3.52.95 \(\int \frac {-512+768 x+x^3+(2-3 x) \log (3)}{-256 x^3+x^3 \log (3)} \, dx\)

Optimal. Leaf size=24 \[ 2-\frac {(2-3 x)^2}{4 x^2}+\frac {x}{-256+\log (3)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.58, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6, 12, 14} \begin {gather*} -\frac {512-\log (9)}{2 x^2 (256-\log (3))}+\frac {3}{x}-\frac {x}{256-\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-512 + 768*x + x^3 + (2 - 3*x)*Log[3])/(-256*x^3 + x^3*Log[3]),x]

[Out]

3/x - x/(256 - Log[3]) - (512 - Log[9])/(2*x^2*(256 - Log[3]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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} &=\int \frac {-512+768 x+x^3+(2-3 x) \log (3)}{x^3 (-256+\log (3))} \, dx\\ &=\frac {\int \frac {-512+768 x+x^3+(2-3 x) \log (3)}{x^3} \, dx}{-256+\log (3)}\\ &=\frac {\int \left (1-\frac {3 (-256+\log (3))}{x^2}+\frac {-512+\log (9)}{x^3}\right ) \, dx}{-256+\log (3)}\\ &=\frac {3}{x}-\frac {x}{256-\log (3)}-\frac {512-\log (9)}{2 x^2 (256-\log (3))}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 1.08 \begin {gather*} \frac {256+x^3+3 x (-256+\log (3))-\log (3)}{x^2 (-256+\log (3))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-512 + 768*x + x^3 + (2 - 3*x)*Log[3])/(-256*x^3 + x^3*Log[3]),x]

[Out]

(256 + x^3 + 3*x*(-256 + Log[3]) - Log[3])/(x^2*(-256 + Log[3]))

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 31, normalized size = 1.29 \begin {gather*} \frac {x^{3} + {\left (3 \, x - 1\right )} \log \relax (3) - 768 \, x + 256}{x^{2} \log \relax (3) - 256 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+2)*log(3)+x^3+768*x-512)/(x^3*log(3)-256*x^3),x, algorithm="fricas")

[Out]

(x^3 + (3*x - 1)*log(3) - 768*x + 256)/(x^2*log(3) - 256*x^2)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 18, normalized size = 0.75 \begin {gather*} \frac {x}{\log \relax (3) - 256} + \frac {3 \, x - 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+2)*log(3)+x^3+768*x-512)/(x^3*log(3)-256*x^3),x, algorithm="giac")

[Out]

x/(log(3) - 256) + (3*x - 1)/x^2

________________________________________________________________________________________

maple [A]  time = 0.06, size = 20, normalized size = 0.83

method	result	size
norman	\(\frac {-1+\frac {x^{3}}{\ln \relax (3)-256}+3 x}{x^{2}}\)	\(20\)
gosper	\(\frac {x^{3}+3 x \ln \relax (3)-\ln \relax (3)-768 x +256}{x^{2} \left (\ln \relax (3)-256\right )}\)	\(28\)
default	\(\frac {x -\frac {-3 \ln \relax (3)+768}{x}-\frac {2 \ln \relax (3)-512}{2 x^{2}}}{\ln \relax (3)-256}\)	\(32\)
risch	\(\frac {x}{\ln \relax (3)-256}+\frac {\left (3 \ln \relax (3)^{2}-1536 \ln \relax (3)+196608\right ) x -\ln \relax (3)^{2}+512 \ln \relax (3)-65536}{\left (\ln \relax (3)-256\right )^{2} x^{2}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+2)*ln(3)+x^3+768*x-512)/(x^3*ln(3)-256*x^3),x,method=_RETURNVERBOSE)

[Out]

(-1+1/(ln(3)-256)*x^3+3*x)/x^2

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 18, normalized size = 0.75 \begin {gather*} \frac {x}{\log \relax (3) - 256} + \frac {3 \, x - 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+2)*log(3)+x^3+768*x-512)/(x^3*log(3)-256*x^3),x, algorithm="maxima")

[Out]

x/(log(3) - 256) + (3*x - 1)/x^2

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 31, normalized size = 1.29 \begin {gather*} \frac {x}{\ln \relax (3)-256}+\frac {x\,\left (\ln \left (27\right )-768\right )-\ln \relax (3)+256}{x^2\,\left (\ln \relax (3)-256\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((768*x - log(3)*(3*x - 2) + x^3 - 512)/(x^3*log(3) - 256*x^3),x)

[Out]

x/(log(3) - 256) + (x*(log(27) - 768) - log(3) + 256)/(x^2*(log(3) - 256))

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 22, normalized size = 0.92 \begin {gather*} \frac {x + \frac {x \left (-768 + 3 \log {\relax (3 )}\right ) - \log {\relax (3 )} + 256}{x^{2}}}{-256 + \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+2)*ln(3)+x**3+768*x-512)/(x**3*ln(3)-256*x**3),x)

[Out]

(x + (x*(-768 + 3*log(3)) - log(3) + 256)/x**2)/(-256 + log(3))

________________________________________________________________________________________