3.2179 \(\int \frac{1+x+x^2}{x} \, dx\)

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

[Out]

x + x^2/2 + Log[x]

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Rubi [A]  time = 0.0027441, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {14} \[ \frac{x^2}{2}+x+\log (x) \]

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*}

Mathematica [A]  time = 0.00061, size = 11, normalized size = 1. \[ \frac{x^2}{2}+x+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + x^2)/x,x]

[Out]

x + x^2/2 + Log[x]

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Maple [A]  time = 0.039, size = 10, normalized size = 0.9 \begin{align*} x+{\frac{{x}^{2}}{2}}+\ln \left ( x \right ) \end{align*}

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 = 0.997564, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} + x + \log \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.6929, size = 30, normalized size = 2.73 \begin{align*} \frac{1}{2} \, x^{2} + x + \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.073268, size = 8, normalized size = 0.73 \begin{align*} \frac{x^{2}}{2} + x + \log{\left (x \right )} \end{align*}

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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Giac [A]  time = 1.14119, size = 14, normalized size = 1.27 \begin{align*} \frac{1}{2} \, x^{2} + x + \log \left ({\left | x \right |}\right ) \end{align*}

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))