3.4.25 \(\int \frac {2+5 x+3 x^2+2 x^3}{1+x+x^2} \, dx\) [325]

Optimal. Leaf size=12 \[ x+x^2+\log \left (1+x+x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1671, 642} \begin {gather*} x^2+\log \left (x^2+x+1\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x + 3*x^2 + 2*x^3)/(1 + x + x^2),x]

[Out]

x + x^2 + Log[1 + x + x^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1671

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {2+5 x+3 x^2+2 x^3}{1+x+x^2} \, dx &=\int \left (1+2 x+\frac {1+2 x}{1+x+x^2}\right ) \, dx\\ &=x+x^2+\int \frac {1+2 x}{1+x+x^2} \, dx\\ &=x+x^2+\log \left (1+x+x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} x+x^2+\log \left (1+x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x + 3*x^2 + 2*x^3)/(1 + x + x^2),x]

[Out]

x + x^2 + Log[1 + x + x^2]

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Maple [A]
time = 0.27, size = 13, normalized size = 1.08

method	result	size
default	\(x +x^{2}+\ln \left (x^{2}+x +1\right )\)	\(13\)
norman	\(x +x^{2}+\ln \left (x^{2}+x +1\right )\)	\(13\)
risch	\(x +x^{2}+\ln \left (x^{2}+x +1\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+3*x^2+5*x+2)/(x^2+x+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3*x^2+5*x+2)/(x^2+x+1),x, algorithm="maxima")

[Out]

x^2 + x + log(x^2 + x + 1)

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Fricas [A]
time = 0.38, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3*x^2+5*x+2)/(x^2+x+1),x, algorithm="fricas")

[Out]

x^2 + x + log(x^2 + x + 1)

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Sympy [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log {\left (x^{2} + x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+3*x**2+5*x+2)/(x**2+x+1),x)

[Out]

x**2 + x + log(x**2 + x + 1)

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Giac [A]
time = 3.68, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3*x^2+5*x+2)/(x^2+x+1),x, algorithm="giac")

[Out]

x^2 + x + log(x^2 + x + 1)

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Mupad [B]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} x+\ln \left (x^2+x+1\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3*x^2 + 2*x^3 + 2)/(x + x^2 + 1),x)

[Out]

x + log(x + x^2 + 1) + x^2

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