3.82.33 \(\int \frac {-4+100 e^{2/5}+12 x^2-2 x^3+\sqrt [5]{e} (-80 x+10 x^2)+e^x (-1+x+2 x^2-2 x^3-x^4+x^5+e^{4/5} (-625+625 x)+e^{3/5} (500 x-500 x^2)+e^{2/5} (50-50 x-150 x^2+150 x^3)+\sqrt [5]{e} (-20 x+20 x^2+20 x^3-20 x^4))}{x^2+625 e^{4/5} x^2-500 e^{3/5} x^3-2 x^4+x^6+e^{2/5} (-50 x^2+150 x^4)+\sqrt [5]{e} (20 x^3-20 x^5)} \, dx\)

Optimal. Leaf size=36 \[ \frac {e^x}{x}-\frac {4-x}{-x+\left (5 \sqrt [5]{e}-x\right )^2 x} \]

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Rubi [B]  time = 2.14, antiderivative size = 482, normalized size of antiderivative = 13.39, number of steps used = 25, number of rules used = 11, integrand size = 200, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6, 6688, 6742, 2197, 614, 616, 31, 740, 800, 632, 638} \begin {gather*} -\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{-x^2+10 \sqrt [5]{e} x-25 e^{2/5}+1}-\frac {2 \left (-5 \sqrt [5]{e} x+25 e^{2/5}+1\right )}{x \left (-x^2+10 \sqrt [5]{e} x-25 e^{2/5}+1\right )}-\frac {40 \sqrt [5]{e} \left (-5 \sqrt [5]{e} x+25 e^{2/5}+1\right )}{\left (1-25 e^{2/5}\right ) \left (-x^2+10 \sqrt [5]{e} x-25 e^{2/5}+1\right )}-\frac {5 \sqrt [5]{e} x-25 e^{2/5}+1}{-x^2+10 \sqrt [5]{e} x-25 e^{2/5}+1}+\frac {e^x}{x}+\frac {2 \left (3+25 e^{2/5}\right )}{\left (1-25 e^{2/5}\right ) x}+\frac {20 \left (2+5 \sqrt [5]{e}\right ) \sqrt [5]{e} \log \left (-x+5 \sqrt [5]{e}+1\right )}{\left (1+5 \sqrt [5]{e}\right )^2}+\frac {5}{2} \sqrt [5]{e} \log \left (-x+5 \sqrt [5]{e}+1\right )+\frac {\left (3+5 \sqrt [5]{e}\right ) \left (1-25 e^{2/5}\right ) \log \left (-x+5 \sqrt [5]{e}+1\right )}{\left (1+5 \sqrt [5]{e}\right )^3}-\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (-x+5 \sqrt [5]{e}+1\right )+\frac {20 \left (2-5 \sqrt [5]{e}\right ) \sqrt [5]{e} \log \left (x-5 \sqrt [5]{e}+1\right )}{\left (1-5 \sqrt [5]{e}\right )^2}-\frac {5}{2} \sqrt [5]{e} \log \left (x-5 \sqrt [5]{e}+1\right )+\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (x-5 \sqrt [5]{e}+1\right )-\frac {\left (3-5 \sqrt [5]{e}\right ) \left (1+5 \sqrt [5]{e}\right ) \log \left (x-5 \sqrt [5]{e}+1\right )}{\left (1-5 \sqrt [5]{e}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(2*(3 + 25*E^(2/5)))/((1 - 25*E^(2/5))*x) + E^x/x - ((6 + 5*E^(1/5))*(5*E^(1/5) - x))/(1 - 25*E^(2/5) + 10*E^(
1/5)*x - x^2) - (40*E^(1/5)*(1 + 25*E^(2/5) - 5*E^(1/5)*x))/((1 - 25*E^(2/5))*(1 - 25*E^(2/5) + 10*E^(1/5)*x -
x^2)) - (2*(1 + 25*E^(2/5) - 5*E^(1/5)*x))/(x*(1 - 25*E^(2/5) + 10*E^(1/5)*x - x^2)) - (1 - 25*E^(2/5) + 5*E^
(1/5)*x)/(1 - 25*E^(2/5) + 10*E^(1/5)*x - x^2) - ((6 + 5*E^(1/5))*Log[1 + 5*E^(1/5) - x])/2 + ((3 + 5*E^(1/5))
*(1 - 25*E^(2/5))*Log[1 + 5*E^(1/5) - x])/(1 + 5*E^(1/5))^3 + (5*E^(1/5)*Log[1 + 5*E^(1/5) - x])/2 + (20*(2 +
5*E^(1/5))*E^(1/5)*Log[1 + 5*E^(1/5) - x])/(1 + 5*E^(1/5))^2 - ((3 - 5*E^(1/5))*(1 + 5*E^(1/5))*Log[1 - 5*E^(1
/5) + x])/(1 - 5*E^(1/5))^2 + ((6 + 5*E^(1/5))*Log[1 - 5*E^(1/5) + x])/2 - (5*E^(1/5)*Log[1 - 5*E^(1/5) + x])/
2 + (20*(2 - 5*E^(1/5))*E^(1/5)*Log[1 - 5*E^(1/5) + x])/(1 - 5*E^(1/5))^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\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 )}{\left (1+625 e^{4/5}\right ) 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\\ &=\int \frac {100 e^{2/5}+625 e^{\frac {4}{5}+x} (-1+x)+10 \sqrt [5]{e} (-8+x) x-500 e^{\frac {3}{5}+x} (-1+x) x-20 e^{\frac {1}{5}+x} (-1+x)^2 x (1+x)+e^x (-1+x)^3 (1+x)^2-2 \left (2-6 x^2+x^3\right )+50 e^{\frac {2}{5}+x} \left (1-x-3 x^2+3 x^3\right )}{x^2 \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )^2} \, dx\\ &=\int \left (\frac {e^x (-1+x)}{x^2}+\frac {12 \left (1+\frac {5 \sqrt [5]{e}}{6}\right )}{\left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2}-\frac {4 \left (1-25 e^{2/5}\right )}{x^2 \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2}-\frac {80 \sqrt [5]{e}}{x \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2}-\frac {2 x}{\left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2} \, dx\right )+\left (2 \left (6+5 \sqrt [5]{e}\right )\right ) \int \frac {1}{\left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2} \, dx-\left (4 \left (1-25 e^{2/5}\right )\right ) \int \frac {1}{x^2 \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2} \, dx-\left (80 \sqrt [5]{e}\right ) \int \frac {1}{x \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )^2} \, dx+\int \frac {e^x (-1+x)}{x^2} \, dx\\ &=\frac {e^x}{x}-\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {40 \sqrt [5]{e} \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{\left (1-25 e^{2/5}\right ) \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {2 \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{x \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {1-25 e^{2/5}+5 \sqrt [5]{e} x}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}+\left (-6-5 \sqrt [5]{e}\right ) \int \frac {1}{-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2} \, dx+\left (5 \sqrt [5]{e}\right ) \int \frac {1}{-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2} \, dx-\frac {\left (20 \sqrt [5]{e}\right ) \int \frac {-4+10 \sqrt [5]{e} x}{x \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )} \, dx}{1-25 e^{2/5}}-\int \frac {-2 \left (3+25 e^{2/5}\right )+20 \sqrt [5]{e} x}{x^2 \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )} \, dx\\ &=\frac {e^x}{x}-\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {40 \sqrt [5]{e} \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{\left (1-25 e^{2/5}\right ) \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {2 \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{x \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {1-25 e^{2/5}+5 \sqrt [5]{e} x}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}+\frac {1}{2} \left (-6-5 \sqrt [5]{e}\right ) \int \frac {1}{-1-5 \sqrt [5]{e}+x} \, dx+\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \int \frac {1}{1-5 \sqrt [5]{e}+x} \, dx+\frac {1}{2} \left (5 \sqrt [5]{e}\right ) \int \frac {1}{-1-5 \sqrt [5]{e}+x} \, dx-\frac {1}{2} \left (5 \sqrt [5]{e}\right ) \int \frac {1}{1-5 \sqrt [5]{e}+x} \, dx-\frac {\left (20 \sqrt [5]{e}\right ) \int \left (-\frac {4}{\left (-1+25 e^{2/5}\right ) x}+\frac {2 \left (-25 \sqrt [5]{e}+125 e^{3/5}+2 x\right )}{\left (-1+25 e^{2/5}\right ) \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )}\right ) \, dx}{1-25 e^{2/5}}-\int \left (-\frac {2 \left (3+25 e^{2/5}\right )}{\left (-1+25 e^{2/5}\right ) x^2}-\frac {80 \sqrt [5]{e}}{\left (-1+25 e^{2/5}\right )^2 x}+\frac {2 \left (-3-350 e^{2/5}+625 e^{4/5}+40 \sqrt [5]{e} x\right )}{\left (-1+25 e^{2/5}\right )^2 \left (-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2\right )}\right ) \, dx\\ &=\frac {2 \left (3+25 e^{2/5}\right )}{\left (1-25 e^{2/5}\right ) x}+\frac {e^x}{x}-\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {40 \sqrt [5]{e} \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{\left (1-25 e^{2/5}\right ) \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {2 \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{x \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {1-25 e^{2/5}+5 \sqrt [5]{e} x}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {5}{2} \sqrt [5]{e} \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1-5 \sqrt [5]{e}+x\right )-\frac {5}{2} \sqrt [5]{e} \log \left (1-5 \sqrt [5]{e}+x\right )-\frac {2 \int \frac {-3-350 e^{2/5}+625 e^{4/5}+40 \sqrt [5]{e} x}{-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2} \, dx}{\left (1-25 e^{2/5}\right )^2}+\frac {\left (40 \sqrt [5]{e}\right ) \int \frac {-25 \sqrt [5]{e}+125 e^{3/5}+2 x}{-1+25 e^{2/5}-10 \sqrt [5]{e} x+x^2} \, dx}{\left (1-25 e^{2/5}\right )^2}\\ &=\frac {2 \left (3+25 e^{2/5}\right )}{\left (1-25 e^{2/5}\right ) x}+\frac {e^x}{x}-\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {40 \sqrt [5]{e} \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{\left (1-25 e^{2/5}\right ) \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {2 \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{x \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {1-25 e^{2/5}+5 \sqrt [5]{e} x}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {5}{2} \sqrt [5]{e} \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1-5 \sqrt [5]{e}+x\right )-\frac {5}{2} \sqrt [5]{e} \log \left (1-5 \sqrt [5]{e}+x\right )-\frac {\left (\left (3-5 \sqrt [5]{e}\right ) \left (1+5 \sqrt [5]{e}\right )\right ) \int \frac {1}{1-5 \sqrt [5]{e}+x} \, dx}{\left (1-5 \sqrt [5]{e}\right )^2}+\frac {\left (\left (1-5 \sqrt [5]{e}\right ) \left (3+5 \sqrt [5]{e}\right )\right ) \int \frac {1}{-1-5 \sqrt [5]{e}+x} \, dx}{\left (1+5 \sqrt [5]{e}\right )^2}+\frac {\left (20 \left (2-5 \sqrt [5]{e}\right ) \sqrt [5]{e}\right ) \int \frac {1}{1-5 \sqrt [5]{e}+x} \, dx}{\left (1-5 \sqrt [5]{e}\right )^2}+\frac {\left (20 \left (2+5 \sqrt [5]{e}\right ) \sqrt [5]{e}\right ) \int \frac {1}{-1-5 \sqrt [5]{e}+x} \, dx}{\left (1+5 \sqrt [5]{e}\right )^2}\\ &=\frac {2 \left (3+25 e^{2/5}\right )}{\left (1-25 e^{2/5}\right ) x}+\frac {e^x}{x}-\frac {\left (6+5 \sqrt [5]{e}\right ) \left (5 \sqrt [5]{e}-x\right )}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}-\frac {40 \sqrt [5]{e} \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{\left (1-25 e^{2/5}\right ) \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {2 \left (1+25 e^{2/5}-5 \sqrt [5]{e} x\right )}{x \left (1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2\right )}-\frac {1-25 e^{2/5}+5 \sqrt [5]{e} x}{1-25 e^{2/5}+10 \sqrt [5]{e} x-x^2}+\frac {\left (1-5 \sqrt [5]{e}\right ) \left (3+5 \sqrt [5]{e}\right ) \log \left (1+5 \sqrt [5]{e}-x\right )}{\left (1+5 \sqrt [5]{e}\right )^2}-\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {5}{2} \sqrt [5]{e} \log \left (1+5 \sqrt [5]{e}-x\right )+\frac {20 \left (2+5 \sqrt [5]{e}\right ) \sqrt [5]{e} \log \left (1+5 \sqrt [5]{e}-x\right )}{\left (1+5 \sqrt [5]{e}\right )^2}-\frac {\left (3-5 \sqrt [5]{e}\right ) \left (1+5 \sqrt [5]{e}\right ) \log \left (1-5 \sqrt [5]{e}+x\right )}{\left (1-5 \sqrt [5]{e}\right )^2}+\frac {1}{2} \left (6+5 \sqrt [5]{e}\right ) \log \left (1-5 \sqrt [5]{e}+x\right )-\frac {5}{2} \sqrt [5]{e} \log \left (1-5 \sqrt [5]{e}+x\right )+\frac {20 \left (2-5 \sqrt [5]{e}\right ) \sqrt [5]{e} \log \left (1-5 \sqrt [5]{e}+x\right )}{\left (1-5 \sqrt [5]{e}\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 57, normalized size = 1.58 \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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)*(
-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]

[Out]

(-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))

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fricas [A]  time = 0.81, size = 42, normalized size = 1.17 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+2
0*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")

[Out]

((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)

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giac [B]  time = 0.72, size = 1602, normalized size = 44.50 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+2
0*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")

[Out]

-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)*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) + 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) - 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) + 93750*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^(8/5)*log(x - 5*e^(1/5) - 1) + 937500*x^2*e^(7/5)*log(x - 5*e^(1/
5) - 1) + 93750*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)*lo
g(x - 5*e^(1/5) - 1) + 7500*x^2*e^(3/5)*log(x - 5*e^(1/5) - 1) - 750*x^2*e^(2/5)*log(x - 5*e^(1/5) - 1) + 7812
50*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) - 93750*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) + 750*x^2*e^(2/5)*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) - 93750*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) - 31250*x^2*e^(x + 6/5) + 3750*x^2*e^(x + 4/5) - 150*x^2*e^(x +
2/5) + 2*x^2*e^x + 1953125*x*e^(9/5)*log(x - 5*e^(1/5) + 1) - 2343750*x*e^(8/5)*log(x - 5*e^(1/5) + 1) - 31250
0*x*e^(7/5)*log(x - 5*e^(1/5) + 1) + 406250*x*e^(6/5)*log(x - 5*e^(1/5) + 1) + 18750*x*e*log(x - 5*e^(1/5) + 1
) - 31250*x*e^(4/5)*log(x - 5*e^(1/5) + 1) - 2500*x*e^(3/5)*log(x - 5*e^(1/5) + 1) + 750*x*e^(2/5)*log(x - 5*e
^(1/5) + 1) + 85*x*e^(1/5)*log(x - 5*e^(1/5) + 1) + 1953125*x*e^(9/5)*log(x - 5*e^(1/5) - 1) - 2343750*x*e^(8/
5)*log(x - 5*e^(1/5) - 1) - 312500*x*e^(7/5)*log(x - 5*e^(1/5) - 1) + 406250*x*e^(6/5)*log(x - 5*e^(1/5) - 1)
+ 18750*x*e*log(x - 5*e^(1/5) - 1) - 31250*x*e^(4/5)*log(x - 5*e^(1/5) - 1) + 1500*x*e^(3/5)*log(x - 5*e^(1/5)
- 1) + 750*x*e^(2/5)*log(x - 5*e^(1/5) - 1) - 75*x*e^(1/5)*log(x - 5*e^(1/5) - 1) - 1953125*x*e^(9/5)*log(-x
+ 5*e^(1/5) + 1) + 2343750*x*e^(8/5)*log(-x + 5*e^(1/5) + 1) + 312500*x*e^(7/5)*log(-x + 5*e^(1/5) + 1) - 4062
50*x*e^(6/5)*log(-x + 5*e^(1/5) + 1) - 18750*x*e*log(-x + 5*e^(1/5) + 1) + 31250*x*e^(4/5)*log(-x + 5*e^(1/5)
+ 1) - 1500*x*e^(3/5)*log(-x + 5*e^(1/5) + 1) - 750*x*e^(2/5)*log(-x + 5*e^(1/5) + 1) + 75*x*e^(1/5)*log(-x +
5*e^(1/5) + 1) - 1953125*x*e^(9/5)*log(-x + 5*e^(1/5) - 1) + 2343750*x*e^(8/5)*log(-x + 5*e^(1/5) - 1) + 31250
0*x*e^(7/5)*log(-x + 5*e^(1/5) - 1) - 406250*x*e^(6/5)*log(-x + 5*e^(1/5) - 1) - 18750*x*e*log(-x + 5*e^(1/5)
- 1) + 31250*x*e^(4/5)*log(-x + 5*e^(1/5) - 1) + 2500*x*e^(3/5)*log(-x + 5*e^(1/5) - 1) - 750*x*e^(2/5)*log(-x
+ 5*e^(1/5) - 1) - 85*x*e^(1/5)*log(-x + 5*e^(1/5) - 1) - 31250*x*e^(6/5) + 3750*x*e^(4/5) - 150*x*e^(2/5) +
312500*x*e^(x + 7/5) - 37500*x*e^(x + 1) + 1500*x*e^(x + 3/5) - 20*x*e^(x + 1/5) + 2*x + 125000*e^(6/5) - 1500
0*e^(4/5) + 600*e^(2/5) - 781250*e^(x + 8/5) + 125000*e^(x + 6/5) - 7500*e^(x + 4/5) + 200*e^(x + 2/5) - 2*e^x
- 8)/(15625*x^3*e^(6/5) - 1875*x^3*e^(4/5) + 75*x^3*e^(2/5) - x^3 - 156250*x^2*e^(7/5) + 18750*x^2*e - 750*x^
2*e^(3/5) + 10*x^2*e^(1/5) + 390625*x*e^(8/5) - 62500*x*e^(6/5) + 3750*x*e^(4/5) - 100*x*e^(2/5) + x)

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maple [A]  time = 0.47, size = 34, normalized size = 0.94

method	result	size
risch	\(\frac {x -4}{\left (25 \,{\mathrm e}^{\frac {2}{5}}-10 x \,{\mathrm e}^{\frac {1}{5}}+x^{2}-1\right ) x}+\frac {{\mathrm e}^{x}}{x}\)	\(34\)
norman	\(\frac {-4+\left (25 \,{\mathrm e}^{\frac {2}{5}}-1\right ) {\mathrm e}^{x}+x +{\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{5}} x}{x \left (25 \,{\mathrm e}^{\frac {2}{5}}-10 x \,{\mathrm e}^{\frac {1}{5}}+x^{2}-1\right )}\)	\(50\)
default	\(\text {Expression too large to display}\)	\(15588\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)/(6
25*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,me
thod=_RETURNVERBOSE)

[Out]

25*(1/25*x-4/25)/(25*exp(2/5)-10*x*exp(1/5)+x^2-1)/x+exp(x)/x

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+2
0*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")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 5.98, size = 32, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp(x)/x + (x - 4)/(x^3 - 10*x^2*exp(1/5) + x*(25*exp(2/5) - 1))

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sympy [B]  time = 4.64, size = 100, normalized size = 2.78 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

-(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) + 500*exp(3/5)) + x*(-1875*exp(4/5) - 1 + 75*exp(2/5) + 15625*exp(6/5))) + e
xp(x)/x

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