3.62.35 \(\int \frac {-18+x^2+2 x^3}{x^3} \, dx\)

Optimal. Leaf size=15 \[ 1+e^8+\frac {9}{x^2}+2 x+\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} \frac {9}{x^2}+2 x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 + x^2 + 2*x^3)/x^3,x]

[Out]

9/x^2 + 2*x + 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 {gather*} \begin {aligned} \text {integral} &=\int \left (2-\frac {18}{x^3}+\frac {1}{x}\right ) \, dx\\ &=\frac {9}{x^2}+2 x+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.73 \begin {gather*} \frac {9}{x^2}+2 x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 + x^2 + 2*x^3)/x^3,x]

[Out]

9/x^2 + 2*x + Log[x]

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fricas [A]  time = 0.44, size = 17, normalized size = 1.13 \begin {gather*} \frac {2 \, x^{3} + x^{2} \log \relax (x) + 9}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2-18)/x^3,x, algorithm="fricas")

[Out]

(2*x^3 + x^2*log(x) + 9)/x^2

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giac [A]  time = 0.12, size = 12, normalized size = 0.80 \begin {gather*} 2 \, x + \frac {9}{x^{2}} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2-18)/x^3,x, algorithm="giac")

[Out]

2*x + 9/x^2 + log(abs(x))

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maple [A]  time = 0.04, size = 12, normalized size = 0.80

method	result	size
default	\(2 x +\frac {9}{x^{2}}+\ln \relax (x )\)	\(12\)
risch	\(2 x +\frac {9}{x^{2}}+\ln \relax (x )\)	\(12\)
norman	\(\frac {2 x^{3}+9}{x^{2}}+\ln \relax (x )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+x^2-18)/x^3,x,method=_RETURNVERBOSE)

[Out]

2*x+9/x^2+ln(x)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.73 \begin {gather*} 2 \, x + \frac {9}{x^{2}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2-18)/x^3,x, algorithm="maxima")

[Out]

2*x + 9/x^2 + log(x)

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mupad [B]  time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} 2\,x+\ln \relax (x)+\frac {9}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 2*x^3 - 18)/x^3,x)

[Out]

2*x + log(x) + 9/x^2

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sympy [A]  time = 0.07, size = 10, normalized size = 0.67 \begin {gather*} 2 x + \log {\relax (x )} + \frac {9}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+x**2-18)/x**3,x)

[Out]

2*x + log(x) + 9/x**2

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