3.46.18 \(\int \frac {90+56 x-5 x^2-30 \log (x)}{576 x^2+240 x^3+25 x^4+(-288 x^2-60 x^3) \log (x)+36 x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ -\log (5)-\frac {5-x}{x (-x+6 (4+x-\log (x)))} \]

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Rubi [F]  time = 0.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {90+56 x-5 x^2-30 \log (x)}{576 x^2+240 x^3+25 x^4+\left (-288 x^2-60 x^3\right ) \log (x)+36 x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(90 + 56*x - 5*x^2 - 30*Log[x])/(576*x^2 + 240*x^3 + 25*x^4 + (-288*x^2 - 60*x^3)*Log[x] + 36*x^2*Log[x]^2
),x]

[Out]

-5*Defer[Int][(24 + 5*x - 6*Log[x])^(-2), x] - 30*Defer[Int][1/(x^2*(24 + 5*x - 6*Log[x])^2), x] + 31*Defer[In
t][1/(x*(24 + 5*x - 6*Log[x])^2), x] + 5*Defer[Int][1/(x^2*(24 + 5*x - 6*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {90+56 x-5 x^2-30 \log (x)}{x^2 (24+5 x-6 \log (x))^2} \, dx\\ &=\int \left (\frac {-30+31 x-5 x^2}{x^2 (24+5 x-6 \log (x))^2}+\frac {5}{x^2 (24+5 x-6 \log (x))}\right ) \, dx\\ &=5 \int \frac {1}{x^2 (24+5 x-6 \log (x))} \, dx+\int \frac {-30+31 x-5 x^2}{x^2 (24+5 x-6 \log (x))^2} \, dx\\ &=5 \int \frac {1}{x^2 (24+5 x-6 \log (x))} \, dx+\int \left (-\frac {5}{(24+5 x-6 \log (x))^2}-\frac {30}{x^2 (24+5 x-6 \log (x))^2}+\frac {31}{x (24+5 x-6 \log (x))^2}\right ) \, dx\\ &=-\left (5 \int \frac {1}{(24+5 x-6 \log (x))^2} \, dx\right )+5 \int \frac {1}{x^2 (24+5 x-6 \log (x))} \, dx-30 \int \frac {1}{x^2 (24+5 x-6 \log (x))^2} \, dx+31 \int \frac {1}{x (24+5 x-6 \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 20, normalized size = 0.67 \begin {gather*} \frac {5-x}{x (-24-5 x+6 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(90 + 56*x - 5*x^2 - 30*Log[x])/(576*x^2 + 240*x^3 + 25*x^4 + (-288*x^2 - 60*x^3)*Log[x] + 36*x^2*Lo
g[x]^2),x]

[Out]

(5 - x)/(x*(-24 - 5*x + 6*Log[x]))

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fricas [A]  time = 0.63, size = 20, normalized size = 0.67 \begin {gather*} \frac {x - 5}{5 \, x^{2} - 6 \, x \log \relax (x) + 24 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*log(x)-5*x^2+56*x+90)/(36*x^2*log(x)^2+(-60*x^3-288*x^2)*log(x)+25*x^4+240*x^3+576*x^2),x, algo
rithm="fricas")

[Out]

(x - 5)/(5*x^2 - 6*x*log(x) + 24*x)

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giac [A]  time = 4.44, size = 20, normalized size = 0.67 \begin {gather*} \frac {x - 5}{5 \, x^{2} - 6 \, x \log \relax (x) + 24 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*log(x)-5*x^2+56*x+90)/(36*x^2*log(x)^2+(-60*x^3-288*x^2)*log(x)+25*x^4+240*x^3+576*x^2),x, algo
rithm="giac")

[Out]

(x - 5)/(5*x^2 - 6*x*log(x) + 24*x)

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maple [A]  time = 0.04, size = 19, normalized size = 0.63

method	result	size
norman	\(\frac {x -5}{x \left (-6 \ln \relax (x )+24+5 x \right )}\)	\(19\)
risch	\(\frac {x -5}{x \left (-6 \ln \relax (x )+24+5 x \right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-30*ln(x)-5*x^2+56*x+90)/(36*x^2*ln(x)^2+(-60*x^3-288*x^2)*ln(x)+25*x^4+240*x^3+576*x^2),x,method=_RETURN
VERBOSE)

[Out]

(x-5)/x/(-6*ln(x)+24+5*x)

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maxima [A]  time = 0.39, size = 20, normalized size = 0.67 \begin {gather*} \frac {x - 5}{5 \, x^{2} - 6 \, x \log \relax (x) + 24 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*log(x)-5*x^2+56*x+90)/(36*x^2*log(x)^2+(-60*x^3-288*x^2)*log(x)+25*x^4+240*x^3+576*x^2),x, algo
rithm="maxima")

[Out]

(x - 5)/(5*x^2 - 6*x*log(x) + 24*x)

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mupad [B]  time = 3.36, size = 18, normalized size = 0.60 \begin {gather*} \frac {x-5}{x\,\left (5\,x-6\,\ln \relax (x)+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((56*x - 30*log(x) - 5*x^2 + 90)/(36*x^2*log(x)^2 - log(x)*(288*x^2 + 60*x^3) + 576*x^2 + 240*x^3 + 25*x^4)
,x)

[Out]

(x - 5)/(x*(5*x - 6*log(x) + 24))

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sympy [A]  time = 0.12, size = 17, normalized size = 0.57 \begin {gather*} \frac {5 - x}{- 5 x^{2} + 6 x \log {\relax (x )} - 24 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*ln(x)-5*x**2+56*x+90)/(36*x**2*ln(x)**2+(-60*x**3-288*x**2)*ln(x)+25*x**4+240*x**3+576*x**2),x)

[Out]

(5 - x)/(-5*x**2 + 6*x*log(x) - 24*x)

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