3.73.25 \(\int \frac {128 x^8+\log (24)}{x \log (24)} \, dx\)

Optimal. Leaf size=13 \[ 1+\frac {16 x^8}{\log (24)}+\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 14} \begin {gather*} \frac {16 x^8}{\log (24)}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(128*x^8 + Log[24])/(x*Log[24]),x]

[Out]

(16*x^8)/Log[24] + 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 {\int \frac {128 x^8+\log (24)}{x} \, dx}{\log (24)}\\ &=\frac {\int \left (128 x^7+\frac {\log (24)}{x}\right ) \, dx}{\log (24)}\\ &=\frac {16 x^8}{\log (24)}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \frac {16 x^8}{\log (24)}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(128*x^8 + Log[24])/(x*Log[24]),x]

[Out]

(16*x^8)/Log[24] + Log[x]

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fricas [A]  time = 1.07, size = 16, normalized size = 1.23 \begin {gather*} \frac {16 \, x^{8} + \log \left (24\right ) \log \relax (x)}{\log \left (24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(24)+128*x^8)/x/log(24),x, algorithm="fricas")

[Out]

(16*x^8 + log(24)*log(x))/log(24)

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giac [A]  time = 0.14, size = 19, normalized size = 1.46 \begin {gather*} \frac {128 \, x^{8} + \log \left (24\right ) \log \left (x^{8}\right )}{8 \, \log \left (24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(24)+128*x^8)/x/log(24),x, algorithm="giac")

[Out]

1/8*(128*x^8 + log(24)*log(x^8))/log(24)

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maple [A]  time = 0.04, size = 13, normalized size = 1.00

method	result	size
norman	\(\frac {16 x^{8}}{\ln \left (24\right )}+\ln \relax (x )\)	\(13\)
default	\(\frac {16 x^{8}+\ln \left (24\right ) \ln \relax (x )}{\ln \left (24\right )}\)	\(17\)
risch	\(\frac {16 x^{8}}{3 \ln \relax (2)+\ln \relax (3)}+\ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(24)+128*x^8)/x/ln(24),x,method=_RETURNVERBOSE)

[Out]

16*x^8/ln(24)+ln(x)

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maxima [A]  time = 0.36, size = 19, normalized size = 1.46 \begin {gather*} \frac {128 \, x^{8} + \log \left (24\right ) \log \left (x^{8}\right )}{8 \, \log \left (24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(24)+128*x^8)/x/log(24),x, algorithm="maxima")

[Out]

1/8*(128*x^8 + log(24)*log(x^8))/log(24)

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mupad [B]  time = 0.03, size = 12, normalized size = 0.92 \begin {gather*} \ln \relax (x)+\frac {16\,x^8}{\ln \left (24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(24) + 128*x^8)/(x*log(24)),x)

[Out]

log(x) + (16*x^8)/log(24)

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sympy [A]  time = 0.10, size = 14, normalized size = 1.08 \begin {gather*} \frac {16 x^{8} + \log {\left (24 \right )} \log {\relax (x )}}{\log {\left (24 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(24)+128*x**8)/x/ln(24),x)

[Out]

(16*x**8 + log(24)*log(x))/log(24)

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