3.80.26 \(\int e^{-1/x} (-4 x^3-20 x^4) \, dx\)

Optimal. Leaf size=27 \[ 4 e^{-1/x} x^2 \left (-x^2+(1-x) x^2\right ) \]

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Rubi [C]  time = 0.07, antiderivative size = 15, normalized size of antiderivative = 0.56, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1593, 2226, 2218} \begin {gather*} -20 \Gamma \left (-5,\frac {1}{x}\right )-4 \Gamma \left (-4,\frac {1}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^3 - 20*x^4)/E^x^(-1),x]

[Out]

-20*Gamma[-5, x^(-1)] - 4*Gamma[-4, x^(-1)]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2218

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

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-1/x} (-4-20 x) x^3 \, dx\\ &=\int \left (-4 e^{-1/x} x^3-20 e^{-1/x} x^4\right ) \, dx\\ &=-\left (4 \int e^{-1/x} x^3 \, dx\right )-20 \int e^{-1/x} x^4 \, dx\\ &=-20 \Gamma \left (-5,\frac {1}{x}\right )-4 \Gamma \left (-4,\frac {1}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.01, size = 15, normalized size = 0.56 \begin {gather*} -4 \left (5 \Gamma \left (-5,\frac {1}{x}\right )+\Gamma \left (-4,\frac {1}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^3 - 20*x^4)/E^x^(-1),x]

[Out]

-4*(5*Gamma[-5, x^(-1)] + Gamma[-4, x^(-1)])

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fricas [A]  time = 1.08, size = 11, normalized size = 0.41 \begin {gather*} -4 \, x^{5} e^{\left (-\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^4-4*x^3)/exp(1/x),x, algorithm="fricas")

[Out]

-4*x^5*e^(-1/x)

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giac [A]  time = 0.12, size = 11, normalized size = 0.41 \begin {gather*} -4 \, x^{5} e^{\left (-\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^4-4*x^3)/exp(1/x),x, algorithm="giac")

[Out]

-4*x^5*e^(-1/x)

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maple [A]  time = 0.06, size = 12, normalized size = 0.44

method	result	size
gosper	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
derivativedivides	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
default	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
norman	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
risch	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
meijerg	\(-4 x^{5}+4 x^{4}-2 x^{3}+\frac {2 x^{2}}{3}-\frac {x}{6}+\frac {1}{30}+\frac {x^{5} \left (-\frac {137}{x^{5}}+\frac {300}{x^{4}}-\frac {600}{x^{3}}+\frac {1200}{x^{2}}-\frac {1800}{x}+1440\right )}{360}-\frac {x^{5} \left (\frac {6}{x^{4}}-\frac {6}{x^{3}}+\frac {12}{x^{2}}-\frac {36}{x}+144\right ) {\mathrm e}^{-\frac {1}{x}}}{36}+\frac {x^{4} \left (\frac {125}{x^{4}}-\frac {240}{x^{3}}+\frac {360}{x^{2}}-\frac {480}{x}+360\right )}{360}-\frac {x^{4} \left (-\frac {5}{x^{3}}+\frac {5}{x^{2}}-\frac {10}{x}+30\right ) {\mathrm e}^{-\frac {1}{x}}}{30}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20*x^4-4*x^3)/exp(1/x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.48, size = 15, normalized size = 0.56 \begin {gather*} -4 \, \Gamma \left (-4, \frac {1}{x}\right ) - 20 \, \Gamma \left (-5, \frac {1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^4-4*x^3)/exp(1/x),x, algorithm="maxima")

[Out]

-4*gamma(-4, 1/x) - 20*gamma(-5, 1/x)

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mupad [B]  time = 4.61, size = 11, normalized size = 0.41 \begin {gather*} -4\,x^5\,{\mathrm {e}}^{-\frac {1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-1/x)*(4*x^3 + 20*x^4),x)

[Out]

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

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sympy [A]  time = 0.09, size = 10, normalized size = 0.37 \begin {gather*} - 4 x^{5} e^{- \frac {1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x**4-4*x**3)/exp(1/x),x)

[Out]

-4*x**5*exp(-1/x)

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