\(\int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx\) [6765]

Optimal result

Integrand size = 22, antiderivative size = 14 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=-15 x^3+\frac {9}{15625 \log ^4(x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {12, 6873, 6874, 2339, 30} \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=\frac {9}{15625 \log ^4(x)}-15 x^3 \]

[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 2339

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 6873

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

Rule 6874

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

Rubi steps \begin{align*} \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 \text {Subst}\left (\int \frac {1}{x^5} \, dx,x,\log (x)\right )}{15625} \\ & = -15 x^3+\frac {9}{15625 \log ^4(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=-15 x^3+\frac {9}{15625 \log ^4(x)} \]

[In]

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

[Out]

-15*x^3 + 9/(15625*Log[x]^4)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(-15 x^{3}+\frac {9}{15625 \ln \left (x \right )^{4}}\)	\(13\)
risch	\(-15 x^{3}+\frac {9}{15625 \ln \left (x \right )^{4}}\)	\(13\)
parts	\(-15 x^{3}+\frac {9}{15625 \ln \left (x \right )^{4}}\)	\(13\)
norman	\(\frac {\frac {9}{15625}-15 x^{3} \ln \left (x \right )^{4}}{\ln \left (x \right )^{4}}\)	\(17\)
parallelrisch	\(-\frac {234375 x^{3} \ln \left (x \right )^{4}-9}{15625 \ln \left (x \right )^{4}}\)	\(18\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=-\frac {3 \, {\left (78125 \, x^{3} \log \left (x\right )^{4} - 3\right )}}{15625 \, \log \left (x\right )^{4}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=- 15 x^{3} + \frac {9}{15625 \log {\left (x \right )}^{4}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=-15 \, x^{3} + \frac {9}{15625 \, \log \left (x\right )^{4}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=-15 \, x^{3} + \frac {9}{15625 \, \log \left (x\right )^{4}} \]

[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

Mupad [B] (verification not implemented)

Time = 12.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-36-703125 x^3 \log ^5(x)}{15625 x \log ^5(x)} \, dx=\frac {9}{15625\,{\ln \left (x\right )}^4}-15\,x^3 \]

[In]

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

[Out]

9/(15625*log(x)^4) - 15*x^3