3.39.12 \(\int \frac {e^3+x+2 x^2}{x^2} \, dx\)

Optimal. Leaf size=20 \[ -6+e-\frac {e^3}{x}+2 x+\log \left (\frac {x}{25}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} 2 x-\frac {e^3}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^3 + x + 2*x^2)/x^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.70 \begin {gather*} -\frac {e^3}{x}+2 x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3 + x + 2*x^2)/x^2,x]

[Out]

-(E^3/x) + 2*x + Log[x]

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fricas [A]  time = 0.50, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, x^{2} + x \log \relax (x) - e^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^2 + x*log(x) - e^3)/x

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giac [A]  time = 0.19, size = 14, normalized size = 0.70 \begin {gather*} 2 \, x - \frac {e^{3}}{x} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - e^3/x + log(abs(x))

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maple [A]  time = 0.03, size = 14, normalized size = 0.70

method	result	size
default	\(2 x +\ln \relax (x )-\frac {{\mathrm e}^{3}}{x}\)	\(14\)
risch	\(2 x +\ln \relax (x )-\frac {{\mathrm e}^{3}}{x}\)	\(14\)
norman	\(\frac {2 x^{2}-{\mathrm e}^{3}}{x}+\ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x+ln(x)-exp(3)/x

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maxima [A]  time = 0.37, size = 13, normalized size = 0.65 \begin {gather*} 2 \, x - \frac {e^{3}}{x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - e^3/x + log(x)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.65 \begin {gather*} 2\,x+\ln \relax (x)-\frac {{\mathrm {e}}^3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(3) + 2*x^2)/x^2,x)

[Out]

2*x + log(x) - exp(3)/x

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sympy [A]  time = 0.08, size = 10, normalized size = 0.50 \begin {gather*} 2 x + \log {\relax (x )} - \frac {e^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(x) - exp(3)/x

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