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

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

[Out]

-x^(-1) + x^2/2 + Log[x]

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Rubi [A]  time = 0.0113741, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^2}{2}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + x^3)/x^2,x]

[Out]

-x^(-1) + x^2/2 + Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} + x + 1}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**3 + x + 1)/x**2, x)

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Mathematica [A]  time = 0.00126265, size = 15, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x + x^3)/x^2,x]

[Out]

-x^(-1) + x^2/2 + Log[x]

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Maple [A]  time = 0.005, size = 14, normalized size = 0.9 \[ -{x}^{-1}+{\frac{{x}^{2}}{2}}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x+1)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.802605, size = 18, normalized size = 1.2 \[ \frac{1}{2} \, x^{2} - \frac{1}{x} + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 - 1/x + log(x)

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Fricas [A]  time = 0.264821, size = 20, normalized size = 1.33 \[ \frac{x^{3} + 2 \, x \log \left (x\right ) - 2}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^3 + 2*x*log(x) - 2)/x

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Sympy [A]  time = 0.124718, size = 10, normalized size = 0.67 \[ \frac{x^{2}}{2} + \log{\left (x \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x+1)/x**2,x)

[Out]

x**2/2 + log(x) - 1/x

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GIAC/XCAS [A]  time = 0.259521, size = 19, normalized size = 1.27 \[ \frac{1}{2} \, x^{2} - \frac{1}{x} +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 - 1/x + ln(abs(x))