3.63.98 \(\int \frac {375 x+(3-375 x) \log (\frac {1}{125} (1-125 x))}{-x^2+125 x^3+(-2 x+250 x^2) \log (\frac {1}{125} (1-125 x))+(-1+125 x) \log ^2(\frac {1}{125} (1-125 x))} \, dx\)

Optimal. Leaf size=21 \[ \frac {3 x}{x+\frac {x^2}{\log \left (\frac {1}{125}-x\right )}} \]

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Rubi [A]  time = 0.18, antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6688, 6711, 32} \begin {gather*} \frac {3}{\frac {x}{\log \left (\frac {1}{125}-x\right )}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(375*x + (3 - 375*x)*Log[(1 - 125*x)/125])/(-x^2 + 125*x^3 + (-2*x + 250*x^2)*Log[(1 - 125*x)/125] + (-1 +
125*x)*Log[(1 - 125*x)/125]^2),x]

[Out]

3/(1 + x/Log[1/125 - x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6688

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

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-375 x-(3-375 x) \log \left (\frac {1}{125}-x\right )}{(1-125 x) \left (x+\log \left (\frac {1}{125}-x\right )\right )^2} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{\log \left (\frac {1}{125}-x\right )}\right )\right )\\ &=\frac {3}{1+\frac {x}{\log \left (\frac {1}{125}-x\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 15, normalized size = 0.71 \begin {gather*} -\frac {3 x}{x+\log \left (\frac {1}{125}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(375*x + (3 - 375*x)*Log[(1 - 125*x)/125])/(-x^2 + 125*x^3 + (-2*x + 250*x^2)*Log[(1 - 125*x)/125] +
(-1 + 125*x)*Log[(1 - 125*x)/125]^2),x]

[Out]

(-3*x)/(x + Log[1/125 - x])

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fricas [A]  time = 0.52, size = 13, normalized size = 0.62 \begin {gather*} -\frac {3 \, x}{x + \log \left (-x + \frac {1}{125}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-375*x+3)*log(-x+1/125)+375*x)/((125*x-1)*log(-x+1/125)^2+(250*x^2-2*x)*log(-x+1/125)+125*x^3-x^2)
,x, algorithm="fricas")

[Out]

-3*x/(x + log(-x + 1/125))

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giac [A]  time = 0.74, size = 13, normalized size = 0.62 \begin {gather*} -\frac {3 \, x}{x + \log \left (-x + \frac {1}{125}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-375*x+3)*log(-x+1/125)+375*x)/((125*x-1)*log(-x+1/125)^2+(250*x^2-2*x)*log(-x+1/125)+125*x^3-x^2)
,x, algorithm="giac")

[Out]

-3*x/(x + log(-x + 1/125))

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maple [A]  time = 0.09, size = 14, normalized size = 0.67

method	result	size
risch	\(-\frac {3 x}{\ln \left (-x +\frac {1}{125}\right )+x}\)	\(14\)
norman	\(\frac {3 \ln \left (-x +\frac {1}{125}\right )}{\ln \left (-x +\frac {1}{125}\right )+x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-375*x+3)*ln(-x+1/125)+375*x)/((125*x-1)*ln(-x+1/125)^2+(250*x^2-2*x)*ln(-x+1/125)+125*x^3-x^2),x,method
=_RETURNVERBOSE)

[Out]

-3*x/(ln(-x+1/125)+x)

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maxima [A]  time = 0.50, size = 17, normalized size = 0.81 \begin {gather*} -\frac {3 \, x}{x - 3 \, \log \relax (5) + \log \left (-125 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-375*x+3)*log(-x+1/125)+375*x)/((125*x-1)*log(-x+1/125)^2+(250*x^2-2*x)*log(-x+1/125)+125*x^3-x^2)
,x, algorithm="maxima")

[Out]

-3*x/(x - 3*log(5) + log(-125*x + 1))

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mupad [B]  time = 0.24, size = 13, normalized size = 0.62 \begin {gather*} -\frac {3\,x}{x+\ln \left (\frac {1}{125}-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(375*x - log(1/125 - x)*(375*x - 3))/(log(1/125 - x)*(2*x - 250*x^2) - log(1/125 - x)^2*(125*x - 1) + x^2
- 125*x^3),x)

[Out]

-(3*x)/(x + log(1/125 - x))

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sympy [A]  time = 0.12, size = 12, normalized size = 0.57 \begin {gather*} - \frac {3 x}{x + \log {\left (\frac {1}{125} - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-375*x+3)*ln(-x+1/125)+375*x)/((125*x-1)*ln(-x+1/125)**2+(250*x**2-2*x)*ln(-x+1/125)+125*x**3-x**2
),x)

[Out]

-3*x/(x + log(1/125 - x))

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