Integrand size = 77, antiderivative size = 24 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=-e^{\frac {625}{(5-x)^2 x^3}}+\frac {e^x}{x} \] Output:
exp(x)/x-exp(625/x^3/(5-x)^2)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=-e^{\frac {625}{(-5+x)^2 x^3}}+\frac {e^x}{x} \] Input:
Integrate[(E^(625/(25*x^3 - 10*x^4 + x^5))*(-9375 + 3125*x) + E^x*(125*x^2 - 200*x^3 + 90*x^4 - 16*x^5 + x^6))/(-125*x^4 + 75*x^5 - 15*x^6 + x^7),x]
Output:
-E^(625/((-5 + x)^2*x^3)) + E^x/x
Time = 0.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2026, 2007, 7239, 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.
\(\displaystyle \int \frac {e^{\frac {625}{x^5-10 x^4+25 x^3}} (3125 x-9375)+e^x \left (x^6-16 x^5+90 x^4-200 x^3+125 x^2\right )}{x^7-15 x^6+75 x^5-125 x^4} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{\frac {625}{x^5-10 x^4+25 x^3}} (3125 x-9375)+e^x \left (x^6-16 x^5+90 x^4-200 x^3+125 x^2\right )}{x^4 \left (x^3-15 x^2+75 x-125\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{\frac {625}{x^5-10 x^4+25 x^3}} (3125 x-9375)+e^x \left (x^6-16 x^5+90 x^4-200 x^3+125 x^2\right )}{(x-5)^3 x^4}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {e^x (x-1)}{x^2}+\frac {3125 e^{\frac {625}{(x-5)^2 x^3}} (x-3)}{(x-5)^3 x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^x}{x}-e^{\frac {625}{(5-x)^2 x^3}}\) |
Input:
Int[(E^(625/(25*x^3 - 10*x^4 + x^5))*(-9375 + 3125*x) + E^x*(125*x^2 - 200 *x^3 + 90*x^4 - 16*x^5 + x^6))/(-125*x^4 + 75*x^5 - 15*x^6 + x^7),x]
Output:
-E^(625/((5 - x)^2*x^3)) + E^x/x
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.91 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{x}-{\mathrm e}^{\frac {625}{x^{3} \left (-5+x \right )^{2}}}\) | \(21\) |
parallelrisch | \(-\frac {18750 \,{\mathrm e}^{\frac {625}{x^{3} \left (x^{2}-10 x +25\right )}} x -18750 \,{\mathrm e}^{x}}{18750 x}\) | \(30\) |
parts | \(\frac {{\mathrm e}^{x}}{x}+\frac {-25 x^{3} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}+10 x^{4} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}-x^{5} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}}{x^{3} \left (-5+x \right )^{2}}\) | \(90\) |
norman | \(\frac {{\mathrm e}^{x} x^{4}-25 x^{3} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}+10 x^{4} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}-x^{5} {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}+25 \,{\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{x} x^{3}}{x^{3} \left (-5+x \right )^{2}}\) | \(103\) |
orering | \(-\frac {x \left (x^{14}-32 x^{13}+437 x^{12}-3310 x^{11}+15125 x^{10}-42500 x^{9}+71875 x^{8}-87500 x^{7}+331250 x^{6}-1781250 x^{5}+4687500 x^{4}-4687500 x^{3}-9765625 x^{2}+58593750 x -87890625\right ) \left (\left (x^{6}-16 x^{5}+90 x^{4}-200 x^{3}+125 x^{2}\right ) {\mathrm e}^{x}+\left (3125 x -9375\right ) {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}\right )}{3125 \left (x^{9}-14 x^{8}+56 x^{7}+40 x^{6}-755 x^{5}+1550 x^{4}+2375 x^{3}-21875 x^{2}+46875 x -28125\right ) \left (x^{7}-15 x^{6}+75 x^{5}-125 x^{4}\right )}+\frac {\left (x^{7}-16 x^{6}+90 x^{5}-200 x^{4}+125 x^{3}+3125 x -9375\right ) x^{5} \left (-5+x \right )^{3} \left (\frac {\left (6 x^{5}-80 x^{4}+360 x^{3}-600 x^{2}+250 x \right ) {\mathrm e}^{x}+\left (x^{6}-16 x^{5}+90 x^{4}-200 x^{3}+125 x^{2}\right ) {\mathrm e}^{x}+3125 \,{\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}-\frac {625 \left (3125 x -9375\right ) \left (5 x^{4}-40 x^{3}+75 x^{2}\right ) {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}}{\left (x^{5}-10 x^{4}+25 x^{3}\right )^{2}}}{x^{7}-15 x^{6}+75 x^{5}-125 x^{4}}-\frac {\left (\left (x^{6}-16 x^{5}+90 x^{4}-200 x^{3}+125 x^{2}\right ) {\mathrm e}^{x}+\left (3125 x -9375\right ) {\mathrm e}^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}}\right ) \left (7 x^{6}-90 x^{5}+375 x^{4}-500 x^{3}\right )}{\left (x^{7}-15 x^{6}+75 x^{5}-125 x^{4}\right )^{2}}\right )}{3125 x^{9}-43750 x^{8}+175000 x^{7}+125000 x^{6}-2359375 x^{5}+4843750 x^{4}+7421875 x^{3}-68359375 x^{2}+146484375 x -87890625}\) | \(529\) |
Input:
int(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5 -10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5-125*x^4),x,method=_RETURNVERBOSE)
Output:
exp(x)/x-exp(625/x^3/(-5+x)^2)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=-\frac {x e^{\left (\frac {625}{x^{5} - 10 \, x^{4} + 25 \, x^{3}}\right )} - e^{x}}{x} \] Input:
integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(62 5/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5-125*x^4),x, algorithm="fricas")
Output:
-(x*e^(625/(x^5 - 10*x^4 + 25*x^3)) - e^x)/x
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=- e^{\frac {625}{x^{5} - 10 x^{4} + 25 x^{3}}} + \frac {e^{x}}{x} \] Input:
integrate(((x**6-16*x**5+90*x**4-200*x**3+125*x**2)*exp(x)+(3125*x-9375)*e xp(625/(x**5-10*x**4+25*x**3)))/(x**7-15*x**6+75*x**5-125*x**4),x)
Output:
-exp(625/(x**5 - 10*x**4 + 25*x**3)) + exp(x)/x
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (20) = 40\).
Time = 0.58 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=-\frac {{\left (x e^{\left (\frac {5}{x^{2} - 10 \, x + 25} + \frac {3}{x} + \frac {10}{x^{2}} + \frac {25}{x^{3}}\right )} - e^{\left (x + \frac {3}{x - 5}\right )}\right )} e^{\left (-\frac {3}{x - 5}\right )}}{x} \] Input:
integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(62 5/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5-125*x^4),x, algorithm="maxima")
Output:
-(x*e^(5/(x^2 - 10*x + 25) + 3/x + 10/x^2 + 25/x^3) - e^(x + 3/(x - 5)))*e ^(-3/(x - 5))/x
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=\frac {e^{x}}{x} - e^{\left (\frac {625}{x^{5} - 10 \, x^{4} + 25 \, x^{3}}\right )} \] Input:
integrate(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(62 5/(x^5-10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5-125*x^4),x, algorithm="giac")
Output:
e^x/x - e^(625/(x^5 - 10*x^4 + 25*x^3))
Time = 2.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=\frac {{\mathrm {e}}^x}{x}-{\mathrm {e}}^{\frac {625}{x^5-10\,x^4+25\,x^3}} \] Input:
int(-(exp(x)*(125*x^2 - 200*x^3 + 90*x^4 - 16*x^5 + x^6) + exp(625/(25*x^3 - 10*x^4 + x^5))*(3125*x - 9375))/(125*x^4 - 75*x^5 + 15*x^6 - x^7),x)
Output:
exp(x)/x - exp(625/(25*x^3 - 10*x^4 + x^5))
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {625}{25 x^3-10 x^4+x^5}} (-9375+3125 x)+e^x \left (125 x^2-200 x^3+90 x^4-16 x^5+x^6\right )}{-125 x^4+75 x^5-15 x^6+x^7} \, dx=\frac {-e^{\frac {625}{x^{5}-10 x^{4}+25 x^{3}}} x +e^{x}}{x} \] Input:
int(((x^6-16*x^5+90*x^4-200*x^3+125*x^2)*exp(x)+(3125*x-9375)*exp(625/(x^5 -10*x^4+25*x^3)))/(x^7-15*x^6+75*x^5-125*x^4),x)
Output:
( - e**(625/(x**5 - 10*x**4 + 25*x**3))*x + e**x)/x