3.27.7 \(\int \frac {1}{4} (625-1250 e^{1-e^5+x^2} x) \, dx\)

Optimal. Leaf size=20 \[ \frac {625}{4} \left (-e^{1-e^5+x^2}+x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2209} \begin {gather*} \frac {625 x}{4}-\frac {625}{4} e^{x^2-e^5+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(625 - 1250*E^(1 - E^5 + x^2)*x)/4,x]

[Out]

(-625*E^(1 - E^5 + x^2))/4 + (625*x)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (625-1250 e^{1-e^5+x^2} x\right ) \, dx\\ &=\frac {625 x}{4}-\frac {625}{2} \int e^{1-e^5+x^2} x \, dx\\ &=-\frac {625}{4} e^{1-e^5+x^2}+\frac {625 x}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {625}{4} \left (-e^{1-e^5+x^2}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(625 - 1250*E^(1 - E^5 + x^2)*x)/4,x]

[Out]

(625*(-E^(1 - E^5 + x^2) + x))/4

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fricas [A]  time = 0.54, size = 16, normalized size = 0.80 \begin {gather*} \frac {625}{4} \, x - \frac {625}{4} \, e^{\left (x^{2} - e^{5} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-625/2*x*exp(-exp(5)+x^2+1)+625/4,x, algorithm="fricas")

[Out]

625/4*x - 625/4*e^(x^2 - e^5 + 1)

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giac [A]  time = 0.53, size = 16, normalized size = 0.80 \begin {gather*} \frac {625}{4} \, x - \frac {625}{4} \, e^{\left (x^{2} - e^{5} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-625/2*x*exp(-exp(5)+x^2+1)+625/4,x, algorithm="giac")

[Out]

625/4*x - 625/4*e^(x^2 - e^5 + 1)

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maple [A]  time = 0.01, size = 17, normalized size = 0.85

method	result	size
default	\(\frac {625 x}{4}-\frac {625 \,{\mathrm e}^{-{\mathrm e}^{5}+x^{2}+1}}{4}\)	\(17\)
norman	\(\frac {625 x}{4}-\frac {625 \,{\mathrm e}^{-{\mathrm e}^{5}+x^{2}+1}}{4}\)	\(17\)
risch	\(\frac {625 x}{4}-\frac {625 \,{\mathrm e}^{-{\mathrm e}^{5}+x^{2}+1}}{4}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-625/2*x*exp(-exp(5)+x^2+1)+625/4,x,method=_RETURNVERBOSE)

[Out]

625/4*x-625/4*exp(-exp(5)+x^2+1)

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maxima [A]  time = 0.38, size = 16, normalized size = 0.80 \begin {gather*} \frac {625}{4} \, x - \frac {625}{4} \, e^{\left (x^{2} - e^{5} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-625/2*x*exp(-exp(5)+x^2+1)+625/4,x, algorithm="maxima")

[Out]

625/4*x - 625/4*e^(x^2 - e^5 + 1)

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mupad [B]  time = 1.57, size = 17, normalized size = 0.85 \begin {gather*} \frac {625\,x}{4}-\frac {625\,{\mathrm {e}}^{-{\mathrm {e}}^5}\,{\mathrm {e}}^{x^2}\,\mathrm {e}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(625/4 - (625*x*exp(x^2 - exp(5) + 1))/2,x)

[Out]

(625*x)/4 - (625*exp(-exp(5))*exp(x^2)*exp(1))/4

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sympy [A]  time = 0.09, size = 17, normalized size = 0.85 \begin {gather*} \frac {625 x}{4} - \frac {625 e^{x^{2} - e^{5} + 1}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-625/2*x*exp(-exp(5)+x**2+1)+625/4,x)

[Out]

625*x/4 - 625*exp(x**2 - exp(5) + 1)/4

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