3.66.65 \(\int \frac {1}{375} e^{-x} (-25+e^x (50-2 x)+25 x) \, dx\)

Optimal. Leaf size=34 \[ \frac {1}{5} \left (2+e^2+x+\frac {1}{3} \left (4-x-e^{-x} x-\frac {x^2}{25}\right )\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 22, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6742, 2194, 2176} \begin {gather*} -\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25 + E^x*(50 - 2*x) + 25*x)/(375*E^x),x]

[Out]

-1/375*(25 - x)^2 - x/(15*E^x)

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{375} \int e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx\\ &=\frac {1}{375} \int \left (-25 e^{-x}-2 (-25+x)+25 e^{-x} x\right ) \, dx\\ &=-\frac {1}{375} (25-x)^2-\frac {1}{15} \int e^{-x} \, dx+\frac {1}{15} \int e^{-x} x \, dx\\ &=\frac {e^{-x}}{15}-\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15}+\frac {1}{15} \int e^{-x} \, dx\\ &=-\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.68 \begin {gather*} \frac {2 x}{15}-\frac {e^{-x} x}{15}-\frac {x^2}{375} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + E^x*(50 - 2*x) + 25*x)/(375*E^x),x]

[Out]

(2*x)/15 - x/(15*E^x) - x^2/375

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fricas [A]  time = 0.84, size = 20, normalized size = 0.59 \begin {gather*} -\frac {1}{375} \, {\left ({\left (x^{2} - 50 \, x\right )} e^{x} + 25 \, x\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/375*((-2*x+50)*exp(x)+25*x-25)/exp(x),x, algorithm="fricas")

[Out]

-1/375*((x^2 - 50*x)*e^x + 25*x)*e^(-x)

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giac [A]  time = 0.18, size = 16, normalized size = 0.47 \begin {gather*} -\frac {1}{375} \, x^{2} - \frac {1}{15} \, x e^{\left (-x\right )} + \frac {2}{15} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/375*((-2*x+50)*exp(x)+25*x-25)/exp(x),x, algorithm="giac")

[Out]

-1/375*x^2 - 1/15*x*e^(-x) + 2/15*x

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maple [A]  time = 0.02, size = 17, normalized size = 0.50

method	result	size
default	\(-\frac {x^{2}}{375}+\frac {2 x}{15}-\frac {x \,{\mathrm e}^{-x}}{15}\)	\(17\)
risch	\(-\frac {x^{2}}{375}+\frac {2 x}{15}-\frac {x \,{\mathrm e}^{-x}}{15}\)	\(17\)
norman	\(\left (-\frac {x}{15}+\frac {2 \,{\mathrm e}^{x} x}{15}-\frac {{\mathrm e}^{x} x^{2}}{375}\right ) {\mathrm e}^{-x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/375*((-2*x+50)*exp(x)+25*x-25)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-1/375*x^2+2/15*x-1/15*x/exp(x)

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maxima [A]  time = 0.38, size = 24, normalized size = 0.71 \begin {gather*} -\frac {1}{375} \, x^{2} - \frac {1}{15} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {2}{15} \, x + \frac {1}{15} \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/375*((-2*x+50)*exp(x)+25*x-25)/exp(x),x, algorithm="maxima")

[Out]

-1/375*x^2 - 1/15*(x + 1)*e^(-x) + 2/15*x + 1/15*e^(-x)

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mupad [B]  time = 4.08, size = 12, normalized size = 0.35 \begin {gather*} -\frac {x\,\left (x+25\,{\mathrm {e}}^{-x}-50\right )}{375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*((exp(x)*(2*x - 50))/375 - x/15 + 1/15),x)

[Out]

-(x*(x + 25*exp(-x) - 50))/375

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sympy [A]  time = 0.09, size = 15, normalized size = 0.44 \begin {gather*} - \frac {x^{2}}{375} + \frac {2 x}{15} - \frac {x e^{- x}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/375*((-2*x+50)*exp(x)+25*x-25)/exp(x),x)

[Out]

-x**2/375 + 2*x/15 - x*exp(-x)/15

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