3.55.34 \(\int \frac {-3-2 x-6 x^2}{3 x} \, dx\)

Optimal. Leaf size=24 \[ \frac {7}{2}-\frac {2 x}{3}-x^2-\log \left (\frac {5}{3}\right )-\log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 14} \begin {gather*} -x^2-\frac {2 x}{3}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 2*x - 6*x^2)/(3*x),x]

[Out]

(-2*x)/3 - x^2 - Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{3} \int \frac {-3-2 x-6 x^2}{x} \, dx\\ &=\frac {1}{3} \int \left (-2-\frac {3}{x}-6 x\right ) \, dx\\ &=-\frac {2 x}{3}-x^2-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 0.62 \begin {gather*} -\frac {2 x}{3}-x^2-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 2*x - 6*x^2)/(3*x),x]

[Out]

(-2*x)/3 - x^2 - Log[x]

________________________________________________________________________________________

fricas [A]  time = 2.00, size = 13, normalized size = 0.54 \begin {gather*} -x^{2} - \frac {2}{3} \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-6*x^2-2*x-3)/x,x, algorithm="fricas")

[Out]

-x^2 - 2/3*x - log(x)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 14, normalized size = 0.58 \begin {gather*} -x^{2} - \frac {2}{3} \, x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-6*x^2-2*x-3)/x,x, algorithm="giac")

[Out]

-x^2 - 2/3*x - log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 14, normalized size = 0.58

method	result	size
default	\(-x^{2}-\frac {2 x}{3}-\ln \relax (x )\)	\(14\)
norman	\(-x^{2}-\frac {2 x}{3}-\ln \relax (x )\)	\(14\)
risch	\(-x^{2}-\frac {2 x}{3}-\ln \relax (x )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-6*x^2-2*x-3)/x,x,method=_RETURNVERBOSE)

[Out]

-x^2-2/3*x-ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 13, normalized size = 0.54 \begin {gather*} -x^{2} - \frac {2}{3} \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-6*x^2-2*x-3)/x,x, algorithm="maxima")

[Out]

-x^2 - 2/3*x - log(x)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.54 \begin {gather*} -\frac {2\,x}{3}-\ln \relax (x)-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x)/3 + 2*x^2 + 1)/x,x)

[Out]

- (2*x)/3 - log(x) - x^2

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 12, normalized size = 0.50 \begin {gather*} - x^{2} - \frac {2 x}{3} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-6*x**2-2*x-3)/x,x)

[Out]

-x**2 - 2*x/3 - log(x)

________________________________________________________________________________________