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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14} \[ \frac {x^2}{2}-\frac {1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^3)/x^2,x]

[Out]

-x^(-1) + x^2/2 + Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {1+x+x^3}{x^2} \, dx &=\int \left (\frac {1}{x^2}+\frac {1}{x}+x\right ) \, dx\\ &=-\frac {1}{x}+\frac {x^2}{2}+\log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {x^2}{2}-\frac {1}{x}+\log (x) \]

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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fricas [A]  time = 0.65, size = 15, normalized size = 1.00 \[ \frac {x^{3} + 2 \, x \log \relax (x) - 2}{2 \, x} \]

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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giac [A]  time = 0.28, size = 14, normalized size = 0.93 \[ \frac {1}{2} \, x^{2} - \frac {1}{x} + \log \left ({\left | x \right |}\right ) \]

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 + log(abs(x))

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {x^{2}}{2}+\ln \relax (x )-\frac {1}{x} \]

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 = 0.95, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, x^{2} - \frac {1}{x} + \log \relax (x) \]

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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mupad [B]  time = 0.02, size = 13, normalized size = 0.87 \[ \ln \relax (x)-\frac {1}{x}+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^3 + 1)/x^2,x)

[Out]

log(x) - 1/x + x^2/2

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sympy [A]  time = 0.08, size = 10, normalized size = 0.67 \[ \frac {x^{2}}{2} + \log {\relax (x )} - \frac {1}{x} \]

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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