3.176 \(\int \frac {x^2}{1+x} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = 0.00, 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]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, x^{2} - x + \log \left (x + 1\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.94, size = 14, normalized size = 0.93 \[ \frac {1}{2} \, x^{2} - x + \log \left ({\left | x + 1 \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {x^{2}}{2}-x +\ln \left (x +1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, x^{2} - x + \log \left (x + 1\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.87 \[ \ln \left (x+1\right )-x+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x + 1),x)

[Out]

log(x + 1) - x + x^2/2

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 10, normalized size = 0.67 \[ \frac {x^{2}}{2} - x + \log {\left (x + 1 \right )} \]

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)

________________________________________________________________________________________