3.83.94 \(\int \frac {5-2 e^2+6 x^2}{2 x^2} \, dx\)

Optimal. Leaf size=24 \[ 1+\frac {e^2+\frac {1}{2} (-5-x)}{x}+3 x+\log (5) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14} \begin {gather*} 3 x-\frac {5-2 e^2}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - 2*E^2 + 6*x^2)/(2*x^2),x]

[Out]

-1/2*(5 - 2*E^2)/x + 3*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 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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {5-2 e^2+6 x^2}{x^2} \, dx\\ &=\frac {1}{2} \int \left (6+\frac {5-2 e^2}{x^2}\right ) \, dx\\ &=-\frac {5-2 e^2}{2 x}+3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.75 \begin {gather*} -\frac {5-2 e^2}{2 x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 2*E^2 + 6*x^2)/(2*x^2),x]

[Out]

-1/2*(5 - 2*E^2)/x + 3*x

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fricas [A]  time = 0.69, size = 16, normalized size = 0.67 \begin {gather*} \frac {6 \, x^{2} + 2 \, e^{2} - 5}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*exp(2)+6*x^2+5)/x^2,x, algorithm="fricas")

[Out]

1/2*(6*x^2 + 2*e^2 - 5)/x

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giac [A]  time = 0.16, size = 15, normalized size = 0.62 \begin {gather*} 3 \, x + \frac {2 \, e^{2} - 5}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*exp(2)+6*x^2+5)/x^2,x, algorithm="giac")

[Out]

3*x + 1/2*(2*e^2 - 5)/x

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maple [A]  time = 0.03, size = 14, normalized size = 0.58

method	result	size
norman	\(\frac {3 x^{2}+{\mathrm e}^{2}-\frac {5}{2}}{x}\)	\(14\)
default	\(3 x -\frac {-2 \,{\mathrm e}^{2}+5}{2 x}\)	\(16\)
risch	\(3 x +\frac {{\mathrm e}^{2}}{x}-\frac {5}{2 x}\)	\(16\)
gosper	\(\frac {6 x^{2}+2 \,{\mathrm e}^{2}-5}{2 x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-2*exp(2)+6*x^2+5)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 15, normalized size = 0.62 \begin {gather*} 3 \, x + \frac {2 \, e^{2} - 5}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*exp(2)+6*x^2+5)/x^2,x, algorithm="maxima")

[Out]

3*x + 1/2*(2*e^2 - 5)/x

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mupad [B]  time = 5.03, size = 12, normalized size = 0.50 \begin {gather*} 3\,x+\frac {{\mathrm {e}}^2-\frac {5}{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - exp(2) + 5/2)/x^2,x)

[Out]

3*x + (exp(2) - 5/2)/x

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sympy [A]  time = 0.12, size = 12, normalized size = 0.50 \begin {gather*} 3 x + \frac {-5 + 2 e^{2}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*exp(2)+6*x**2+5)/x**2,x)

[Out]

3*x + (-5 + 2*exp(2))/(2*x)

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