3.7.13 \(\int \frac {-3+e^5+x^2}{x^2} \, dx\)

Optimal. Leaf size=13 \[ \frac {3-e^5}{x}+x \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14} \begin {gather*} x+\frac {3-e^5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + E^5 + x^2)/x^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {3-e^5}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^5 + x^2)/x^2,x]

[Out]

(3 - E^5)/x + x

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fricas [A]  time = 0.54, size = 13, normalized size = 1.00 \begin {gather*} \frac {x^{2} - e^{5} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - e^5 + 3)/x

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giac [A]  time = 0.37, size = 11, normalized size = 0.85 \begin {gather*} x - \frac {e^{5} - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - (e^5 - 3)/x

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maple [A]  time = 0.02, size = 12, normalized size = 0.92

method	result	size
default	\(x -\frac {{\mathrm e}^{5}-3}{x}\)	\(12\)
norman	\(\frac {x^{2}+3-{\mathrm e}^{5}}{x}\)	\(14\)
gosper	\(-\frac {-x^{2}+{\mathrm e}^{5}-3}{x}\)	\(15\)
risch	\(\frac {3}{x}-\frac {{\mathrm e}^{5}}{x}+x\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-(exp(5)-3)/x

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maxima [A]  time = 0.44, size = 11, normalized size = 0.85 \begin {gather*} x - \frac {e^{5} - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - (e^5 - 3)/x

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mupad [B]  time = 0.03, size = 11, normalized size = 0.85 \begin {gather*} x-\frac {{\mathrm {e}}^5-3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5) + x^2 - 3)/x^2,x)

[Out]

x - (exp(5) - 3)/x

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sympy [A]  time = 0.08, size = 7, normalized size = 0.54 \begin {gather*} x + \frac {3 - e^{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + (3 - exp(5))/x

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