) + E^(2/5)*(50 - 50*x - 150*x^2 + 150*x^3) + E^(1/5)*(-2 0*x + 20*x^2 + 20*x^3 - 20*x^4)))/(x^2 + 625*E^(4/5)*x^2 - 500*E^(3/5)*x^3 - 2*x^4 + x^6 + E^(2/5)*(-50*x^2 + 150*x^4) + E^(1/5)*(20*x^3 - 20*x^5)), x]
Output:
(-4 + 25*E^(2/5 + x) + x - 10*E^(1/5 + x)*x + E^x*(-1 + x^2))/(x*(-1 + 25* E^(2/5) - 10*E^(1/5)*x + x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 9.62 (sec) , antiderivative size = 1104, normalized size of antiderivative = 30.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6, 2026, 2463, 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.
| \(\displaystyle \int \frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{x^6-2 x^4-500 e^{3/5} x^3+625 e^{4/5} x^2+x^2+\sqrt [5]{e} \left (20 x^3-20 x^5\right )+e^{2/5} \left (150 x^4-50 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{x^6-2 x^4-500 e^{3/5} x^3+\left (1+625 e^{4/5}\right ) x^2+\sqrt [5]{e} \left (20 x^3-20 x^5\right )+e^{2/5} \left (150 x^4-50 x^2\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{x^2 \left (x^4-20 \sqrt [5]{e} x^3-2 \left (1-75 e^{2/5}\right ) x^2+20 \left (1-25 e^{2/5}\right ) \sqrt [5]{e} x+\left (1-25 e^{2/5}\right )^2\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{4 \left (-x+5 \sqrt [5]{e}-1\right ) x^2}+\frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{4 \left (-x+5 \sqrt [5]{e}+1\right ) x^2}+\frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{4 \left (-x+5 \sqrt [5]{e}-1\right )^2 x^2}+\frac {-2 x^3+12 x^2+\sqrt [5]{e} \left (10 x^2-80 x\right )+e^x \left (x^5-x^4-2 x^3+2 x^2+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (150 x^3-150 x^2-50 x+50\right )+\sqrt [5]{e} \left (-20 x^4+20 x^3+20 x^2-20 x\right )+x+e^{4/5} (625 x-625)-1\right )+100 e^{2/5}-4}{4 \left (-x+5 \sqrt [5]{e}+1\right )^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} \left (2+15 \sqrt [5]{e}\right ) e^x x+\frac {e^x x}{2}+\frac {15}{4} e^{x+\frac {1}{5}} x+\frac {1}{4} \left (2-5 \sqrt [5]{e}-75 e^{2/5}\right ) e^x+\frac {1}{4} \left (2+15 \sqrt [5]{e}\right ) e^x-\frac {1}{4} \left (3+10 \sqrt [5]{e}\right ) e^x+\frac {1}{4} \left (1-10 \sqrt [5]{e}\right ) e^x-\frac {e^x}{2}+\frac {25}{4} \left (1+3 \sqrt [5]{e}\right ) e^{x+\frac {1}{5}}-\frac {15}{4} e^{x+\frac {1}{5}}-\frac {5}{4} \left (3-10 \sqrt [5]{e}-25 e^{2/5}\right ) \sqrt [5]{e} \operatorname {ExpIntegralEi}(x)+\frac {1}{4} \left (2-5 \sqrt [5]{e}-100 e^{2/5}-125 e^{3/5}\right ) \operatorname {ExpIntegralEi}(x)+\frac {1}{4} \left (3+20 \sqrt [5]{e}+25 e^{2/5}\right ) \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (1-25 e^{2/5}\right ) \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (1-5 \sqrt [5]{e}\right ) \left (1+5 \sqrt [5]{e}\right )^2 \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (1+5 \sqrt [5]{e}\right )^2 \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (1-5 \sqrt [5]{e}\right )^2 \left (1+5 \sqrt [5]{e}\right ) \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (1-5 \sqrt [5]{e}\right )^2 \operatorname {ExpIntegralEi}(x)+\frac {20 \sqrt [5]{e} \log \left (-x+5 \sqrt [5]{e}+1\right )}{\left (1+5 \sqrt [5]{e}\right )^2}-\frac {\left (3-5 \sqrt [5]{e}\right ) \log \left (-x+5 \sqrt [5]{e}+1\right )}{2 \left (1+5 \sqrt [5]{e}\right )}+\frac {2 \left (1-5 \sqrt [5]{e}\right ) \log \left (-x+5 \sqrt [5]{e}+1\right )}{\left (1+5 \sqrt [5]{e}\right )^2}-\frac {1}{2} \log \left (-x+5 \sqrt [5]{e}+1\right )-\frac {20 \sqrt [5]{e} \log (x)}{\left (1+5 \sqrt [5]{e}\right )^2}-\frac {20 \sqrt [5]{e} \log (x)}{\left (1-5 \sqrt [5]{e}\right )^2}-\frac {\left (1+15 \sqrt [5]{e}\right ) \log (x)}{1+5 \sqrt [5]{e}}+\frac {2 \left (1+5 \sqrt [5]{e}\right ) \log (x)}{\left (1-5 \sqrt [5]{e}\right )^2}-\frac {2 \left (1-5 \sqrt [5]{e}\right ) \log (x)}{\left (1+5 \sqrt [5]{e}\right )^2}+\frac {\left (1-15 \sqrt [5]{e}\right ) \log (x)}{1-5 \sqrt [5]{e}}+\frac {20 \sqrt [5]{e} \log \left (x-5 \sqrt [5]{e}+1\right )}{\left (1-5 \sqrt [5]{e}\right )^2}-\frac {2 \left (1+5 \sqrt [5]{e}\right ) \log \left (x-5 \sqrt [5]{e}+1\right )}{\left (1-5 \sqrt [5]{e}\right )^2}+\frac {5 \left (1-\sqrt [5]{e}\right ) \log \left (x-5 \sqrt [5]{e}+1\right )}{2 \left (1-5 \sqrt [5]{e}\right )}-\frac {1}{2} \log \left (x-5 \sqrt [5]{e}+1\right )-\frac {5 \left (7-5 \sqrt [5]{e}\right ) \sqrt [5]{e}}{2 \left (1+5 \sqrt [5]{e}\right ) \left (-x+5 \sqrt [5]{e}+1\right )}-\frac {1+5 \sqrt [5]{e}}{2 \left (-x+5 \sqrt [5]{e}+1\right )}-\frac {1-5 \sqrt [5]{e}}{\left (1+5 \sqrt [5]{e}\right ) \left (-x+5 \sqrt [5]{e}+1\right )}+\frac {3}{-x+5 \sqrt [5]{e}+1}-\frac {5 \left (9-5 \sqrt [5]{e}\right ) \sqrt [5]{e}}{2 \left (1-5 \sqrt [5]{e}\right ) \left (x-5 \sqrt [5]{e}+1\right )}+\frac {1+5 \sqrt [5]{e}}{\left (1-5 \sqrt [5]{e}\right ) \left (x-5 \sqrt [5]{e}+1\right )}-\frac {1-5 \sqrt [5]{e}}{2 \left (x-5 \sqrt [5]{e}+1\right )}-\frac {3}{x-5 \sqrt [5]{e}+1}+\frac {\left (1-5 \sqrt [5]{e}\right ) \left (1+5 \sqrt [5]{e}\right )^2 e^x}{4 x}+\frac {\left (1+5 \sqrt [5]{e}\right )^2 e^x}{4 x}+\frac {\left (1-5 \sqrt [5]{e}\right )^2 \left (1+5 \sqrt [5]{e}\right ) e^x}{4 x}+\frac {\left (1-5 \sqrt [5]{e}\right )^2 e^x}{4 x}+\frac {1+5 \sqrt [5]{e}}{\left (1-5 \sqrt [5]{e}\right ) x}+\frac {1+5 \sqrt [5]{e}}{x}+\frac {1-5 \sqrt [5]{e}}{\left (1+5 \sqrt [5]{e}\right ) x}+\frac {1-5 \sqrt [5]{e}}{x}\) |
Input:
Int[(-4 + 100*E^(2/5) + 12*x^2 - 2*x^3 + E^(1/5)*(-80*x + 10*x^2) + E^x*(- 1 + x + 2*x^2 - 2*x^3 - x^4 + x^5 + E^(4/5)*(-625 + 625*x) + E^(3/5)*(500* x - 500*x^2) + E^(2/5)*(50 - 50*x - 150*x^2 + 150*x^3) + E^(1/5)*(-20*x + 20*x^2 + 20*x^3 - 20*x^4)))/(x^2 + 625*E^(4/5)*x^2 - 500*E^(3/5)*x^3 - 2*x ^4 + x^6 + E^(2/5)*(-50*x^2 + 150*x^4) + E^(1/5)*(20*x^3 - 20*x^5)),x]
Output:
-1/2*E^x + ((1 - 10*E^(1/5))*E^x)/4 - ((3 + 10*E^(1/5))*E^x)/4 + ((2 + 15* E^(1/5))*E^x)/4 + ((2 - 5*E^(1/5) - 75*E^(2/5))*E^x)/4 - (15*E^(1/5 + x))/ 4 + (25*(1 + 3*E^(1/5))*E^(1/5 + x))/4 + 3/(1 + 5*E^(1/5) - x) - (1 - 5*E^ (1/5))/((1 + 5*E^(1/5))*(1 + 5*E^(1/5) - x)) - (1 + 5*E^(1/5))/(2*(1 + 5*E ^(1/5) - x)) - (5*(7 - 5*E^(1/5))*E^(1/5))/(2*(1 + 5*E^(1/5))*(1 + 5*E^(1/ 5) - x)) + (1 - 5*E^(1/5))/x + (1 - 5*E^(1/5))/((1 + 5*E^(1/5))*x) + (1 + 5*E^(1/5))/x + (1 + 5*E^(1/5))/((1 - 5*E^(1/5))*x) + ((1 - 5*E^(1/5))^2*E^ x)/(4*x) + ((1 - 5*E^(1/5))^2*(1 + 5*E^(1/5))*E^x)/(4*x) + ((1 + 5*E^(1/5) )^2*E^x)/(4*x) + ((1 - 5*E^(1/5))*(1 + 5*E^(1/5))^2*E^x)/(4*x) + (E^x*x)/2 - ((2 + 15*E^(1/5))*E^x*x)/4 + (15*E^(1/5 + x)*x)/4 - 3/(1 - 5*E^(1/5) + x) - (1 - 5*E^(1/5))/(2*(1 - 5*E^(1/5) + x)) + (1 + 5*E^(1/5))/((1 - 5*E^( 1/5))*(1 - 5*E^(1/5) + x)) - (5*(9 - 5*E^(1/5))*E^(1/5))/(2*(1 - 5*E^(1/5) )*(1 - 5*E^(1/5) + x)) - ((1 - 5*E^(1/5))^2*ExpIntegralEi[x])/4 - ((1 - 5* E^(1/5))^2*(1 + 5*E^(1/5))*ExpIntegralEi[x])/4 - ((1 + 5*E^(1/5))^2*ExpInt egralEi[x])/4 - ((1 - 5*E^(1/5))*(1 + 5*E^(1/5))^2*ExpIntegralEi[x])/4 - ( (1 - 25*E^(2/5))*ExpIntegralEi[x])/4 + ((3 + 20*E^(1/5) + 25*E^(2/5))*ExpI ntegralEi[x])/4 + ((2 - 5*E^(1/5) - 100*E^(2/5) - 125*E^(3/5))*ExpIntegral Ei[x])/4 - (5*(3 - 10*E^(1/5) - 25*E^(2/5))*E^(1/5)*ExpIntegralEi[x])/4 - Log[1 + 5*E^(1/5) - x]/2 + (2*(1 - 5*E^(1/5))*Log[1 + 5*E^(1/5) - x])/(1 + 5*E^(1/5))^2 - ((3 - 5*E^(1/5))*Log[1 + 5*E^(1/5) - x])/(2*(1 + 5*E^(1...
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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 3.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {x -4}{\left (25 \,{\mathrm e}^{\frac {2}{5}}-10 \,{\mathrm e}^{\frac {1}{5}} x +x^{2}-1\right ) x}+\frac {{\mathrm e}^{x}}{x}\) | \(34\) |
| norman | \(\frac {-4+\left (25 \,{\mathrm e}^{\frac {2}{5}}-1\right ) {\mathrm e}^{x}+x +{\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{5}} x}{x \left (25 \,{\mathrm e}^{\frac {2}{5}}-10 \,{\mathrm e}^{\frac {1}{5}} x +x^{2}-1\right )}\) | \(50\) |
| parallelrisch | \(\frac {-4+25 \,{\mathrm e}^{x} {\mathrm e}^{\frac {2}{5}}-10 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{5}} x +{\mathrm e}^{x} x^{2}+x -{\mathrm e}^{x}}{x \left (25 \,{\mathrm e}^{\frac {2}{5}}-10 \,{\mathrm e}^{\frac {1}{5}} x +x^{2}-1\right )}\) | \(51\) |
| parts | \(\frac {{\mathrm e}^{x}}{x}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-20 \,{\mathrm e}^{\frac {1}{5}} \textit {\_Z}^{3}+\left (150 \,{\mathrm e}^{\frac {2}{5}}-2\right ) \textit {\_Z}^{2}+\left (20 \,{\mathrm e}^{\frac {1}{5}}-500 \,{\mathrm e}^{\frac {3}{5}}\right ) \textit {\_Z} +625 \,{\mathrm e}^{\frac {4}{5}}-50 \,{\mathrm e}^{\frac {2}{5}}+1\right )}{\sum }\frac {\left (2+2 \left (1-15625 \,{\mathrm e}^{\frac {6}{5}}+1875 \,{\mathrm e}^{\frac {4}{5}}-75 \,{\mathrm e}^{\frac {2}{5}}\right ) \textit {\_R}^{2}+\left (62500 \,{\mathrm e}^{\frac {6}{5}}-390625 \,{\mathrm e}^{\frac {8}{5}}-75000 \,{\mathrm e}-40 \,{\mathrm e}^{\frac {1}{5}}-3750 \,{\mathrm e}^{\frac {4}{5}}+3000 \,{\mathrm e}^{\frac {3}{5}}+100 \,{\mathrm e}^{\frac {2}{5}}+625000 \,{\mathrm e}^{\frac {7}{5}}-1\right ) \textit {\_R} +250000 \,{\mathrm e}^{\frac {6}{5}}-2343750 \,{\mathrm e}^{\frac {8}{5}}+18750 \,{\mathrm e}+5 \,{\mathrm e}^{\frac {1}{5}}-7500 \,{\mathrm e}^{\frac {4}{5}}-500 \,{\mathrm e}^{\frac {3}{5}}+1953125 \,{\mathrm e}^{\frac {9}{5}}-312500 \,{\mathrm e}^{\frac {7}{5}}\right ) \ln \left (x -\textit {\_R} \right )}{125 \,{\mathrm e}^{\frac {3}{5}}-75 \textit {\_R} \,{\mathrm e}^{\frac {2}{5}}+15 \,{\mathrm e}^{\frac {1}{5}} \textit {\_R}^{2}-\textit {\_R}^{3}-5 \,{\mathrm e}^{\frac {1}{5}}+\textit {\_R}}}{2 \left (625 \,{\mathrm e}^{\frac {4}{5}}-50 \,{\mathrm e}^{\frac {2}{5}}+1\right )^{2}}-\frac {2 \left (-2+31250 \,{\mathrm e}^{\frac {6}{5}}-3750 \,{\mathrm e}^{\frac {4}{5}}+150 \,{\mathrm e}^{\frac {2}{5}}\right )}{\left (625 \,{\mathrm e}^{\frac {4}{5}}-50 \,{\mathrm e}^{\frac {2}{5}}+1\right )^{2} x}\) | \(223\) |
| default | \(\text {Expression too large to display}\) | \(15560\) |
Input:
int((((625*x-625)*exp(1/5)^4+(-500*x^2+500*x)*exp(1/5)^3+(150*x^3-150*x^2- 50*x+50)*exp(1/5)^2+(-20*x^4+20*x^3+20*x^2-20*x)*exp(1/5)+x^5-x^4-2*x^3+2* x^2+x-1)*exp(x)+100*exp(1/5)^2+(10*x^2-80*x)*exp(1/5)-2*x^3+12*x^2-4)/(625 *x^2*exp(1/5)^4-500*x^3*exp(1/5)^3+(150*x^4-50*x^2)*exp(1/5)^2+(-20*x^5+20 *x^3)*exp(1/5)+x^6-2*x^4+x^2),x,method=_RETURNVERBOSE)
Output:
25*(1/25*x-4/25)/(25*exp(2/5)-10*exp(1/5)*x+x^2-1)/x+exp(x)/x
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\frac {{\left (x^{2} - 10 \, x e^{\frac {1}{5}} + 25 \, e^{\frac {2}{5}} - 1\right )} e^{x} + x - 4}{x^{3} - 10 \, x^{2} e^{\frac {1}{5}} + 25 \, x e^{\frac {2}{5}} - x} \] Input:
integrate((((625*x-625)*exp(1/5)^4+(-500*x^2+500*x)*exp(1/5)^3+(150*x^3-15 0*x^2-50*x+50)*exp(1/5)^2+(-20*x^4+20*x^3+20*x^2-20*x)*exp(1/5)+x^5-x^4-2* x^3+2*x^2+x-1)*exp(x)+100*exp(1/5)^2+(10*x^2-80*x)*exp(1/5)-2*x^3+12*x^2-4 )/(625*x^2*exp(1/5)^4-500*x^3*exp(1/5)^3+(150*x^4-50*x^2)*exp(1/5)^2+(-20* x^5+20*x^3)*exp(1/5)+x^6-2*x^4+x^2),x, algorithm="fricas")
Output:
((x^2 - 10*x*e^(1/5) + 25*e^(2/5) - 1)*e^x + x - 4)/(x^3 - 10*x^2*e^(1/5) + 25*x*e^(2/5) - x)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (20) = 40\).
Time = 2.95 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.78 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=- \frac {x \left (- 625 e^{\frac {4}{5}} - 1 + 50 e^{\frac {2}{5}}\right ) - 200 e^{\frac {2}{5}} + 4 + 2500 e^{\frac {4}{5}}}{x^{3} \left (- 50 e^{\frac {2}{5}} + 1 + 625 e^{\frac {4}{5}}\right ) + x^{2} \left (- 6250 e - 10 e^{\frac {1}{5}} + 500 e^{\frac {3}{5}}\right ) + x \left (- 1875 e^{\frac {4}{5}} - 1 + 75 e^{\frac {2}{5}} + 15625 e^{\frac {6}{5}}\right )} + \frac {e^{x}}{x} \] Input:
integrate((((625*x-625)*exp(1/5)**4+(-500*x**2+500*x)*exp(1/5)**3+(150*x** 3-150*x**2-50*x+50)*exp(1/5)**2+(-20*x**4+20*x**3+20*x**2-20*x)*exp(1/5)+x **5-x**4-2*x**3+2*x**2+x-1)*exp(x)+100*exp(1/5)**2+(10*x**2-80*x)*exp(1/5) -2*x**3+12*x**2-4)/(625*x**2*exp(1/5)**4-500*x**3*exp(1/5)**3+(150*x**4-50 *x**2)*exp(1/5)**2+(-20*x**5+20*x**3)*exp(1/5)+x**6-2*x**4+x**2),x)
Output:
-(x*(-625*exp(4/5) - 1 + 50*exp(2/5)) - 200*exp(2/5) + 4 + 2500*exp(4/5))/ (x**3*(-50*exp(2/5) + 1 + 625*exp(4/5)) + x**2*(-6250*E - 10*exp(1/5) + 50 0*exp(3/5)) + x*(-1875*exp(4/5) - 1 + 75*exp(2/5) + 15625*exp(6/5))) + exp (x)/x
Exception generated. \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((((625*x-625)*exp(1/5)^4+(-500*x^2+500*x)*exp(1/5)^3+(150*x^3-15 0*x^2-50*x+50)*exp(1/5)^2+(-20*x^4+20*x^3+20*x^2-20*x)*exp(1/5)+x^5-x^4-2* x^3+2*x^2+x-1)*exp(x)+100*exp(1/5)^2+(10*x^2-80*x)*exp(1/5)-2*x^3+12*x^2-4 )/(625*x^2*exp(1/5)^4-500*x^3*exp(1/5)^3+(150*x^4-50*x^2)*exp(1/5)^2+(-20* x^5+20*x^3)*exp(1/5)+x^6-2*x^4+x^2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 1602 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 1602, normalized size of antiderivative = 44.50 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\text {Too large to display} \] Input:
integrate((((625*x-625)*exp(1/5)^4+(-500*x^2+500*x)*exp(1/5)^3+(150*x^3-15 0*x^2-50*x+50)*exp(1/5)^2+(-20*x^4+20*x^3+20*x^2-20*x)*exp(1/5)+x^5-x^4-2* x^3+2*x^2+x-1)*exp(x)+100*exp(1/5)^2+(10*x^2-80*x)*exp(1/5)-2*x^3+12*x^2-4 )/(625*x^2*exp(1/5)^4-500*x^3*exp(1/5)^3+(150*x^4-50*x^2)*exp(1/5)^2+(-20* x^5+20*x^3)*exp(1/5)+x^6-2*x^4+x^2),x, algorithm="giac")
Output:
-1/2*(78125*x^3*e^(7/5)*log(x - 5*e^(1/5) + 1) - 93750*x^3*e^(6/5)*log(x - 5*e^(1/5) + 1) - 9375*x^3*e*log(x - 5*e^(1/5) + 1) + 12500*x^3*e^(4/5)*lo g(x - 5*e^(1/5) + 1) + 375*x^3*e^(3/5)*log(x - 5*e^(1/5) + 1) - 750*x^3*e^ (2/5)*log(x - 5*e^(1/5) + 1) - 85*x^3*e^(1/5)*log(x - 5*e^(1/5) + 1) + 781 25*x^3*e^(7/5)*log(x - 5*e^(1/5) - 1) - 93750*x^3*e^(6/5)*log(x - 5*e^(1/5 ) - 1) - 9375*x^3*e*log(x - 5*e^(1/5) - 1) + 12500*x^3*e^(4/5)*log(x - 5*e ^(1/5) - 1) + 375*x^3*e^(3/5)*log(x - 5*e^(1/5) - 1) - 750*x^3*e^(2/5)*log (x - 5*e^(1/5) - 1) + 75*x^3*e^(1/5)*log(x - 5*e^(1/5) - 1) - 78125*x^3*e^ (7/5)*log(-x + 5*e^(1/5) + 1) + 93750*x^3*e^(6/5)*log(-x + 5*e^(1/5) + 1) + 9375*x^3*e*log(-x + 5*e^(1/5) + 1) - 12500*x^3*e^(4/5)*log(-x + 5*e^(1/5 ) + 1) - 375*x^3*e^(3/5)*log(-x + 5*e^(1/5) + 1) + 750*x^3*e^(2/5)*log(-x + 5*e^(1/5) + 1) - 75*x^3*e^(1/5)*log(-x + 5*e^(1/5) + 1) - 78125*x^3*e^(7 /5)*log(-x + 5*e^(1/5) - 1) + 93750*x^3*e^(6/5)*log(-x + 5*e^(1/5) - 1) + 9375*x^3*e*log(-x + 5*e^(1/5) - 1) - 12500*x^3*e^(4/5)*log(-x + 5*e^(1/5) - 1) - 375*x^3*e^(3/5)*log(-x + 5*e^(1/5) - 1) + 750*x^3*e^(2/5)*log(-x + 5*e^(1/5) - 1) + 85*x^3*e^(1/5)*log(-x + 5*e^(1/5) - 1) - 781250*x^2*e^(8/ 5)*log(x - 5*e^(1/5) + 1) + 937500*x^2*e^(7/5)*log(x - 5*e^(1/5) + 1) + 93 750*x^2*e^(6/5)*log(x - 5*e^(1/5) + 1) - 125000*x^2*e*log(x - 5*e^(1/5) + 1) - 3750*x^2*e^(4/5)*log(x - 5*e^(1/5) + 1) + 7500*x^2*e^(3/5)*log(x - 5* e^(1/5) + 1) + 850*x^2*e^(2/5)*log(x - 5*e^(1/5) + 1) - 781250*x^2*e^(8...
Time = 3.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\frac {{\mathrm {e}}^x}{x}+\frac {x-4}{x^3-10\,{\mathrm {e}}^{1/5}\,x^2+\left (25\,{\mathrm {e}}^{2/5}-1\right )\,x} \] Input:
int((100*exp(2/5) - exp(1/5)*(80*x - 10*x^2) + exp(x)*(x + exp(3/5)*(500*x - 500*x^2) - exp(2/5)*(50*x + 150*x^2 - 150*x^3 - 50) - exp(1/5)*(20*x - 20*x^2 - 20*x^3 + 20*x^4) + 2*x^2 - 2*x^3 - x^4 + x^5 + exp(4/5)*(625*x - 625) - 1) + 12*x^2 - 2*x^3 - 4)/(exp(1/5)*(20*x^3 - 20*x^5) - exp(2/5)*(50 *x^2 - 150*x^4) + 625*x^2*exp(4/5) - 500*x^3*exp(3/5) + x^2 - 2*x^4 + x^6) ,x)
Output:
exp(x)/x + (x - 4)/(x^3 - 10*x^2*exp(1/5) + x*(25*exp(2/5) - 1))
Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 4.44 \[ \int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} \left (-80 x+10 x^2\right )+e^x \left (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} \left (500 x-500 x^2\right )+e^{2/5} \left (50-50 x-150 x^2+150 x^3\right )+\sqrt [5]{e} \left (-20 x+20 x^2+20 x^3-20 x^4\right )\right )}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} \left (-50 x^2+150 x^4\right )+\sqrt [5]{e} \left (20 x^3-20 x^5\right )} \, dx=\frac {625 e^{x +\frac {4}{5}} x^{2}-1875 e^{x +\frac {4}{5}}+625 e^{\frac {4}{5}} x -2500 e^{\frac {4}{5}}+500 e^{x +\frac {3}{5}} x -50 e^{x +\frac {2}{5}} x^{2}+75 e^{x +\frac {2}{5}}-50 e^{\frac {2}{5}} x +200 e^{\frac {2}{5}}+15625 e^{x +\frac {1}{5}} e -10 e^{x +\frac {1}{5}} x -6250 e^{x} e x +e^{x} x^{2}-e^{x}+x -4}{x \left (625 e^{\frac {4}{5}} x^{2}-1875 e^{\frac {4}{5}}+500 e^{\frac {3}{5}} x -50 e^{\frac {2}{5}} x^{2}+75 e^{\frac {2}{5}}+15625 e^{\frac {6}{5}}-10 e^{\frac {1}{5}} x -6250 e x +x^{2}-1\right )} \] Input:
int((((625*x-625)*exp(1/5)^4+(-500*x^2+500*x)*exp(1/5)^3+(150*x^3-150*x^2- 50*x+50)*exp(1/5)^2+(-20*x^4+20*x^3+20*x^2-20*x)*exp(1/5)+x^5-x^4-2*x^3+2* x^2+x-1)*exp(x)+100*exp(1/5)^2+(10*x^2-80*x)*exp(1/5)-2*x^3+12*x^2-4)/(625 *x^2*exp(1/5)^4-500*x^3*exp(1/5)^3+(150*x^4-50*x^2)*exp(1/5)^2+(-20*x^5+20 *x^3)*exp(1/5)+x^6-2*x^4+x^2),x)
Output:
(625*e**((5*x + 4)/5)*x**2 - 1875*e**((5*x + 4)/5) + 625*e**(4/5)*x - 2500 *e**(4/5) + 500*e**((5*x + 3)/5)*x - 50*e**((5*x + 2)/5)*x**2 + 75*e**((5* x + 2)/5) - 50*e**(2/5)*x + 200*e**(2/5) + 15625*e**((5*x + 1)/5)*e - 10*e **((5*x + 1)/5)*x - 6250*e**x*e*x + e**x*x**2 - e**x + x - 4)/(x*(625*e**( 4/5)*x**2 - 1875*e**(4/5) + 500*e**(3/5)*x - 50*e**(2/5)*x**2 + 75*e**(2/5 ) + 15625*e**(1/5)*e - 10*e**(1/5)*x - 6250*e*x + x**2 - 1))