3.52.75 \(\int \frac {-4 e^{4/x}+14 x^2+e^e x^2}{x^2} \, dx\)

Optimal. Leaf size=17 \[ -4+e^{4/x}+14 x+e^e x \]

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 14, 2209} \begin {gather*} \left (14+e^e\right ) x+e^{4/x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^(4/x) + 14*x^2 + E^E*x^2)/x^2,x]

[Out]

E^(4/x) + (14 + E^E)*x

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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]]

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} &=\int \frac {-4 e^{4/x}+\left (14+e^e\right ) x^2}{x^2} \, dx\\ &=\int \left (14 \left (1+\frac {e^e}{14}\right )-\frac {4 e^{4/x}}{x^2}\right ) \, dx\\ &=\left (14+e^e\right ) x-4 \int \frac {e^{4/x}}{x^2} \, dx\\ &=e^{4/x}+\left (14+e^e\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} e^{4/x}+14 x+e^e x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^(4/x) + 14*x^2 + E^E*x^2)/x^2,x]

[Out]

E^(4/x) + 14*x + E^E*x

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fricas [A]  time = 0.59, size = 15, normalized size = 0.88 \begin {gather*} x e^{e} + 14 \, x + e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(exp(1))-4*exp(4/x)+14*x^2)/x^2,x, algorithm="fricas")

[Out]

x*e^e + 14*x + e^(4/x)

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giac [A]  time = 0.12, size = 17, normalized size = 1.00 \begin {gather*} x {\left (\frac {e^{\frac {4}{x}}}{x} + e^{e} + 14\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(exp(1))-4*exp(4/x)+14*x^2)/x^2,x, algorithm="giac")

[Out]

x*(e^(4/x)/x + e^e + 14)

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maple [A]  time = 0.08, size = 16, normalized size = 0.94

method	result	size
derivativedivides	\(14 x +x \,{\mathrm e}^{{\mathrm e}}+{\mathrm e}^{\frac {4}{x}}\)	\(16\)
default	\(14 x +x \,{\mathrm e}^{{\mathrm e}}+{\mathrm e}^{\frac {4}{x}}\)	\(16\)
risch	\(14 x +x \,{\mathrm e}^{{\mathrm e}}+{\mathrm e}^{\frac {4}{x}}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{\frac {4}{x}}+\left ({\mathrm e}^{{\mathrm e}}+14\right ) x^{2}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(exp(1))-4*exp(4/x)+14*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

14*x+x*exp(exp(1))+exp(4/x)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} x e^{e} + 14 \, x + e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(exp(1))-4*exp(4/x)+14*x^2)/x^2,x, algorithm="maxima")

[Out]

x*e^e + 14*x + e^(4/x)

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mupad [B]  time = 3.22, size = 15, normalized size = 0.88 \begin {gather*} 14\,x+{\mathrm {e}}^{4/x}+x\,{\mathrm {e}}^{\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(exp(1)) - 4*exp(4/x) + 14*x^2)/x^2,x)

[Out]

14*x + exp(4/x) + x*exp(exp(1))

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sympy [A]  time = 0.10, size = 12, normalized size = 0.71 \begin {gather*} x \left (14 + e^{e}\right ) + e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(exp(1))-4*exp(4/x)+14*x**2)/x**2,x)

[Out]

x*(14 + exp(E)) + exp(4/x)

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