3.2170 \(\int \frac{1+2 x+x^2}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{3 x^3}-\frac{1}{x^2}-\frac{1}{x} \]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

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Rubi [A]  time = 0.0153259, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{1}{3 x^3}-\frac{1}{x^2}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x + x^2)/x^4,x]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

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Rubi in Sympy [A]  time = 3.23781, size = 10, normalized size = 0.56 \[ - \frac{\left (x + 1\right )^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2*x+1)/x**4,x)

[Out]

-(x + 1)**3/(3*x**3)

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Mathematica [A]  time = 0.00157848, size = 18, normalized size = 1. \[ -\frac{1}{3 x^3}-\frac{1}{x^2}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x + x^2)/x^4,x]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}}}-{x}^{-2}-{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2*x+1)/x^4,x)

[Out]

-1/3/x^3-1/x^2-1/x

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Maxima [A]  time = 0.799498, size = 20, normalized size = 1.11 \[ -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)/x^4,x, algorithm="maxima")

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3

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Fricas [A]  time = 0.198716, size = 20, normalized size = 1.11 \[ -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)/x^4,x, algorithm="fricas")

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3

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Sympy [A]  time = 0.166582, size = 15, normalized size = 0.83 \[ - \frac{3 x^{2} + 3 x + 1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+2*x+1)/x**4,x)

[Out]

-(3*x**2 + 3*x + 1)/(3*x**3)

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GIAC/XCAS [A]  time = 0.201347, size = 20, normalized size = 1.11 \[ -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)/x^4,x, algorithm="giac")

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3