3.80.91 \(\int \frac {-9 x-456 x^2+840 x^4+(-54 x+420 x^2+1680 x^3) \log (x)+(-27+840 x^2) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ 5+\left (420+\frac {9}{x}\right ) \left (3+2 x+\frac {x}{x+\log (x)}\right ) \]

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Rubi [F]  time = 0.48, 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 {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9*x - 456*x^2 + 840*x^4 + (-54*x + 420*x^2 + 1680*x^3)*Log[x] + (-27 + 840*x^2)*Log[x]^2)/(x^4 + 2*x^3*L
og[x] + x^2*Log[x]^2),x]

[Out]

27/x + 840*x - 429*Defer[Int][(x + Log[x])^(-2), x] - 9*Defer[Int][1/(x*(x + Log[x])^2), x] - 420*Defer[Int][x
/(x + Log[x])^2, x] + 420*Defer[Int][(x + Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^2 (x+\log (x))^2} \, dx\\ &=\int \left (\frac {3 \left (-9+280 x^2\right )}{x^2}-\frac {3 \left (3+143 x+140 x^2\right )}{x (x+\log (x))^2}+\frac {420}{x+\log (x)}\right ) \, dx\\ &=3 \int \frac {-9+280 x^2}{x^2} \, dx-3 \int \frac {3+143 x+140 x^2}{x (x+\log (x))^2} \, dx+420 \int \frac {1}{x+\log (x)} \, dx\\ &=3 \int \left (280-\frac {9}{x^2}\right ) \, dx-3 \int \left (\frac {143}{(x+\log (x))^2}+\frac {3}{x (x+\log (x))^2}+\frac {140 x}{(x+\log (x))^2}\right ) \, dx+420 \int \frac {1}{x+\log (x)} \, dx\\ &=\frac {27}{x}+840 x-9 \int \frac {1}{x (x+\log (x))^2} \, dx-420 \int \frac {x}{(x+\log (x))^2} \, dx+420 \int \frac {1}{x+\log (x)} \, dx-429 \int \frac {1}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 23, normalized size = 1.00 \begin {gather*} 3 \left (\frac {9}{x}+280 x+\frac {3+140 x}{x+\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x - 456*x^2 + 840*x^4 + (-54*x + 420*x^2 + 1680*x^3)*Log[x] + (-27 + 840*x^2)*Log[x]^2)/(x^4 + 2
*x^3*Log[x] + x^2*Log[x]^2),x]

[Out]

3*(9/x + 280*x + (3 + 140*x)/(x + Log[x]))

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fricas [A]  time = 0.62, size = 36, normalized size = 1.57 \begin {gather*} \frac {3 \, {\left (280 \, x^{3} + 140 \, x^{2} + {\left (280 \, x^{2} + 9\right )} \log \relax (x) + 12 \, x\right )}}{x^{2} + x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((840*x^2-27)*log(x)^2+(1680*x^3+420*x^2-54*x)*log(x)+840*x^4-456*x^2-9*x)/(x^2*log(x)^2+2*x^3*log(x
)+x^4),x, algorithm="fricas")

[Out]

3*(280*x^3 + 140*x^2 + (280*x^2 + 9)*log(x) + 12*x)/(x^2 + x*log(x))

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giac [A]  time = 0.13, size = 22, normalized size = 0.96 \begin {gather*} 840 \, x + \frac {3 \, {\left (140 \, x + 3\right )}}{x + \log \relax (x)} + \frac {27}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((840*x^2-27)*log(x)^2+(1680*x^3+420*x^2-54*x)*log(x)+840*x^4-456*x^2-9*x)/(x^2*log(x)^2+2*x^3*log(x
)+x^4),x, algorithm="giac")

[Out]

840*x + 3*(140*x + 3)/(x + log(x)) + 27/x

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maple [A]  time = 0.04, size = 27, normalized size = 1.17

method	result	size
risch	\(\frac {840 x^{2}+27}{x}+\frac {420 x +9}{x +\ln \relax (x )}\)	\(27\)
norman	\(\frac {-420 x \ln \relax (x )+840 x^{2} \ln \relax (x )+36 x +840 x^{3}+27 \ln \relax (x )}{\left (x +\ln \relax (x )\right ) x}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((840*x^2-27)*ln(x)^2+(1680*x^3+420*x^2-54*x)*ln(x)+840*x^4-456*x^2-9*x)/(x^2*ln(x)^2+2*x^3*ln(x)+x^4),x,m
ethod=_RETURNVERBOSE)

[Out]

3*(280*x^2+9)/x+3*(140*x+3)/(x+ln(x))

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maxima [A]  time = 0.39, size = 36, normalized size = 1.57 \begin {gather*} \frac {3 \, {\left (280 \, x^{3} + 140 \, x^{2} + {\left (280 \, x^{2} + 9\right )} \log \relax (x) + 12 \, x\right )}}{x^{2} + x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((840*x^2-27)*log(x)^2+(1680*x^3+420*x^2-54*x)*log(x)+840*x^4-456*x^2-9*x)/(x^2*log(x)^2+2*x^3*log(x
)+x^4),x, algorithm="maxima")

[Out]

3*(280*x^3 + 140*x^2 + (280*x^2 + 9)*log(x) + 12*x)/(x^2 + x*log(x))

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mupad [B]  time = 5.18, size = 28, normalized size = 1.22 \begin {gather*} 840\,x+\frac {27\,\ln \relax (x)-x\,\left (420\,\ln \relax (x)-36\right )}{x\,\left (x+\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(840*x^2 - 27) - 9*x - 456*x^2 + 840*x^4 + log(x)*(420*x^2 - 54*x + 1680*x^3))/(2*x^3*log(x) + x
^2*log(x)^2 + x^4),x)

[Out]

840*x + (27*log(x) - x*(420*log(x) - 36))/(x*(x + log(x)))

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sympy [A]  time = 0.12, size = 15, normalized size = 0.65 \begin {gather*} 840 x + \frac {420 x + 9}{x + \log {\relax (x )}} + \frac {27}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((840*x**2-27)*ln(x)**2+(1680*x**3+420*x**2-54*x)*ln(x)+840*x**4-456*x**2-9*x)/(x**2*ln(x)**2+2*x**3
*ln(x)+x**4),x)

[Out]

840*x + (420*x + 9)/(x + log(x)) + 27/x

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