3.68.65 \(\int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx\)

Optimal. Leaf size=14 \[ -15 x^3+\frac {9}{15625 \log ^4(x)} \]

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Rubi [A]  time = 0.18, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {12, 6741, 6742, 2302, 30} \begin {gather*} \frac {9}{15625 \log ^4(x)}-15 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36 - 703125*x^3*Log[x]^5)/(15625*x*Log[x]^5),x]

[Out]

-15*x^3 + 9/(15625*Log[x]^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-36-703125 x^3 \log ^5(x)}{x \log ^5(x)} \, dx}{15625}\\ &=\frac {\int \frac {9 \left (-4-78125 x^3 \log ^5(x)\right )}{x \log ^5(x)} \, dx}{15625}\\ &=\frac {9 \int \frac {-4-78125 x^3 \log ^5(x)}{x \log ^5(x)} \, dx}{15625}\\ &=\frac {9 \int \left (-78125 x^2-\frac {4}{x \log ^5(x)}\right ) \, dx}{15625}\\ &=-15 x^3-\frac {36 \int \frac {1}{x \log ^5(x)} \, dx}{15625}\\ &=-15 x^3-\frac {36 \operatorname {Subst}\left (\int \frac {1}{x^5} \, dx,x,\log (x)\right )}{15625}\\ &=-15 x^3+\frac {9}{15625 \log ^4(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -15 x^3+\frac {9}{15625 \log ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 - 703125*x^3*Log[x]^5)/(15625*x*Log[x]^5),x]

[Out]

-15*x^3 + 9/(15625*Log[x]^4)

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fricas [A]  time = 0.47, size = 17, normalized size = 1.21 \begin {gather*} -\frac {3 \, {\left (78125 \, x^{3} \log \relax (x)^{4} - 3\right )}}{15625 \, \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15625*(-703125*x^3*log(x)^5-36)/x/log(x)^5,x, algorithm="fricas")

[Out]

-3/15625*(78125*x^3*log(x)^4 - 3)/log(x)^4

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giac [A]  time = 0.24, size = 12, normalized size = 0.86 \begin {gather*} -15 \, x^{3} + \frac {9}{15625 \, \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15625*(-703125*x^3*log(x)^5-36)/x/log(x)^5,x, algorithm="giac")

[Out]

-15*x^3 + 9/15625/log(x)^4

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maple [A]  time = 0.02, size = 13, normalized size = 0.93

method	result	size
default	\(-15 x^{3}+\frac {9}{15625 \ln \relax (x )^{4}}\)	\(13\)
risch	\(-15 x^{3}+\frac {9}{15625 \ln \relax (x )^{4}}\)	\(13\)
norman	\(\frac {\frac {9}{15625}-15 x^{3} \ln \relax (x )^{4}}{\ln \relax (x )^{4}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15625*(-703125*x^3*ln(x)^5-36)/x/ln(x)^5,x,method=_RETURNVERBOSE)

[Out]

-15*x^3+9/15625/ln(x)^4

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maxima [A]  time = 0.37, size = 12, normalized size = 0.86 \begin {gather*} -15 \, x^{3} + \frac {9}{15625 \, \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15625*(-703125*x^3*log(x)^5-36)/x/log(x)^5,x, algorithm="maxima")

[Out]

-15*x^3 + 9/15625/log(x)^4

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mupad [B]  time = 4.11, size = 12, normalized size = 0.86 \begin {gather*} \frac {9}{15625\,{\ln \relax (x)}^4}-15\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(45*x^3*log(x)^5 + 36/15625)/(x*log(x)^5),x)

[Out]

9/(15625*log(x)^4) - 15*x^3

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sympy [A]  time = 0.10, size = 12, normalized size = 0.86 \begin {gather*} - 15 x^{3} + \frac {9}{15625 \log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15625*(-703125*x**3*ln(x)**5-36)/x/ln(x)**5,x)

[Out]

-15*x**3 + 9/(15625*log(x)**4)

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