3.19.48 \(\int \frac {1+x+x^2}{x} \, dx\)

Optimal. Leaf size=11 \[ \frac {x^2}{2}+x+\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14} \begin {gather*} \frac {x^2}{2}+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^2)/x,x]

[Out]

x + x^2/2 + 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 {align*} \int \frac {1+x+x^2}{x} \, dx &=\int \left (1+\frac {1}{x}+x\right ) \, dx\\ &=x+\frac {x^2}{2}+\log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + x^2)/x,x]

[Out]

x + x^2/2 + Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x+x^2}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x + x^2)/x,x]

[Out]

IntegrateAlgebraic[(1 + x + x^2)/x, x]

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fricas [A]  time = 0.39, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/x,x, algorithm="fricas")

[Out]

1/2*x^2 + x + log(x)

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giac [A]  time = 0.15, size = 10, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/x,x, algorithm="giac")

[Out]

1/2*x^2 + x + log(abs(x))

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maple [A]  time = 0.05, size = 10, normalized size = 0.91 \begin {gather*} \frac {x^{2}}{2}+x +\ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x+1)/x,x)

[Out]

x+1/2*x^2+ln(x)

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maxima [A]  time = 0.91, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/x,x, algorithm="maxima")

[Out]

1/2*x^2 + x + log(x)

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mupad [B]  time = 0.02, size = 9, normalized size = 0.82 \begin {gather*} x+\ln \relax (x)+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^2 + 1)/x,x)

[Out]

x + log(x) + x^2/2

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sympy [A]  time = 0.08, size = 8, normalized size = 0.73 \begin {gather*} \frac {x^{2}}{2} + x + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+x+1)/x,x)

[Out]

x**2/2 + x + log(x)

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