### 3.194 $$\int \frac{x^2}{(1+x)^3} \, dx$$

Optimal. Leaf size=21 $\frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1)$

[Out]

-1/(2*(1 + x)^2) + 2/(1 + x) + Log[1 + x]

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Rubi [A]  time = 0.0072975, antiderivative size = 21, 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} $\frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/(1 + x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0083873, size = 21, normalized size = 1. $\frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/(1 + x)^3,x]

[Out]

-1/(2*(1 + x)^2) + 2/(1 + x) + Log[1 + x]

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Maple [A]  time = 0.005, size = 20, normalized size = 1. \begin{align*} -{\frac{1}{2\, \left ( 1+x \right ) ^{2}}}+2\, \left ( 1+x \right ) ^{-1}+\ln \left ( 1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.922966, size = 30, normalized size = 1.43 \begin{align*} \frac{4 \, x + 3}{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/(1+x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.80456, size = 84, normalized size = 4. \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 4 \, x + 3}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.086027, size = 19, normalized size = 0.9 \begin{align*} \frac{4 x + 3}{2 x^{2} + 4 x + 2} + \log{\left (x + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*x + 3)/(2*x**2 + 4*x + 2) + log(x + 1)

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Giac [A]  time = 1.05038, size = 24, normalized size = 1.14 \begin{align*} \frac{4 \, x + 3}{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/(1+x)^3,x, algorithm="giac")

[Out]

1/2*(4*x + 3)/(x + 1)^2 + log(abs(x + 1))