3.2167 \(\int \frac{3+3 x+2 x^2}{(1+x)^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0301152, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.19602, size = 17, normalized size = 0.89 \[ 2 \log{\left (x + 1 \right )} + \frac{1}{x + 1} - \frac{1}{\left (x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x + 1) + 1/(x + 1) - 1/(x + 1)**2

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Mathematica [A]  time = 0.0149365, size = 19, normalized size = 1. \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 20, normalized size = 1.1 \[ - \left ( 1+x \right ) ^{-2}+ \left ( 1+x \right ) ^{-1}+2\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+3*x+3)/(1+x)^3,x)

[Out]

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

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Maxima [A]  time = 0.793498, size = 26, normalized size = 1.37 \[ \frac{x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x^2 + 2*x + 1) + 2*log(x + 1)

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Fricas [A]  time = 0.213295, size = 36, normalized size = 1.89 \[ \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.206183, size = 15, normalized size = 0.79 \[ \frac{x}{x^{2} + 2 x + 1} + 2 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x**2 + 2*x + 1) + 2*log(x + 1)

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GIAC/XCAS [A]  time = 0.203072, size = 20, normalized size = 1.05 \[ \frac{x}{{\left (x + 1\right )}^{2}} + 2 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x + 1)^2 + 2*ln(abs(x + 1))