3.486 \(\int \frac{7-2 x+3 x^2-x^3+x^4}{2+x} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^4}{4}-x^3+\frac{9 x^2}{2}-20 x+47 \log (x+2) \]

[Out]

-20*x + (9*x^2)/2 - x^3 + x^4/4 + 47*Log[2 + x]

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Rubi [A]  time = 0.020162, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1850} \[ \frac{x^4}{4}-x^3+\frac{9 x^2}{2}-20 x+47 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(7 - 2*x + 3*x^2 - x^3 + x^4)/(2 + x),x]

[Out]

-20*x + (9*x^2)/2 - x^3 + x^4/4 + 47*Log[2 + x]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{7-2 x+3 x^2-x^3+x^4}{2+x} \, dx &=\int \left (-20+9 x-3 x^2+x^3+\frac{47}{2+x}\right ) \, dx\\ &=-20 x+\frac{9 x^2}{2}-x^3+\frac{x^4}{4}+47 \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0078275, size = 30, normalized size = 1.03 \[ \frac{x^4}{4}-x^3+\frac{9 x^2}{2}-20 x+47 \log (x+2)-70 \]

Antiderivative was successfully verified.

[In]

Integrate[(7 - 2*x + 3*x^2 - x^3 + x^4)/(2 + x),x]

[Out]

-70 - 20*x + (9*x^2)/2 - x^3 + x^4/4 + 47*Log[2 + x]

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Maple [A]  time = 0.003, size = 26, normalized size = 0.9 \begin{align*} -20\,x+{\frac{9\,{x}^{2}}{2}}-{x}^{3}+{\frac{{x}^{4}}{4}}+47\,\ln \left ( 2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^3+3*x^2-2*x+7)/(2+x),x)

[Out]

-20*x+9/2*x^2-x^3+1/4*x^4+47*ln(2+x)

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Maxima [A]  time = 1.32159, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{4} \, x^{4} - x^{3} + \frac{9}{2} \, x^{2} - 20 \, x + 47 \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+3*x^2-2*x+7)/(2+x),x, algorithm="maxima")

[Out]

1/4*x^4 - x^3 + 9/2*x^2 - 20*x + 47*log(x + 2)

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Fricas [A]  time = 0.943417, size = 65, normalized size = 2.24 \begin{align*} \frac{1}{4} \, x^{4} - x^{3} + \frac{9}{2} \, x^{2} - 20 \, x + 47 \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+3*x^2-2*x+7)/(2+x),x, algorithm="fricas")

[Out]

1/4*x^4 - x^3 + 9/2*x^2 - 20*x + 47*log(x + 2)

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Sympy [A]  time = 0.074734, size = 24, normalized size = 0.83 \begin{align*} \frac{x^{4}}{4} - x^{3} + \frac{9 x^{2}}{2} - 20 x + 47 \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x**3+3*x**2-2*x+7)/(2+x),x)

[Out]

x**4/4 - x**3 + 9*x**2/2 - 20*x + 47*log(x + 2)

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Giac [A]  time = 1.09244, size = 35, normalized size = 1.21 \begin{align*} \frac{1}{4} \, x^{4} - x^{3} + \frac{9}{2} \, x^{2} - 20 \, x + 47 \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+3*x^2-2*x+7)/(2+x),x, algorithm="giac")

[Out]

1/4*x^4 - x^3 + 9/2*x^2 - 20*x + 47*log(abs(x + 2))