3.59.29 \(\int \frac {4 \log ^3(5 x)-7 \log ^4(5 x)}{3 x^8} \, dx\)

Optimal. Leaf size=20 \[ 3+\log \left (2 e^{\frac {\log ^4(5 x)}{3 x^7}}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 2305, 2304} \begin {gather*} \frac {\log ^4(5 x)}{3 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*Log[5*x]^3 - 7*Log[5*x]^4)/(3*x^8),x]

[Out]

Log[5*x]^4/(3*x^7)

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]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {4 \log ^3(5 x)-7 \log ^4(5 x)}{x^8} \, dx\\ &=\frac {1}{3} \int \left (\frac {4 \log ^3(5 x)}{x^8}-\frac {7 \log ^4(5 x)}{x^8}\right ) \, dx\\ &=\frac {4}{3} \int \frac {\log ^3(5 x)}{x^8} \, dx-\frac {7}{3} \int \frac {\log ^4(5 x)}{x^8} \, dx\\ &=-\frac {4 \log ^3(5 x)}{21 x^7}+\frac {\log ^4(5 x)}{3 x^7}+\frac {4}{7} \int \frac {\log ^2(5 x)}{x^8} \, dx-\frac {4}{3} \int \frac {\log ^3(5 x)}{x^8} \, dx\\ &=-\frac {4 \log ^2(5 x)}{49 x^7}+\frac {\log ^4(5 x)}{3 x^7}+\frac {8}{49} \int \frac {\log (5 x)}{x^8} \, dx-\frac {4}{7} \int \frac {\log ^2(5 x)}{x^8} \, dx\\ &=-\frac {8}{2401 x^7}-\frac {8 \log (5 x)}{343 x^7}+\frac {\log ^4(5 x)}{3 x^7}-\frac {8}{49} \int \frac {\log (5 x)}{x^8} \, dx\\ &=\frac {\log ^4(5 x)}{3 x^7}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.65 \begin {gather*} \frac {\log ^4(5 x)}{3 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*Log[5*x]^3 - 7*Log[5*x]^4)/(3*x^8),x]

[Out]

Log[5*x]^4/(3*x^7)

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fricas [A]  time = 1.15, size = 11, normalized size = 0.55 \begin {gather*} \frac {\log \left (5 \, x\right )^{4}}{3 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-7*log(5*x)^4+4*log(5*x)^3)/x^8,x, algorithm="fricas")

[Out]

1/3*log(5*x)^4/x^7

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giac [A]  time = 0.16, size = 11, normalized size = 0.55 \begin {gather*} \frac {\log \left (5 \, x\right )^{4}}{3 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-7*log(5*x)^4+4*log(5*x)^3)/x^8,x, algorithm="giac")

[Out]

1/3*log(5*x)^4/x^7

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maple [A]  time = 0.04, size = 12, normalized size = 0.60

method	result	size
derivativedivides	\(\frac {\ln \left (5 x \right )^{4}}{3 x^{7}}\)	\(12\)
default	\(\frac {\ln \left (5 x \right )^{4}}{3 x^{7}}\)	\(12\)
risch	\(\frac {\ln \left (5 x \right )^{4}}{3 x^{7}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-7*ln(5*x)^4+4*ln(5*x)^3)/x^8,x,method=_RETURNVERBOSE)

[Out]

1/3*ln(5*x)^4/x^7

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maxima [B]  time = 0.36, size = 67, normalized size = 3.35 \begin {gather*} \frac {2401 \, \log \left (5 \, x\right )^{4} + 1372 \, \log \left (5 \, x\right )^{3} + 588 \, \log \left (5 \, x\right )^{2} + 168 \, \log \left (5 \, x\right ) + 24}{7203 \, x^{7}} - \frac {4 \, {\left (343 \, \log \left (5 \, x\right )^{3} + 147 \, \log \left (5 \, x\right )^{2} + 42 \, \log \left (5 \, x\right ) + 6\right )}}{7203 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-7*log(5*x)^4+4*log(5*x)^3)/x^8,x, algorithm="maxima")

[Out]

1/7203*(2401*log(5*x)^4 + 1372*log(5*x)^3 + 588*log(5*x)^2 + 168*log(5*x) + 24)/x^7 - 4/7203*(343*log(5*x)^3 +
147*log(5*x)^2 + 42*log(5*x) + 6)/x^7

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mupad [B]  time = 4.02, size = 11, normalized size = 0.55 \begin {gather*} \frac {{\ln \left (5\,x\right )}^4}{3\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*log(5*x)^3)/3 - (7*log(5*x)^4)/3)/x^8,x)

[Out]

log(5*x)^4/(3*x^7)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.50 \begin {gather*} \frac {\log {\left (5 x \right )}^{4}}{3 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-7*ln(5*x)**4+4*ln(5*x)**3)/x**8,x)

[Out]

log(5*x)**4/(3*x**7)

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