3.2.92 \(\int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx\) [192]

Optimal. Leaf size=15 \[ \frac {1}{3} \log \left (4+3 x^2+x^3\right ) \]

[Out]

1/3*ln(x^3+3*x^2+4)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1601} \begin {gather*} \frac {1}{3} \log \left (x^3+3 x^2+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x + x^2)/(4 + 3*x^2 + x^3),x]

[Out]

Log[4 + 3*x^2 + x^3]/3

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (4+3 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[4 + 3*x^2 + x^3]/3

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Maple [A]
time = 0.01, size = 14, normalized size = 0.93

method	result	size
default	\(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\)	\(14\)
norman	\(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\)	\(14\)
risch	\(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(x^3+3*x^2+4)

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Maxima [A]
time = 2.16, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(x^3 + 3*x^2 + 4)

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Fricas [A]
time = 0.43, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(x^3 + 3*x^2 + 4)

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Sympy [A]
time = 0.02, size = 12, normalized size = 0.80 \begin {gather*} \frac {\log {\left (x^{3} + 3 x^{2} + 4 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**3 + 3*x**2 + 4)/3

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Giac [A]
time = 0.46, size = 14, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, \log \left ({\left | x^{3} + 3 \, x^{2} + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(abs(x^3 + 3*x^2 + 4))

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Mupad [B]
time = 0.05, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x^3+3\,x^2+4\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2)/(3*x^2 + x^3 + 4),x)

[Out]

log(3*x^2 + x^3 + 4)/3

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