3.31.69 \(\int \frac {250-75 x^4-100 \log (x)}{-6 x-x^5+4 x \log (x)} \, dx\)

Optimal. Leaf size=20 \[ 5 \left (4-5 \log \left (\frac {x}{6+x^4-4 \log (x)}\right )\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6741, 12, 6742, 6684} \begin {gather*} 25 \log \left (x^4-4 \log (x)+6\right )-25 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(250 - 75*x^4 - 100*Log[x])/(-6*x - x^5 + 4*x*Log[x]),x]

[Out]

-25*Log[x] + 25*Log[6 + x^4 - 4*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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

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} &=\int \frac {25 \left (-10+3 x^4+4 \log (x)\right )}{6 x+x^5-4 x \log (x)} \, dx\\ &=25 \int \frac {-10+3 x^4+4 \log (x)}{6 x+x^5-4 x \log (x)} \, dx\\ &=25 \int \left (-\frac {1}{x}+\frac {4 \left (-1+x^4\right )}{x \left (6+x^4-4 \log (x)\right )}\right ) \, dx\\ &=-25 \log (x)+100 \int \frac {-1+x^4}{x \left (6+x^4-4 \log (x)\right )} \, dx\\ &=-25 \log (x)+25 \log \left (6+x^4-4 \log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 17, normalized size = 0.85 \begin {gather*} 25 \left (-\log (x)+\log \left (6+x^4-4 \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(250 - 75*x^4 - 100*Log[x])/(-6*x - x^5 + 4*x*Log[x]),x]

[Out]

25*(-Log[x] + Log[6 + x^4 - 4*Log[x]])

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fricas [A]  time = 0.62, size = 19, normalized size = 0.95 \begin {gather*} 25 \, \log \left (-x^{4} + 4 \, \log \relax (x) - 6\right ) - 25 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x)-75*x^4+250)/(4*x*log(x)-x^5-6*x),x, algorithm="fricas")

[Out]

25*log(-x^4 + 4*log(x) - 6) - 25*log(x)

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giac [A]  time = 0.19, size = 19, normalized size = 0.95 \begin {gather*} 25 \, \log \left (-x^{4} + 4 \, \log \relax (x) - 6\right ) - 25 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x)-75*x^4+250)/(4*x*log(x)-x^5-6*x),x, algorithm="giac")

[Out]

25*log(-x^4 + 4*log(x) - 6) - 25*log(x)

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maple [A]  time = 0.02, size = 18, normalized size = 0.90

method	result	size
norman	\(-25 \ln \relax (x )+25 \ln \left (6+x^{4}-4 \ln \relax (x )\right )\)	\(18\)
risch	\(-25 \ln \relax (x )+25 \ln \left (-\frac {x^{4}}{4}+\ln \relax (x )-\frac {3}{2}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-100*ln(x)-75*x^4+250)/(4*x*ln(x)-x^5-6*x),x,method=_RETURNVERBOSE)

[Out]

-25*ln(x)+25*ln(6+x^4-4*ln(x))

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maxima [A]  time = 0.62, size = 17, normalized size = 0.85 \begin {gather*} 25 \, \log \left (-\frac {1}{4} \, x^{4} + \log \relax (x) - \frac {3}{2}\right ) - 25 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x)-75*x^4+250)/(4*x*log(x)-x^5-6*x),x, algorithm="maxima")

[Out]

25*log(-1/4*x^4 + log(x) - 3/2) - 25*log(x)

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mupad [B]  time = 1.84, size = 17, normalized size = 0.85 \begin {gather*} 25\,\ln \left (x^4-4\,\ln \relax (x)+6\right )-25\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*log(x) + 75*x^4 - 250)/(6*x - 4*x*log(x) + x^5),x)

[Out]

25*log(x^4 - 4*log(x) + 6) - 25*log(x)

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sympy [A]  time = 0.13, size = 19, normalized size = 0.95 \begin {gather*} - 25 \log {\relax (x )} + 25 \log {\left (- \frac {x^{4}}{4} + \log {\relax (x )} - \frac {3}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*ln(x)-75*x**4+250)/(4*x*ln(x)-x**5-6*x),x)

[Out]

-25*log(x) + 25*log(-x**4/4 + log(x) - 3/2)

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