3.41.20 $\int \frac{e^{-2e}(16+e^{2e}(-5+2x^3))}{x^2}dx$

Optimal. Leaf size=29 $$-\frac{-5+16e^{-2e}}{x}+x(-\frac{\sqrt{e}}{x}+x)$$

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 2, integrand size = 24, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.083, Rules used = {12, 14} $$\begin{array}{c}{x}^2+\frac{5-16e^{-2e}}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(16 + E^(2*E)*(-5 + 2*x^3))/(E^(2*E)*x^2),x]

[Out]

(5 - 16/E^(2*E))/x + x^2

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =e^{-2e}\int \frac{16+e^{2e}(-5+2x^3)}{x^2}dx\\ & =e^{-2e}\int (-\frac{-16+5e^{2e}}{x^2}+2e^{2e}x)dx\\ & =\frac{5-16e^{-2e}}{x}+x^2\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 18, normalized size = 0.62 $$\begin{array}{c}-\frac{-5+16e^{-2e}}{x}+x^2\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(16 + E^(2*E)*(-5 + 2*x^3))/(E^(2*E)*x^2),x]

[Out]

-((-5 + 16/E^(2*E))/x) + x^2

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 22, normalized size = 0.76 $$\begin{array}{c}\frac{((x^3+5)e^{\left(2e\right)}-16)e^{(-2e)}}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-5)*exp(exp(1))^2+16)/x^2/exp(exp(1))^2,x, algorithm="fricas")

[Out]

((x^3 + 5)*e^(2*e) - 16)*e^(-2*e)/x

________________________________________________________________________________________

giac [A]  time = 0.17, size = 29, normalized size = 1.00 $$\begin{array}{c}(x^2{e}^{\left(2e\right)}+\frac{5e^{\left(2e\right)}-16}{x})e^{(-2e)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-5)*exp(exp(1))^2+16)/x^2/exp(exp(1))^2,x, algorithm="giac")

[Out]

(x^2*e^(2*e) + (5*e^(2*e) - 16)/x)*e^(-2*e)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 28, normalized size = 0.97

method	result	size
gosper	$\frac{(x^3{\mathrm{e}}^{2\mathrm{e}}+5{\mathrm{e}}^{2\mathrm{e}}-16){\mathrm{e}}^{-2\mathrm{e}}}{x}$	$28$
risch	$x^2+\frac{5{\mathrm{e}}^{-2\mathrm{e}}{\mathrm{e}}^{2\mathrm{e}}}{x}-\frac{16{\mathrm{e}}^{-2\mathrm{e}}}{x}$	$30$
default	${\mathrm{e}}^{-2\mathrm{e}}({\mathrm{e}}^{2\mathrm{e}}{x}^2-\frac{-5{\mathrm{e}}^{2\mathrm{e}}+16}{x})$	$31$
norman	$\frac{(x^3{\mathrm{e}}^{\mathrm{e}}+{\mathrm{e}}^{-\mathrm{e}}(5{\mathrm{e}}^{2\mathrm{e}}-16)){\mathrm{e}}^{-\mathrm{e}}}{x}$	$33$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3-5)*exp(exp(1))^2+16)/x^2/exp(exp(1))^2,x,method=_RETURNVERBOSE)

[Out]

(x^3*exp(exp(1))^2+5*exp(exp(1))^2-16)/exp(exp(1))^2/x

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 29, normalized size = 1.00 $$\begin{array}{c}(x^2{e}^{\left(2e\right)}+\frac{5e^{\left(2e\right)}-16}{x})e^{(-2e)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-5)*exp(exp(1))^2+16)/x^2/exp(exp(1))^2,x, algorithm="maxima")

[Out]

(x^2*e^(2*e) + (5*e^(2*e) - 16)/x)*e^(-2*e)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 18, normalized size = 0.62 $$\begin{array}{c}{x}^2-\frac{16{\mathrm{e}}^{-2\mathrm{e}}-5}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*exp(1))*(exp(2*exp(1))*(2*x^3 - 5) + 16))/x^2,x)

[Out]

x^2 - (16*exp(-2*exp(1)) - 5)/x

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 27, normalized size = 0.93 $$\begin{array}{c}\frac{x^2{e}^{2e}+\frac{-16+5e^{2e}}{x}}{e^{2e}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3-5)*exp(exp(1))**2+16)/x**2/exp(exp(1))**2,x)

[Out]

(x**2*exp(2*E) + (-16 + 5*exp(2*E))/x)*exp(-2*E)

________________________________________________________________________________________