3.41.39 \(\int \frac {-15+5 x-4 x^4+(20 x+8 x^5) \log (x)-4 x^6 \log ^2(x)}{x^4-2 x^5 \log (x)+x^6 \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ 1-4 x+\frac {5}{x^2 \left (x-x^2 \log (x)\right )} \]

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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 {-15+5 x-4 x^4+\left (20 x+8 x^5\right ) \log (x)-4 x^6 \log ^2(x)}{x^4-2 x^5 \log (x)+x^6 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15 + 5*x - 4*x^4 + (20*x + 8*x^5)*Log[x] - 4*x^6*Log[x]^2)/(x^4 - 2*x^5*Log[x] + x^6*Log[x]^2),x]

[Out]

-4*x + 5*Defer[Int][1/(x^4*(-1 + x*Log[x])^2), x] + 5*Defer[Int][1/(x^3*(-1 + x*Log[x])^2), x] + 20*Defer[Int]
[1/(x^4*(-1 + x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-15+5 x-4 x^4+\left (20 x+8 x^5\right ) \log (x)-4 x^6 \log ^2(x)}{x^4 (1-x \log (x))^2} \, dx\\ &=\int \left (-4+\frac {5 (1+x)}{x^4 (-1+x \log (x))^2}+\frac {20}{x^4 (-1+x \log (x))}\right ) \, dx\\ &=-4 x+5 \int \frac {1+x}{x^4 (-1+x \log (x))^2} \, dx+20 \int \frac {1}{x^4 (-1+x \log (x))} \, dx\\ &=-4 x+5 \int \left (\frac {1}{x^4 (-1+x \log (x))^2}+\frac {1}{x^3 (-1+x \log (x))^2}\right ) \, dx+20 \int \frac {1}{x^4 (-1+x \log (x))} \, dx\\ &=-4 x+5 \int \frac {1}{x^4 (-1+x \log (x))^2} \, dx+5 \int \frac {1}{x^3 (-1+x \log (x))^2} \, dx+20 \int \frac {1}{x^4 (-1+x \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 17, normalized size = 0.81 \begin {gather*} -4 x-\frac {5}{x^3 (-1+x \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 + 5*x - 4*x^4 + (20*x + 8*x^5)*Log[x] - 4*x^6*Log[x]^2)/(x^4 - 2*x^5*Log[x] + x^6*Log[x]^2),x]

[Out]

-4*x - 5/(x^3*(-1 + x*Log[x]))

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fricas [A]  time = 0.54, size = 30, normalized size = 1.43 \begin {gather*} -\frac {4 \, x^{5} \log \relax (x) - 4 \, x^{4} + 5}{x^{4} \log \relax (x) - x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6*log(x)^2+(8*x^5+20*x)*log(x)-4*x^4+5*x-15)/(x^6*log(x)^2-2*x^5*log(x)+x^4),x, algorithm="fri
cas")

[Out]

-(4*x^5*log(x) - 4*x^4 + 5)/(x^4*log(x) - x^3)

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giac [A]  time = 0.14, size = 20, normalized size = 0.95 \begin {gather*} -4 \, x - \frac {5}{x^{4} \log \relax (x) - x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6*log(x)^2+(8*x^5+20*x)*log(x)-4*x^4+5*x-15)/(x^6*log(x)^2-2*x^5*log(x)+x^4),x, algorithm="gia
c")

[Out]

-4*x - 5/(x^4*log(x) - x^3)

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maple [A]  time = 0.03, size = 18, normalized size = 0.86

method	result	size
risch	\(-4 x -\frac {5}{x^{3} \left (x \ln \relax (x )-1\right )}\)	\(18\)
norman	\(\frac {-5+4 x^{4}-4 x^{5} \ln \relax (x )}{x^{3} \left (x \ln \relax (x )-1\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^6*ln(x)^2+(8*x^5+20*x)*ln(x)-4*x^4+5*x-15)/(x^6*ln(x)^2-2*x^5*ln(x)+x^4),x,method=_RETURNVERBOSE)

[Out]

-4*x-5/x^3/(x*ln(x)-1)

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maxima [A]  time = 0.39, size = 30, normalized size = 1.43 \begin {gather*} -\frac {4 \, x^{5} \log \relax (x) - 4 \, x^{4} + 5}{x^{4} \log \relax (x) - x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6*log(x)^2+(8*x^5+20*x)*log(x)-4*x^4+5*x-15)/(x^6*log(x)^2-2*x^5*log(x)+x^4),x, algorithm="max
ima")

[Out]

-(4*x^5*log(x) - 4*x^4 + 5)/(x^4*log(x) - x^3)

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mupad [B]  time = 3.22, size = 17, normalized size = 0.81 \begin {gather*} -4\,x-\frac {5}{x^3\,\left (x\,\ln \relax (x)-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^6*log(x)^2 - 5*x - log(x)*(20*x + 8*x^5) + 4*x^4 + 15)/(x^6*log(x)^2 - 2*x^5*log(x) + x^4),x)

[Out]

- 4*x - 5/(x^3*(x*log(x) - 1))

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sympy [A]  time = 0.10, size = 15, normalized size = 0.71 \begin {gather*} - 4 x - \frac {5}{x^{4} \log {\relax (x )} - x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**6*ln(x)**2+(8*x**5+20*x)*ln(x)-4*x**4+5*x-15)/(x**6*ln(x)**2-2*x**5*ln(x)+x**4),x)

[Out]

-4*x - 5/(x**4*log(x) - x**3)

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