\(\int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} (1562500-1250000000 x+1500000000 x^2-400000000 x^3)}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx\) [7432]

Optimal result

Integrand size = 79, antiderivative size = 26 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 e^{\frac {4}{3 \left (-x+16 x^2 (-5+2 x)^2\right )}} \]

[Out]

390625*exp(1/3/(16*(-5+2*x)^2*x^2-x))^4

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6838} \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 e^{-\frac {4}{3 \left (-64 x^4+320 x^3-400 x^2+x\right )}} \]

[In]

Int[(E^(4/(-3*x + 1200*x^2 - 960*x^3 + 192*x^4))*(1562500 - 1250000000*x + 1500000000*x^2 - 400000000*x^3))/(3
*x^2 - 2400*x^3 + 481920*x^4 - 768384*x^5 + 460800*x^6 - 122880*x^7 + 12288*x^8),x]

[Out]

390625/E^(4/(3*(x - 400*x^2 + 320*x^3 - 64*x^4)))

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = 390625 e^{-\frac {4}{3 \left (x-400 x^2+320 x^3-64 x^4\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 e^{\frac {4}{3 x \left (-1+400 x-320 x^2+64 x^3\right )}} \]

[In]

Integrate[(E^(4/(-3*x + 1200*x^2 - 960*x^3 + 192*x^4))*(1562500 - 1250000000*x + 1500000000*x^2 - 400000000*x^
3))/(3*x^2 - 2400*x^3 + 481920*x^4 - 768384*x^5 + 460800*x^6 - 122880*x^7 + 12288*x^8),x]

[Out]

390625*E^(4/(3*x*(-1 + 400*x - 320*x^2 + 64*x^3)))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

method	result	size
risch	\(390625 \,{\mathrm e}^{\frac {4}{3 x \left (64 x^{3}-320 x^{2}+400 x -1\right )}}\)	\(26\)
gosper	\(390625 \,{\mathrm e}^{\frac {4}{3 x \left (64 x^{3}-320 x^{2}+400 x -1\right )}}\)	\(28\)

[In]

int((-400000000*x^3+1500000000*x^2-1250000000*x+1562500)*exp(1/(192*x^4-960*x^3+1200*x^2-3*x))^4/(12288*x^8-12
2880*x^7+460800*x^6-768384*x^5+481920*x^4-2400*x^3+3*x^2),x,method=_RETURNVERBOSE)

[Out]

390625*exp(4/3/x/(64*x^3-320*x^2+400*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 \, e^{\left (\frac {4}{3 \, {\left (64 \, x^{4} - 320 \, x^{3} + 400 \, x^{2} - x\right )}}\right )} \]

[In]

integrate((-400000000*x^3+1500000000*x^2-1250000000*x+1562500)*exp(1/(192*x^4-960*x^3+1200*x^2-3*x))^4/(12288*
x^8-122880*x^7+460800*x^6-768384*x^5+481920*x^4-2400*x^3+3*x^2),x, algorithm="fricas")

[Out]

390625*e^(4/3/(64*x^4 - 320*x^3 + 400*x^2 - x))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 e^{\frac {4}{192 x^{4} - 960 x^{3} + 1200 x^{2} - 3 x}} \]

[In]

integrate((-400000000*x**3+1500000000*x**2-1250000000*x+1562500)*exp(1/(192*x**4-960*x**3+1200*x**2-3*x))**4/(
12288*x**8-122880*x**7+460800*x**6-768384*x**5+481920*x**4-2400*x**3+3*x**2),x)

[Out]

390625*exp(4/(192*x**4 - 960*x**3 + 1200*x**2 - 3*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (23) = 46\).

Time = 0.68 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 \, e^{\left (\frac {256 \, x^{2}}{3 \, {\left (64 \, x^{3} - 320 \, x^{2} + 400 \, x - 1\right )}} - \frac {1280 \, x}{3 \, {\left (64 \, x^{3} - 320 \, x^{2} + 400 \, x - 1\right )}} + \frac {1600}{3 \, {\left (64 \, x^{3} - 320 \, x^{2} + 400 \, x - 1\right )}} - \frac {4}{3 \, x}\right )} \]

[In]

integrate((-400000000*x^3+1500000000*x^2-1250000000*x+1562500)*exp(1/(192*x^4-960*x^3+1200*x^2-3*x))^4/(12288*
x^8-122880*x^7+460800*x^6-768384*x^5+481920*x^4-2400*x^3+3*x^2),x, algorithm="maxima")

[Out]

390625*e^(256/3*x^2/(64*x^3 - 320*x^2 + 400*x - 1) - 1280/3*x/(64*x^3 - 320*x^2 + 400*x - 1) + 1600/3/(64*x^3
- 320*x^2 + 400*x - 1) - 4/3/x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625 \, e^{\left (\frac {4}{3 \, {\left (64 \, x^{4} - 320 \, x^{3} + 400 \, x^{2} - x\right )}}\right )} \]

[In]

integrate((-400000000*x^3+1500000000*x^2-1250000000*x+1562500)*exp(1/(192*x^4-960*x^3+1200*x^2-3*x))^4/(12288*
x^8-122880*x^7+460800*x^6-768384*x^5+481920*x^4-2400*x^3+3*x^2),x, algorithm="giac")

[Out]

390625*e^(4/3/(64*x^4 - 320*x^3 + 400*x^2 - x))

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4}{-3 x+1200 x^2-960 x^3+192 x^4}} \left (1562500-1250000000 x+1500000000 x^2-400000000 x^3\right )}{3 x^2-2400 x^3+481920 x^4-768384 x^5+460800 x^6-122880 x^7+12288 x^8} \, dx=390625\,{\mathrm {e}}^{-\frac {4}{3\,\left (-64\,x^4+320\,x^3-400\,x^2+x\right )}} \]

[In]

int(-(exp(-4/(3*x - 1200*x^2 + 960*x^3 - 192*x^4))*(1250000000*x - 1500000000*x^2 + 400000000*x^3 - 1562500))/
(3*x^2 - 2400*x^3 + 481920*x^4 - 768384*x^5 + 460800*x^6 - 122880*x^7 + 12288*x^8),x)

[Out]

390625*exp(-4/(3*(x - 400*x^2 + 320*x^3 - 64*x^4)))