3.473 \(\int \frac{-2+x^2+x^3}{x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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Rubi [A]  time = 0.0125453, antiderivative size = 15, 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 \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(-2 + x^2 + x^3)/x^4,x]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} + x^{2} - 2}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x**2-2)/x**4,x)

[Out]

Integral((x**3 + x**2 - 2)/x**4, x)

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Mathematica [A]  time = 0.00273745, size = 15, normalized size = 1. \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + x^2 + x^3)/x^4,x]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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Maple [A]  time = 0.008, size = 14, normalized size = 0.9 \[{\frac{2}{3\,{x}^{3}}}-{x}^{-1}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2-2)/x^4,x)

[Out]

2/3/x^3-1/x+ln(x)

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Maxima [A]  time = 0.78114, size = 20, normalized size = 1.33 \[ -\frac{3 \, x^{2} - 2}{3 \, x^{3}} + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 - 2)/x^4,x, algorithm="maxima")

[Out]

-1/3*(3*x^2 - 2)/x^3 + log(x)

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Fricas [A]  time = 0.263128, size = 26, normalized size = 1.73 \[ \frac{3 \, x^{3} \log \left (x\right ) - 3 \, x^{2} + 2}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 - 2)/x^4,x, algorithm="fricas")

[Out]

1/3*(3*x^3*log(x) - 3*x^2 + 2)/x^3

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Sympy [A]  time = 0.169219, size = 14, normalized size = 0.93 \[ \log{\left (x \right )} - \frac{3 x^{2} - 2}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x**2-2)/x**4,x)

[Out]

log(x) - (3*x**2 - 2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.260572, size = 22, normalized size = 1.47 \[ -\frac{3 \, x^{2} - 2}{3 \, x^{3}} +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 - 2)/x^4,x, algorithm="giac")

[Out]

-1/3*(3*x^2 - 2)/x^3 + ln(abs(x))