3.36.86 \(\int \frac {384+3 x^2}{-1152+384 x+x^3} \, dx\)

Optimal. Leaf size=17 \[ \log \left (-2+x+\frac {1}{2} \left (-2+\frac {x^3}{192}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.65, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1587} \begin {gather*} \log \left (-x^3-384 x+1152\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(384 + 3*x^2)/(-1152 + 384*x + x^3),x]

[Out]

Log[1152 - 384*x - x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (1152-384 x-x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 0.53 \begin {gather*} \log \left (-1152+384 x+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384 + 3*x^2)/(-1152 + 384*x + x^3),x]

[Out]

Log[-1152 + 384*x + x^3]

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fricas [A]  time = 0.69, size = 9, normalized size = 0.53 \begin {gather*} \log \left (x^{3} + 384 \, x - 1152\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+384)/(x^3+384*x-1152),x, algorithm="fricas")

[Out]

log(x^3 + 384*x - 1152)

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giac [A]  time = 0.13, size = 10, normalized size = 0.59 \begin {gather*} \log \left ({\left | x^{3} + 384 \, x - 1152 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+384)/(x^3+384*x-1152),x, algorithm="giac")

[Out]

log(abs(x^3 + 384*x - 1152))

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maple [A]  time = 0.02, size = 10, normalized size = 0.59

method	result	size
derivativedivides	\(\ln \left (x^{3}+384 x -1152\right )\)	\(10\)
default	\(\ln \left (x^{3}+384 x -1152\right )\)	\(10\)
norman	\(\ln \left (x^{3}+384 x -1152\right )\)	\(10\)
risch	\(\ln \left (x^{3}+384 x -1152\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+384)/(x^3+384*x-1152),x,method=_RETURNVERBOSE)

[Out]

ln(x^3+384*x-1152)

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maxima [A]  time = 0.37, size = 9, normalized size = 0.53 \begin {gather*} \log \left (x^{3} + 384 \, x - 1152\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+384)/(x^3+384*x-1152),x, algorithm="maxima")

[Out]

log(x^3 + 384*x - 1152)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.53 \begin {gather*} \ln \left (x^3+384\,x-1152\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 384)/(384*x + x^3 - 1152),x)

[Out]

log(384*x + x^3 - 1152)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.47 \begin {gather*} \log {\left (x^{3} + 384 x - 1152 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+384)/(x**3+384*x-1152),x)

[Out]

log(x**3 + 384*x - 1152)

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