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

Optimal. Leaf size=9 \[ \frac {x^2}{x+1} \]

[Out]

x^2/(1+x)

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Rubi [A]  time = 0.01, antiderivative size = 7, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {683} \[ x+\frac {1}{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

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

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.78 \[ x+\frac {1}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

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fricas [A]  time = 0.54, size = 12, normalized size = 1.33 \[ \frac {x^{2} + x + 1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + x + 1)/(x + 1)

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giac [A]  time = 0.30, size = 8, normalized size = 0.89 \[ x + \frac {1}{x + 1} + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/(x + 1) + 1

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maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ x +\frac {1}{x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/(x+1)

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maxima [A]  time = 0.95, size = 7, normalized size = 0.78 \[ x + \frac {1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/(x + 1)

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mupad [B]  time = 0.02, size = 7, normalized size = 0.78 \[ x+\frac {1}{x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2)/(x + 1)^2,x)

[Out]

x + 1/(x + 1)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.56 \[ x + \frac {1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/(x + 1)

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