3.2166 \(\int \frac{2+2 x+x^2}{(1+x)^3} \, dx\)

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 = 0.0227111, 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 \[ \log (x+1)-\frac{1}{2 (x+1)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(1 + x)^2) + Log[1 + x]

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Rubi in Sympy [A]  time = 3.54138, size = 12, normalized size = 0.86 \[ \log{\left (x + 1 \right )} - \frac{1}{2 \left (x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2*x+2)/(1+x)**3,x)

[Out]

log(x + 1) - 1/(2*(x + 1)**2)

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Mathematica [A]  time = 0.0103546, 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 = 0.009, size = 13, normalized size = 0.9 \[ -{\frac{1}{2\, \left ( 1+x \right ) ^{2}}}+\ln \left ( 1+x \right ) \]

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 = 0.812455, size = 23, normalized size = 1.64 \[ -\frac{1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 2)/(x + 1)^3,x, algorithm="maxima")

[Out]

-1/2/(x^2 + 2*x + 1) + log(x + 1)

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Fricas [A]  time = 0.207747, size = 38, normalized size = 2.71 \[ \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) - 1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 2)/(x + 1)^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 = 0.197706, size = 15, normalized size = 1.07 \[ \log{\left (x + 1 \right )} - \frac{1}{2 x^{2} + 4 x + 2} \]

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/XCAS [A]  time = 0.20443, size = 18, normalized size = 1.29 \[ -\frac{1}{2 \,{\left (x + 1\right )}^{2}} +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 2)/(x + 1)^3,x, algorithm="giac")

[Out]

-1/2/(x + 1)^2 + ln(abs(x + 1))