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 = 0.00739929, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^2}{x+1} \]

Antiderivative was successfully verified.

[In]  Int[(2*x + x^2)/(1 + x)^2,x]

[Out]

x^2/(1 + x)

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Rubi in Sympy [A]  time = 3.01009, size = 5, normalized size = 0.56 \[ \frac{x^{2}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2*x)/(1+x)**2,x)

[Out]

x**2/(x + 1)

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Mathematica [A]  time = 0.00467687, 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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Maple [A]  time = 0.005, size = 8, normalized size = 0.9 \[ x+ \left ( 1+x \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2*x)/(1+x)^2,x)

[Out]

x+1/(1+x)

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Maxima [A]  time = 0.775601, size = 9, normalized size = 1. \[ x + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 1/(x + 1)

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Fricas [A]  time = 0.254757, size = 16, normalized size = 1.78 \[ \frac{x^{2} + x + 1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^2 + x + 1)/(x + 1)

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Sympy [A]  time = 0.123863, 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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GIAC/XCAS [A]  time = 0.25882, size = 11, normalized size = 1.22 \[ x + \frac{1}{x + 1} + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 1/(x + 1) + 1