3.2177 \(\int \frac {2+2 x+x^2}{(1+x)^3} \, dx\)

Optimal. Leaf size=14 \[ \log (x+1)-\frac {1}{2 (x+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {683} \[ \log (x+1)-\frac {1}{2 (x+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x + x^2)/(1 + x)^3,x]

[Out]

-1/(2*(1 + x)^2) + Log[1 + x]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \[ \log (x+1)-\frac {1}{2 (x+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(1 + x)^2 + Log[1 + x]

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fricas [B]  time = 0.86, size = 28, normalized size = 2.00 \[ \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) - 1}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 13, normalized size = 0.93 \[ -\frac {1}{2 \, {\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x + 1)^2 + log(abs(x + 1))

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maple [A]  time = 0.06, size = 13, normalized size = 0.93 \[ \ln \left (x +1\right )-\frac {1}{2 \left (x +1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.76, size = 17, normalized size = 1.21 \[ -\frac {1}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x^2 + 2*x + 1) + log(x + 1)

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mupad [B]  time = 0.03, size = 12, normalized size = 0.86 \[ \ln \left (x+1\right )-\frac {1}{2\,{\left (x+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2 + 2)/(x + 1)^3,x)

[Out]

log(x + 1) - 1/(2*(x + 1)^2)

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sympy [A]  time = 0.10, size = 15, normalized size = 1.07 \[ \log {\left (x + 1 \right )} - \frac {1}{2 x^{2} + 4 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1) - 1/(2*x**2 + 4*x + 2)

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