3.45.77 \(\int (-1+e^{-2 e^{\frac {625+x \log (2)}{x}} x^2+2 x^3} (e^{\frac {625+x \log (2)}{x}} (1250-4 x)+6 x^2)) \, dx\)

Optimal. Leaf size=22 \[ e^{2 x^2 \left (-2 e^{625/x}+x\right )}-x \]

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Rubi [A]  time = 0.11, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6706} \begin {gather*} e^{2 x^3-4 e^{625/x} x^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^(-2*E^((625 + x*Log[2])/x)*x^2 + 2*x^3)*(E^((625 + x*Log[2])/x)*(1250 - 4*x) + 6*x^2),x]

[Out]

E^(-4*E^(625/x)*x^2 + 2*x^3) - x

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} &=-x+\int e^{-2 e^{\frac {625+x \log (2)}{x}} x^2+2 x^3} \left (e^{\frac {625+x \log (2)}{x}} (1250-4 x)+6 x^2\right ) \, dx\\ &=e^{-4 e^{625/x} x^2+2 x^3}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 24, normalized size = 1.09 \begin {gather*} e^{-4 e^{625/x} x^2+2 x^3}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^(-2*E^((625 + x*Log[2])/x)*x^2 + 2*x^3)*(E^((625 + x*Log[2])/x)*(1250 - 4*x) + 6*x^2),x]

[Out]

E^(-4*E^(625/x)*x^2 + 2*x^3) - x

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fricas [A]  time = 0.70, size = 27, normalized size = 1.23 \begin {gather*} -x + e^{\left (2 \, x^{3} - 2 \, x^{2} e^{\left (\frac {x \log \relax (2) + 625}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+1250)*exp((x*log(2)+625)/x)+6*x^2)*exp(-2*x^2*exp((x*log(2)+625)/x)+2*x^3)-1,x, algorithm="fr
icas")

[Out]

-x + e^(2*x^3 - 2*x^2*e^((x*log(2) + 625)/x))

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giac [A]  time = 0.22, size = 25, normalized size = 1.14 \begin {gather*} -x + e^{\left (2 \, x^{3} - 2 \, x^{2} e^{\left (\frac {625}{x} + \log \relax (2)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+1250)*exp((x*log(2)+625)/x)+6*x^2)*exp(-2*x^2*exp((x*log(2)+625)/x)+2*x^3)-1,x, algorithm="gi
ac")

[Out]

-x + e^(2*x^3 - 2*x^2*e^(625/x + log(2)))

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maple [A]  time = 0.07, size = 21, normalized size = 0.95

method	result	size
risch	\(-x +{\mathrm e}^{2 x^{2} \left (-2 \,{\mathrm e}^{\frac {625}{x}}+x \right )}\)	\(21\)
default	\(-x +{\mathrm e}^{-2 x^{2} {\mathrm e}^{\frac {x \ln \relax (2)+625}{x}}+2 x^{3}}\)	\(28\)
norman	\(-x +{\mathrm e}^{-2 x^{2} {\mathrm e}^{\frac {x \ln \relax (2)+625}{x}}+2 x^{3}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x+1250)*exp((x*ln(2)+625)/x)+6*x^2)*exp(-2*x^2*exp((x*ln(2)+625)/x)+2*x^3)-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(2*x^2*(-2*exp(625/x)+x))

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maxima [A]  time = 0.43, size = 22, normalized size = 1.00 \begin {gather*} -x + e^{\left (2 \, x^{3} - 4 \, x^{2} e^{\frac {625}{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+1250)*exp((x*log(2)+625)/x)+6*x^2)*exp(-2*x^2*exp((x*log(2)+625)/x)+2*x^3)-1,x, algorithm="ma
xima")

[Out]

-x + e^(2*x^3 - 4*x^2*e^(625/x))

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mupad [B]  time = 3.38, size = 23, normalized size = 1.05 \begin {gather*} {\mathrm {e}}^{-4\,x^2\,{\mathrm {e}}^{625/x}}\,{\mathrm {e}}^{2\,x^3}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- exp(2*x^3 - 2*x^2*exp((x*log(2) + 625)/x))*(exp((x*log(2) + 625)/x)*(4*x - 1250) - 6*x^2) - 1,x)

[Out]

exp(-4*x^2*exp(625/x))*exp(2*x^3) - x

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sympy [A]  time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} - x + e^{2 x^{3} - 2 x^{2} e^{\frac {x \log {\relax (2 )} + 625}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+1250)*exp((x*ln(2)+625)/x)+6*x**2)*exp(-2*x**2*exp((x*ln(2)+625)/x)+2*x**3)-1,x)

[Out]

-x + exp(2*x**3 - 2*x**2*exp((x*log(2) + 625)/x))

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