3.5.70 \(\int \frac {2 x+x^2}{(1+x)^2} \, dx\) [470]

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

[Out]

x^2/(1+x)

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Rubi [A]
time = 0.00, 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 = {697} \begin {gather*} x+\frac {1}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

Rule 697

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 \begin {gather*} x+\frac {1}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

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Maple [A]
time = 0.20, size = 8, normalized size = 0.89

method	result	size
default	\(x +\frac {1}{1+x}\)	\(8\)
risch	\(x +\frac {1}{1+x}\)	\(8\)
gosper	\(\frac {x^{2}}{1+x}\)	\(10\)
norman	\(\frac {x^{2}}{1+x}\)	\(10\)
meijerg	\(\frac {x \left (3 x +6\right )}{3+3 x}-\frac {2 x}{1+x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2*x)/(1+x)^2,x,method=_RETURNVERBOSE)

[Out]

x+1/(1+x)

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Maxima [A]
time = 0.26, size = 7, normalized size = 0.78 \begin {gather*} x + \frac {1}{x + 1} \end {gather*}

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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Fricas [A]
time = 0.39, size = 12, normalized size = 1.33 \begin {gather*} \frac {x^{2} + x + 1}{x + 1} \end {gather*}

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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Sympy [A]
time = 0.02, size = 5, normalized size = 0.56 \begin {gather*} x + \frac {1}{x + 1} \end {gather*}

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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Giac [A]
time = 5.36, size = 8, normalized size = 0.89 \begin {gather*} x + \frac {1}{x + 1} + 1 \end {gather*}

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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Mupad [B]
time = 0.02, size = 7, normalized size = 0.78 \begin {gather*} x+\frac {1}{x+1} \end {gather*}

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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