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

Optimal. Leaf size=8 \[ -1+x+x^2+\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.88, 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^2+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x + 2*x^2)/x,x]

[Out]

x + x^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 {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {1}{x}+2 x\right ) \, dx\\ &=x+x^2+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.88 \begin {gather*} x+x^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + 2*x^2)/x,x]

[Out]

x + x^2 + Log[x]

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fricas [A]  time = 0.63, size = 7, normalized size = 0.88 \begin {gather*} x^{2} + x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x + log(x)

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giac [A]  time = 0.44, size = 8, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x + log(abs(x))

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maple [A]  time = 0.02, size = 8, normalized size = 1.00

method	result	size
default	\(\ln \relax (x )+x^{2}+x\)	\(8\)
norman	\(\ln \relax (x )+x^{2}+x\)	\(8\)
risch	\(\ln \relax (x )+x^{2}+x\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+x+1)/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)+x^2+x

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maxima [A]  time = 0.38, size = 7, normalized size = 0.88 \begin {gather*} x^{2} + x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + x + log(x)

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mupad [B]  time = 0.02, size = 7, normalized size = 0.88 \begin {gather*} x+\ln \relax (x)+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*x^2 + 1)/x,x)

[Out]

x + log(x) + x^2

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sympy [A]  time = 0.06, size = 7, normalized size = 0.88 \begin {gather*} x^{2} + x + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 + x + log(x)

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