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

Optimal. Leaf size=26 \[ 390625 e^{\frac {4}{3 \left (-x+16 x^2 (-5+2 x)^2\right )}} \]

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Rubi [A]  time = 0.30, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6706} \begin {gather*} 390625 e^{-\frac {4}{3 \left (-64 x^4+320 x^3-400 x^2+x\right )}} \end {gather*}

Antiderivative was successfully verified.

[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 6706

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

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Mathematica [A]  time = 0.38, size = 28, normalized size = 1.08 \begin {gather*} 390625 e^{\frac {4}{3 x \left (-1+400 x-320 x^2+64 x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)))

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fricas [A]  time = 0.66, size = 26, normalized size = 1.00 \begin {gather*} 390625 \, e^{\left (\frac {4}{3 \, {\left (64 \, x^{4} - 320 \, x^{3} + 400 \, x^{2} - x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.17, size = 26, normalized size = 1.00 \begin {gather*} 390625 \, e^{\left (\frac {4}{3 \, {\left (64 \, x^{4} - 320 \, x^{3} + 400 \, x^{2} - x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.08, size = 26, normalized size = 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	\(390625 \,{\mathrm e}^{\frac {4}{3 x \left (64 x^{3}-320 x^{2}+400 x -1\right )}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [B]  time = 0.89, size = 70, normalized size = 2.69 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.04, size = 26, normalized size = 1.00 \begin {gather*} 390625\,{\mathrm {e}}^{-\frac {4}{3\,\left (-64\,x^4+320\,x^3-400\,x^2+x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 0.30, size = 22, normalized size = 0.85 \begin {gather*} 390625 e^{\frac {4}{192 x^{4} - 960 x^{3} + 1200 x^{2} - 3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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