3.353 \(\int \frac{4+3 x+x^2}{x+x^2} \, dx\)

Optimal. Leaf size=12 \[ x+4 \log (x)-2 \log (x+1) \]

[Out]

x + 4*Log[x] - 2*Log[1 + x]

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Rubi [A]  time = 0.0176414, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1593, 893} \[ x+4 \log (x)-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(4 + 3*x + x^2)/(x + x^2),x]

[Out]

x + 4*Log[x] - 2*Log[1 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{4+3 x+x^2}{x+x^2} \, dx &=\int \frac{4+3 x+x^2}{x (1+x)} \, dx\\ &=\int \left (1+\frac{4}{x}-\frac{2}{1+x}\right ) \, dx\\ &=x+4 \log (x)-2 \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0032719, size = 12, normalized size = 1. \[ x+4 \log (x)-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 3*x + x^2)/(x + x^2),x]

[Out]

x + 4*Log[x] - 2*Log[1 + x]

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Maple [A]  time = 0.004, size = 13, normalized size = 1.1 \begin{align*} x+4\,\ln \left ( x \right ) -2\,\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+3*x+4)/(x^2+x),x)

[Out]

x+4*ln(x)-2*ln(1+x)

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Maxima [A]  time = 1.00843, size = 16, normalized size = 1.33 \begin{align*} x - 2 \, \log \left (x + 1\right ) + 4 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3*x+4)/(x^2+x),x, algorithm="maxima")

[Out]

x - 2*log(x + 1) + 4*log(x)

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Fricas [A]  time = 1.44783, size = 39, normalized size = 3.25 \begin{align*} x - 2 \, \log \left (x + 1\right ) + 4 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3*x+4)/(x^2+x),x, algorithm="fricas")

[Out]

x - 2*log(x + 1) + 4*log(x)

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Sympy [A]  time = 0.092957, size = 12, normalized size = 1. \begin{align*} x + 4 \log{\left (x \right )} - 2 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+3*x+4)/(x**2+x),x)

[Out]

x + 4*log(x) - 2*log(x + 1)

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Giac [A]  time = 1.22549, size = 19, normalized size = 1.58 \begin{align*} x - 2 \, \log \left ({\left | x + 1 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3*x+4)/(x^2+x),x, algorithm="giac")

[Out]

x - 2*log(abs(x + 1)) + 4*log(abs(x))