3.22.42 \(\int \frac {-72 x^2+3 x^2 \log (4)}{-24 x^3+(5+x^3) \log (4)} \, dx\)

Optimal. Leaf size=16 \[ \log \left (5+x^2 \left (x-\frac {24 x}{\log (4)}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 12, 1587} \begin {gather*} \log \left (24 x^3-\left (x^3+5\right ) \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-72*x^2 + 3*x^2*Log[4])/(-24*x^3 + (5 + x^3)*Log[4]),x]

[Out]

Log[24*x^3 - (5 + x^3)*Log[4]]

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 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} &=\int \frac {x^2 (-72+3 \log (4))}{-24 x^3+\left (5+x^3\right ) \log (4)} \, dx\\ &=(-72+3 \log (4)) \int \frac {x^2}{-24 x^3+\left (5+x^3\right ) \log (4)} \, dx\\ &=\log \left (24 x^3-\left (5+x^3\right ) \log (4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.62 \begin {gather*} \frac {3 (-24+\log (4)) \log \left (x^3 (-24+\log (4))+5 \log (4)\right )}{-72+\log (64)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72*x^2 + 3*x^2*Log[4])/(-24*x^3 + (5 + x^3)*Log[4]),x]

[Out]

(3*(-24 + Log[4])*Log[x^3*(-24 + Log[4]) + 5*Log[4]])/(-72 + Log[64])

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fricas [A]  time = 0.87, size = 15, normalized size = 0.94 \begin {gather*} \log \left (-12 \, x^{3} + {\left (x^{3} + 5\right )} \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(2)-72*x^2)/(2*(x^3+5)*log(2)-24*x^3),x, algorithm="fricas")

[Out]

log(-12*x^3 + (x^3 + 5)*log(2))

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giac [A]  time = 0.23, size = 19, normalized size = 1.19 \begin {gather*} \log \left ({\left | -x^{3} \log \relax (2) + 12 \, x^{3} - 5 \, \log \relax (2) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(2)-72*x^2)/(2*(x^3+5)*log(2)-24*x^3),x, algorithm="giac")

[Out]

log(abs(-x^3*log(2) + 12*x^3 - 5*log(2)))

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maple [A]  time = 0.40, size = 15, normalized size = 0.94

method	result	size
risch	\(\ln \left (x^{3} \left (\ln \relax (2)-12\right )+5 \ln \relax (2)\right )\)	\(15\)
derivativedivides	\(\ln \left (2 \left (x^{3}+5\right ) \ln \relax (2)-24 x^{3}\right )\)	\(17\)
default	\(\ln \left (x^{3} \ln \relax (2)-12 x^{3}+5 \ln \relax (2)\right )\)	\(18\)
norman	\(\ln \left (x^{3} \ln \relax (2)-12 x^{3}+5 \ln \relax (2)\right )\)	\(18\)
meijerg	\(-\frac {5 \left (\frac {3 \ln \relax (2)}{5}-\frac {36}{5}\right ) \ln \left (1-\frac {x^{3} \left (12-\ln \relax (2)\right )}{5 \ln \relax (2)}\right )}{3 \left (12-\ln \relax (2)\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2*ln(2)-72*x^2)/(2*(x^3+5)*ln(2)-24*x^3),x,method=_RETURNVERBOSE)

[Out]

ln(x^3*(ln(2)-12)+5*ln(2))

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maxima [A]  time = 0.35, size = 16, normalized size = 1.00 \begin {gather*} \log \left (12 \, x^{3} - {\left (x^{3} + 5\right )} \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(2)-72*x^2)/(2*(x^3+5)*log(2)-24*x^3),x, algorithm="maxima")

[Out]

log(12*x^3 - (x^3 + 5)*log(2))

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mupad [B]  time = 0.13, size = 12, normalized size = 0.75 \begin {gather*} \ln \left (\left (\ln \relax (2)-12\right )\,x^3+\ln \left (32\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2*log(2) - 72*x^2)/(2*log(2)*(x^3 + 5) - 24*x^3),x)

[Out]

log(log(32) + x^3*(log(2) - 12))

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sympy [A]  time = 0.24, size = 27, normalized size = 1.69 \begin {gather*} \frac {\left (-36 + 3 \log {\relax (2 )}\right ) \log {\left (x^{3} \left (-12 + \log {\relax (2 )}\right ) + 5 \log {\relax (2 )} \right )}}{3 \left (-12 + \log {\relax (2 )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**2*ln(2)-72*x**2)/(2*(x**3+5)*ln(2)-24*x**3),x)

[Out]

(-36 + 3*log(2))*log(x**3*(-12 + log(2)) + 5*log(2))/(3*(-12 + log(2)))

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