### 3.192 $$\int \frac{2 x+x^2}{4+3 x^2+x^3} \, dx$$

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

[Out]

Log[4 + 3*x^2 + x^3]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0090738, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {1587} $\frac{1}{3} \log \left (x^3+3 x^2+4\right )$

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 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; 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*}

Mathematica [A]  time = 0.0042837, size = 15, normalized size = 1. $\frac{1}{3} \log \left (x^3+3 x^2+4\right )$

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{3}+3\,{x}^{2}+4 \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.927364, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.84928, size = 35, normalized size = 2.33 \begin{align*} \frac{1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.080973, size = 12, normalized size = 0.8 \begin{align*} \frac{\log{\left (x^{3} + 3 x^{2} + 4 \right )}}{3} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.06531, size = 19, normalized size = 1.27 \begin{align*} \frac{1}{3} \, \log \left ({\left | x^{3} + 3 \, x^{2} + 4 \right |}\right ) \end{align*}

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))