3.2175 \(\int \frac{2+2 x+x^2}{2+x} \, dx\)

Optimal. Leaf size=14 \[ \frac{x^2}{2}+2 \log (x+2) \]

[Out]

x^2/2 + 2*Log[2 + x]

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Rubi [A]  time = 0.008025, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {698} \[ \frac{x^2}{2}+2 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x + x^2)/(2 + x),x]

[Out]

x^2/2 + 2*Log[2 + x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0034979, size = 15, normalized size = 1.07 \[ \frac{1}{2} \left (x^2+4 \log (x+2)-4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x + x^2)/(2 + x),x]

[Out]

(-4 + x^2 + 4*Log[2 + x])/2

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Maple [A]  time = 0.039, size = 13, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+2\,\ln \left ( 2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.983942, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*log(x + 2)

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Fricas [A]  time = 2.01117, size = 32, normalized size = 2.29 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*log(x + 2)

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Sympy [A]  time = 0.077718, size = 10, normalized size = 0.71 \begin{align*} \frac{x^{2}}{2} + 2 \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + 2*log(x + 2)

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Giac [A]  time = 1.11415, size = 18, normalized size = 1.29 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*log(abs(x + 2))