3.1.28 \(\int \frac {5-x+x^2}{x^2} \, dx\)

Optimal. Leaf size=12 \[ 3-\frac {5}{x}+x-\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.92, 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 {5}{x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x + x^2)/x^2,x]

[Out]

-5/x + 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 (1+\frac {5}{x^2}-\frac {1}{x}\right ) \, dx\\ &=-\frac {5}{x}+x-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} -\frac {5}{x}+x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x + x^2)/x^2,x]

[Out]

-5/x + x - Log[x]

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fricas [A]  time = 0.88, size = 14, normalized size = 1.17 \begin {gather*} \frac {x^{2} - x \log \relax (x) - 5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - x*log(x) - 5)/x

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giac [A]  time = 0.40, size = 12, normalized size = 1.00 \begin {gather*} x - \frac {5}{x} - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 5/x - log(abs(x))

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maple [A]  time = 0.02, size = 12, normalized size = 1.00

method	result	size
default	\(x -\ln \relax (x )-\frac {5}{x}\)	\(12\)
risch	\(x -\ln \relax (x )-\frac {5}{x}\)	\(12\)
norman	\(\frac {x^{2}-5}{x}-\ln \relax (x )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-ln(x)-5/x

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maxima [A]  time = 0.41, size = 11, normalized size = 0.92 \begin {gather*} x - \frac {5}{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 5/x - log(x)

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mupad [B]  time = 0.02, size = 11, normalized size = 0.92 \begin {gather*} x-\ln \relax (x)-\frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - x + 5)/x^2,x)

[Out]

x - log(x) - 5/x

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sympy [A]  time = 0.06, size = 7, normalized size = 0.58 \begin {gather*} x - \log {\relax (x )} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x) - 5/x

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