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

Optimal. Leaf size=17 \[ \frac {x^2}{2}-2 x+6 \log (x+2) \]

[Out]

-2*x+1/2*x^2+6*ln(2+x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {697} \[ \frac {x^2}{2}-2 x+6 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x^2)/(2 + x),x]

[Out]

-2*x + x^2/2 + 6*Log[2 + x]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {2+x^2}{2+x} \, dx &=\int \left (-2+x+\frac {6}{2+x}\right ) \, dx\\ &=-2 x+\frac {x^2}{2}+6 \log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.06 \[ \frac {x^2}{2}-2 x+6 \log (x+2)-6 \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x^2)/(2 + x),x]

[Out]

-6 - 2*x + x^2/2 + 6*Log[2 + x]

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fricas [A]  time = 0.58, size = 15, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 2*x + 6*log(x + 2)

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giac [A]  time = 0.37, size = 16, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 2*x + 6*log(abs(x + 2))

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {x^{2}}{2}-2 x +6 \ln \left (x +2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x+1/2*x^2+6*ln(x+2)

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maxima [A]  time = 0.44, size = 15, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 2*x + 6*log(x + 2)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.88 \[ 6\,\ln \left (x+2\right )-2\,x+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 2)/(x + 2),x)

[Out]

6*log(x + 2) - 2*x + x^2/2

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sympy [A]  time = 0.07, size = 14, normalized size = 0.82 \[ \frac {x^{2}}{2} - 2 x + 6 \log {\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - 2*x + 6*log(x + 2)

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