### 3.176 $$\int \frac{x^2}{1+x} \, dx$$

Optimal. Leaf size=15 $\frac{x^2}{2}-x+\log (x+1)$

[Out]

-x + x^2/2 + Log[1 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.00575, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {43} $\frac{x^2}{2}-x+\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/(1 + x),x]

[Out]

-x + x^2/2 + Log[1 + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0030243, size = 19, normalized size = 1.27 $\frac{1}{2} (x+1)^2-2 (x+1)+\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/(1 + x),x]

[Out]

-2*(1 + x) + (1 + x)^2/2 + Log[1 + x]

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*} -x+{\frac{{x}^{2}}{2}}+\ln \left ( 1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+x),x)

[Out]

-x+1/2*x^2+ln(1+x)

________________________________________________________________________________________

Maxima [A]  time = 0.950728, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x + log(x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.80622, size = 35, normalized size = 2.33 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x + log(x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.067622, size = 10, normalized size = 0.67 \begin{align*} \frac{x^{2}}{2} - x + \log{\left (x + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - x + log(x + 1)

________________________________________________________________________________________

Giac [A]  time = 1.04549, size = 19, normalized size = 1.27 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x + log(abs(x + 1))