\(\int e^{-1/x} (-4 x^3-20 x^4) \, dx\) [7926]

Optimal result

Integrand size = 19, antiderivative size = 27 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=4 e^{-1/x} x^2 \left (-x^2+(1-x) x^2\right ) \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1607, 2258, 2250} \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-20 \Gamma \left (-5,\frac {1}{x}\right )-4 \Gamma \left (-4,\frac {1}{x}\right ) \]

[In]

Int[(-4*x^3 - 20*x^4)/E^x^(-1),x]

[Out]

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

Rule 1607

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 2250

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

Rule 2258

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{align*} \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{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-4 \left (5 \Gamma \left (-5,\frac {1}{x}\right )+\Gamma \left (-4,\frac {1}{x}\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
parallelrisch	\(-4 x^{5} {\mathrm e}^{-\frac {1}{x}}\)	\(12\)
meijerg	\(-4 x^{5}+4 x^{4}-2 x^{3}+\frac {2 x^{2}}{3}-\frac {x}{6}+\frac {1}{30}+\frac {x^{5} \left (-\frac {137}{x^{5}}+\frac {300}{x^{4}}-\frac {600}{x^{3}}+\frac {1200}{x^{2}}-\frac {1800}{x}+1440\right )}{360}-\frac {x^{5} \left (\frac {6}{x^{4}}-\frac {6}{x^{3}}+\frac {12}{x^{2}}-\frac {36}{x}+144\right ) {\mathrm e}^{-\frac {1}{x}}}{36}+\frac {x^{4} \left (\frac {125}{x^{4}}-\frac {240}{x^{3}}+\frac {360}{x^{2}}-\frac {480}{x}+360\right )}{360}-\frac {x^{4} \left (-\frac {5}{x^{3}}+\frac {5}{x^{2}}-\frac {10}{x}+30\right ) {\mathrm e}^{-\frac {1}{x}}}{30}\)	\(146\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-4 \, x^{5} e^{\left (-\frac {1}{x}\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.37 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=- 4 x^{5} e^{- \frac {1}{x}} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-4 \, \Gamma \left (-4, \frac {1}{x}\right ) - 20 \, \Gamma \left (-5, \frac {1}{x}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-4 \, x^{5} e^{\left (-\frac {1}{x}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.95 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int e^{-1/x} \left (-4 x^3-20 x^4\right ) \, dx=-4\,x^5\,{\mathrm {e}}^{-\frac {1}{x}} \]

[In]

int(-exp(-1/x)*(4*x^3 + 20*x^4),x)

[Out]

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