3.87.35 \(\int \frac {-15-15 x+(-15-10 x) \log (x)}{x^4+(6 x^3+2 x^4) \log (x)+(9 x^2+6 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {4}{x+\frac {1}{5} x (-5+4 (x+(3+x) \log (x)))} \]

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Rubi [F]  time = 1.03, 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 {-15-15 x+(-15-10 x) \log (x)}{x^4+\left (6 x^3+2 x^4\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15 - 15*x + (-15 - 10*x)*Log[x])/(x^4 + (6*x^3 + 2*x^4)*Log[x] + (9*x^2 + 6*x^3 + x^4)*Log[x]^2),x]

[Out]

-15*Defer[Int][1/(x^2*(x + 3*Log[x] + x*Log[x])^2), x] - 10*Defer[Int][1/(x*(x + 3*Log[x] + x*Log[x])^2), x] +
5*Defer[Int][1/((3 + x)*(x + 3*Log[x] + x*Log[x])^2), x] - 5*Defer[Int][1/(x^2*(x + 3*Log[x] + x*Log[x])), x]
- (5*Defer[Int][1/(x*(x + 3*Log[x] + x*Log[x])), x])/3 + (5*Defer[Int][1/((3 + x)*(x + 3*Log[x] + x*Log[x])),
x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 (-3 (1+x)-(3+2 x) \log (x))}{x^2 (x+(3+x) \log (x))^2} \, dx\\ &=5 \int \frac {-3 (1+x)-(3+2 x) \log (x)}{x^2 (x+(3+x) \log (x))^2} \, dx\\ &=5 \int \left (\frac {-9-9 x-x^2}{x^2 (3+x) (x+3 \log (x)+x \log (x))^2}+\frac {-3-2 x}{x^2 (3+x) (x+3 \log (x)+x \log (x))}\right ) \, dx\\ &=5 \int \frac {-9-9 x-x^2}{x^2 (3+x) (x+3 \log (x)+x \log (x))^2} \, dx+5 \int \frac {-3-2 x}{x^2 (3+x) (x+3 \log (x)+x \log (x))} \, dx\\ &=5 \int \left (-\frac {3}{x^2 (x+3 \log (x)+x \log (x))^2}-\frac {2}{x (x+3 \log (x)+x \log (x))^2}+\frac {1}{(3+x) (x+3 \log (x)+x \log (x))^2}\right ) \, dx+5 \int \left (-\frac {1}{x^2 (x+3 \log (x)+x \log (x))}-\frac {1}{3 x (x+3 \log (x)+x \log (x))}+\frac {1}{3 (3+x) (x+3 \log (x)+x \log (x))}\right ) \, dx\\ &=-\left (\frac {5}{3} \int \frac {1}{x (x+3 \log (x)+x \log (x))} \, dx\right )+\frac {5}{3} \int \frac {1}{(3+x) (x+3 \log (x)+x \log (x))} \, dx+5 \int \frac {1}{(3+x) (x+3 \log (x)+x \log (x))^2} \, dx-5 \int \frac {1}{x^2 (x+3 \log (x)+x \log (x))} \, dx-10 \int \frac {1}{x (x+3 \log (x)+x \log (x))^2} \, dx-15 \int \frac {1}{x^2 (x+3 \log (x)+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 17, normalized size = 0.74 \begin {gather*} \frac {5}{x (x+3 \log (x)+x \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 - 15*x + (-15 - 10*x)*Log[x])/(x^4 + (6*x^3 + 2*x^4)*Log[x] + (9*x^2 + 6*x^3 + x^4)*Log[x]^2),x
]

[Out]

5/(x*(x + 3*Log[x] + x*Log[x]))

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fricas [A]  time = 0.70, size = 18, normalized size = 0.78 \begin {gather*} \frac {5}{x^{2} + {\left (x^{2} + 3 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x-15)*log(x)-15*x-15)/((x^4+6*x^3+9*x^2)*log(x)^2+(2*x^4+6*x^3)*log(x)+x^4),x, algorithm="fric
as")

[Out]

5/(x^2 + (x^2 + 3*x)*log(x))

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giac [A]  time = 0.23, size = 19, normalized size = 0.83 \begin {gather*} \frac {5}{x^{2} \log \relax (x) + x^{2} + 3 \, x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x-15)*log(x)-15*x-15)/((x^4+6*x^3+9*x^2)*log(x)^2+(2*x^4+6*x^3)*log(x)+x^4),x, algorithm="giac
")

[Out]

5/(x^2*log(x) + x^2 + 3*x*log(x))

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maple [A]  time = 0.04, size = 18, normalized size = 0.78

method	result	size
norman	\(\frac {5}{x \left (x \ln \relax (x )+3 \ln \relax (x )+x \right )}\)	\(18\)
risch	\(\frac {5}{x \left (x \ln \relax (x )+3 \ln \relax (x )+x \right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x-15)*ln(x)-15*x-15)/((x^4+6*x^3+9*x^2)*ln(x)^2+(2*x^4+6*x^3)*ln(x)+x^4),x,method=_RETURNVERBOSE)

[Out]

5/x/(x*ln(x)+3*ln(x)+x)

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maxima [A]  time = 0.40, size = 18, normalized size = 0.78 \begin {gather*} \frac {5}{x^{2} + {\left (x^{2} + 3 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x-15)*log(x)-15*x-15)/((x^4+6*x^3+9*x^2)*log(x)^2+(2*x^4+6*x^3)*log(x)+x^4),x, algorithm="maxi
ma")

[Out]

5/(x^2 + (x^2 + 3*x)*log(x))

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mupad [B]  time = 5.50, size = 34, normalized size = 1.48 \begin {gather*} \frac {5\,\left (x^2+9\,x+9\right )}{\left (x+\ln \relax (x)\,\left (x+3\right )\right )\,\left (x^3+9\,x^2+9\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(15*x + log(x)*(10*x + 15) + 15)/(log(x)*(6*x^3 + 2*x^4) + log(x)^2*(9*x^2 + 6*x^3 + x^4) + x^4),x)

[Out]

(5*(9*x + x^2 + 9))/((x + log(x)*(x + 3))*(9*x + 9*x^2 + x^3))

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sympy [A]  time = 0.16, size = 14, normalized size = 0.61 \begin {gather*} \frac {5}{x^{2} + \left (x^{2} + 3 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x-15)*ln(x)-15*x-15)/((x**4+6*x**3+9*x**2)*ln(x)**2+(2*x**4+6*x**3)*ln(x)+x**4),x)

[Out]

5/(x**2 + (x**2 + 3*x)*log(x))

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