### 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) + Log[1 + x]

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Rubi [A]  time = 0.0071718, antiderivative size = 14, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0050636, size = 14, normalized size = 1. $\log (x+1)-\frac{1}{2 (x+1)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*(1 + x)^2) + Log[1 + x]

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Maple [A]  time = 0.045, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( 1+x \right ) ^{2}}}+\ln \left ( 1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.990921, size = 23, normalized size = 1.64 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \end{align*}

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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Fricas [B]  time = 1.81557, size = 76, normalized size = 5.43 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) - 1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

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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Sympy [A]  time = 0.101164, size = 15, normalized size = 1.07 \begin{align*} \log{\left (x + 1 \right )} - \frac{1}{2 x^{2} + 4 x + 2} \end{align*}

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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Giac [A]  time = 1.09139, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{2 \,{\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

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))