Integrand size = 84, antiderivative size = 24 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 e^{-\frac {10}{3-\frac {625}{4} (2-x)^2 x^3}} \] Output:
7/exp(5/(3-x^3*(25-25/2*x)^2))^2
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \] Input:
Integrate[(E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))*(-2100000*x^2 + 28 00000*x^3 - 875000*x^4))/(144 - 60000*x^3 + 60000*x^4 - 15000*x^5 + 625000 0*x^6 - 12500000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10),x]
Output:
7*E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))
Time = 0.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2028, 2463, 7257}
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.
\(\displaystyle \int \frac {e^{\frac {40}{625 x^5-2500 x^4+2500 x^3-12}} \left (-875000 x^4+2800000 x^3-2100000 x^2\right )}{390625 x^{10}-3125000 x^9+9375000 x^8-12500000 x^7+6250000 x^6-15000 x^5+60000 x^4-60000 x^3+144} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {e^{\frac {40}{625 x^5-2500 x^4+2500 x^3-12}} x^2 \left (-875000 x^2+2800000 x-2100000\right )}{390625 x^{10}-3125000 x^9+9375000 x^8-12500000 x^7+6250000 x^6-15000 x^5+60000 x^4-60000 x^3+144}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^{\frac {40}{625 x^5-2500 x^4+2500 x^3-12}} x^2 \left (-875000 x^2+2800000 x-2100000\right )}{\left (625 x^5-2500 x^4+2500 x^3-12\right )^2}dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle 7 e^{-\frac {40}{-625 x^5+2500 x^4-2500 x^3+12}}\) |
Input:
Int[(E^(40/(-12 + 2500*x^3 - 2500*x^4 + 625*x^5))*(-2100000*x^2 + 2800000* x^3 - 875000*x^4))/(144 - 60000*x^3 + 60000*x^4 - 15000*x^5 + 6250000*x^6 - 12500000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10),x]
Output:
7/E^(40/(12 - 2500*x^3 + 2500*x^4 - 625*x^5))
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
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]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 0.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
risch | \(7 \,{\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}\) | \(25\) |
gosper | \(7 \,{\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}\) | \(27\) |
norman | \(\frac {\left (4375 x^{5}-17500 x^{4}+17500 x^{3}-84\right ) {\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}\) | \(62\) |
parallelrisch | \(\frac {\left (18900000000 x^{5}-75600000000 x^{4}+75600000000 x^{3}-362880000\right ) {\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}}{2700000000 x^{5}-10800000000 x^{4}+10800000000 x^{3}-51840000}\) | \(63\) |
orering | \(-\frac {\left (625 x^{5}-2500 x^{4}+2500 x^{3}-12\right )^{2} \left (-875000 x^{4}+2800000 x^{3}-2100000 x^{2}\right ) {\mathrm e}^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}}}{25000 \left (5 x -6\right ) \left (-2+x \right ) x^{2} \left (390625 x^{10}-3125000 x^{9}+9375000 x^{8}-12500000 x^{7}+6250000 x^{6}-15000 x^{5}+60000 x^{4}-60000 x^{3}+144\right )}\) | \(121\) |
Input:
int((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000 *x^8-12500000*x^7+6250000*x^6-15000*x^5+60000*x^4-60000*x^3+144)/exp(-20/( 625*x^5-2500*x^4+2500*x^3-12))^2,x,method=_RETURNVERBOSE)
Output:
7*exp(40/(625*x^5-2500*x^4+2500*x^3-12))
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \] Input:
integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9 375000*x^8-12500000*x^7+6250000*x^6-15000*x^5+60000*x^4-60000*x^3+144)/exp (-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="fricas")
Output:
7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 e^{\frac {40}{625 x^{5} - 2500 x^{4} + 2500 x^{3} - 12}} \] Input:
integrate((-875000*x**4+2800000*x**3-2100000*x**2)/(390625*x**10-3125000*x **9+9375000*x**8-12500000*x**7+6250000*x**6-15000*x**5+60000*x**4-60000*x* *3+144)/exp(-20/(625*x**5-2500*x**4+2500*x**3-12))**2,x)
Output:
7*exp(40/(625*x**5 - 2500*x**4 + 2500*x**3 - 12))
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \] Input:
integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9 375000*x^8-12500000*x^7+6250000*x^6-15000*x^5+60000*x^4-60000*x^3+144)/exp (-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="maxima")
Output:
7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 \, e^{\left (\frac {40}{625 \, x^{5} - 2500 \, x^{4} + 2500 \, x^{3} - 12}\right )} \] Input:
integrate((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9 375000*x^8-12500000*x^7+6250000*x^6-15000*x^5+60000*x^4-60000*x^3+144)/exp (-20/(625*x^5-2500*x^4+2500*x^3-12))^2,x, algorithm="giac")
Output:
7*e^(40/(625*x^5 - 2500*x^4 + 2500*x^3 - 12))
Time = 3.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7\,{\mathrm {e}}^{\frac {40}{625\,x^5-2500\,x^4+2500\,x^3-12}} \] Input:
int(-(exp(40/(2500*x^3 - 2500*x^4 + 625*x^5 - 12))*(2100000*x^2 - 2800000* x^3 + 875000*x^4))/(60000*x^4 - 60000*x^3 - 15000*x^5 + 6250000*x^6 - 1250 0000*x^7 + 9375000*x^8 - 3125000*x^9 + 390625*x^10 + 144),x)
Output:
7*exp(40/(2500*x^3 - 2500*x^4 + 625*x^5 - 12))
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {40}{-12+2500 x^3-2500 x^4+625 x^5}} \left (-2100000 x^2+2800000 x^3-875000 x^4\right )}{144-60000 x^3+60000 x^4-15000 x^5+6250000 x^6-12500000 x^7+9375000 x^8-3125000 x^9+390625 x^{10}} \, dx=7 e^{\frac {40}{625 x^{5}-2500 x^{4}+2500 x^{3}-12}} \] Input:
int((-875000*x^4+2800000*x^3-2100000*x^2)/(390625*x^10-3125000*x^9+9375000 *x^8-12500000*x^7+6250000*x^6-15000*x^5+60000*x^4-60000*x^3+144)/exp(-20/( 625*x^5-2500*x^4+2500*x^3-12))^2,x)
Output:
7*e**(40/(625*x**5 - 2500*x**4 + 2500*x**3 - 12))