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^3-1/x+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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14} \[ \frac {2}{3 x^3}-\frac {1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + x^2 + x^3)/x^4,x]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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 {-2+x^2+x^3}{x^4} \, dx &=\int \left (-\frac {2}{x^4}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx\\ &=\frac {2}{3 x^3}-\frac {1}{x}+\log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {2}{3 x^3}-\frac {1}{x}+\log (x) \]

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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fricas [A]  time = 0.50, size = 19, normalized size = 1.27 \[ \frac {3 \, x^{3} \log \relax (x) - 3 \, x^{2} + 2}{3 \, x^{3}} \]

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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giac [A]  time = 0.27, size = 16, normalized size = 1.07 \[ -\frac {3 \, x^{2} - 2}{3 \, x^{3}} + \log \left ({\left | x \right |}\right ) \]

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 + log(abs(x))

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \ln \relax (x )-\frac {1}{x}+\frac {2}{3 x^{3}} \]

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 = 0.97, size = 15, normalized size = 1.00 \[ -\frac {3 \, x^{2} - 2}{3 \, x^{3}} + \log \relax (x) \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.87 \[ \ln \relax (x)-\frac {x^2-\frac {2}{3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + x^3 - 2)/x^4,x)

[Out]

log(x) - (x^2 - 2/3)/x^3

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sympy [A]  time = 0.09, size = 14, normalized size = 0.93 \[ \log {\relax (x )} + \frac {2 - 3 x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+x**2-2)/x**4,x)

[Out]

log(x) + (2 - 3*x**2)/(3*x**3)

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