3.101.69 \(\int \frac {300-60 x-25 x^2-3 x^3}{-300 x+12 x^2-25 x^3+x^4} \, dx\)

Optimal. Leaf size=16 \[ \log \left (\frac {\frac {12}{x}+x}{(25-x)^4}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2074, 260} \begin {gather*} \log \left (x^2+12\right )-4 \log (25-x)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(300 - 60*x - 25*x^2 - 3*x^3)/(-300*x + 12*x^2 - 25*x^3 + x^4),x]

[Out]

-4*Log[25 - x] - Log[x] + Log[12 + x^2]

Rule 260

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

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{-25+x}-\frac {1}{x}+\frac {2 x}{12+x^2}\right ) \, dx\\ &=-4 \log (25-x)-\log (x)+2 \int \frac {x}{12+x^2} \, dx\\ &=-4 \log (25-x)-\log (x)+\log \left (12+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.19 \begin {gather*} -4 \log (25-x)-\log (x)+\log \left (12+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(300 - 60*x - 25*x^2 - 3*x^3)/(-300*x + 12*x^2 - 25*x^3 + x^4),x]

[Out]

-4*Log[25 - x] - Log[x] + Log[12 + x^2]

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fricas [A]  time = 0.62, size = 17, normalized size = 1.06 \begin {gather*} \log \left (x^{2} + 12\right ) - 4 \, \log \left (x - 25\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^3-25*x^2-60*x+300)/(x^4-25*x^3+12*x^2-300*x),x, algorithm="fricas")

[Out]

log(x^2 + 12) - 4*log(x - 25) - log(x)

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giac [A]  time = 0.14, size = 19, normalized size = 1.19 \begin {gather*} \log \left (x^{2} + 12\right ) - 4 \, \log \left ({\left | x - 25 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^3-25*x^2-60*x+300)/(x^4-25*x^3+12*x^2-300*x),x, algorithm="giac")

[Out]

log(x^2 + 12) - 4*log(abs(x - 25)) - log(abs(x))

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maple [A]  time = 0.03, size = 18, normalized size = 1.12

method	result	size
default	\(-\ln \relax (x )-4 \ln \left (x -25\right )+\ln \left (x^{2}+12\right )\)	\(18\)
norman	\(-\ln \relax (x )-4 \ln \left (x -25\right )+\ln \left (x^{2}+12\right )\)	\(18\)
risch	\(-\ln \relax (x )-4 \ln \left (x -25\right )+\ln \left (x^{2}+12\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^3-25*x^2-60*x+300)/(x^4-25*x^3+12*x^2-300*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-4*ln(x-25)+ln(x^2+12)

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maxima [A]  time = 0.37, size = 17, normalized size = 1.06 \begin {gather*} \log \left (x^{2} + 12\right ) - 4 \, \log \left (x - 25\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^3-25*x^2-60*x+300)/(x^4-25*x^3+12*x^2-300*x),x, algorithm="maxima")

[Out]

log(x^2 + 12) - 4*log(x - 25) - log(x)

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mupad [B]  time = 0.13, size = 17, normalized size = 1.06 \begin {gather*} \ln \left (x^2+12\right )-4\,\ln \left (x-25\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((60*x + 25*x^2 + 3*x^3 - 300)/(300*x - 12*x^2 + 25*x^3 - x^4),x)

[Out]

log(x^2 + 12) - 4*log(x - 25) - log(x)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.94 \begin {gather*} - \log {\relax (x )} - 4 \log {\left (x - 25 \right )} + \log {\left (x^{2} + 12 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**3-25*x**2-60*x+300)/(x**4-25*x**3+12*x**2-300*x),x)

[Out]

-log(x) - 4*log(x - 25) + log(x**2 + 12)

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