\(\int \frac {-9 x-456 x^2+840 x^4+(-54 x+420 x^2+1680 x^3) \log (x)+(-27+840 x^2) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx\) [7991]

Optimal result

Integrand size = 65, antiderivative size = 23 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=5+\left (420+\frac {9}{x}\right ) \left (3+2 x+\frac {x}{x+\log (x)}\right ) \]

[Out]

5+(9/x+420)*(x/(x+ln(x))+3+2*x)

Rubi [F]

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

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=3 \left (\frac {9}{x}+280 x+\frac {3+140 x}{x+\log (x)}\right ) \]

[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]))

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {840 x^{2}+27}{x}+\frac {420 x +9}{x +\ln \left (x \right )}\)	\(27\)
norman	\(\frac {-420 x \ln \left (x \right )+840 x^{2} \ln \left (x \right )+36 x +840 x^{3}+27 \ln \left (x \right )}{\left (x +\ln \left (x \right )\right ) x}\)	\(36\)
parallelrisch	\(\frac {840 x^{3}+840 x^{2} \ln \left (x \right )+420 x^{2}+36 x +27 \ln \left (x \right )}{x \left (x +\ln \left (x \right )\right )}\)	\(36\)
default	\(\frac {840 x^{3}+840 x^{2} \ln \left (x \right )+420 x^{2}+36 x +27 \ln \left (x \right )}{x \left (x +\ln \left (x \right )\right )}\)	\(37\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=\frac {3 \, {\left (280 \, x^{3} + 140 \, x^{2} + {\left (280 \, x^{2} + 9\right )} \log \left (x\right ) + 12 \, x\right )}}{x^{2} + x \log \left (x\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=840 x + \frac {420 x + 9}{x + \log {\left (x \right )}} + \frac {27}{x} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=\frac {3 \, {\left (280 \, x^{3} + 140 \, x^{2} + {\left (280 \, x^{2} + 9\right )} \log \left (x\right ) + 12 \, x\right )}}{x^{2} + x \log \left (x\right )} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=840 \, x + \frac {3 \, {\left (140 \, x + 3\right )}}{x + \log \left (x\right )} + \frac {27}{x} \]

[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

Mupad [B] (verification not implemented)

Time = 12.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-9 x-456 x^2+840 x^4+\left (-54 x+420 x^2+1680 x^3\right ) \log (x)+\left (-27+840 x^2\right ) \log ^2(x)}{x^4+2 x^3 \log (x)+x^2 \log ^2(x)} \, dx=840\,x+\frac {27\,\ln \left (x\right )-x\,\left (420\,\ln \left (x\right )-36\right )}{x\,\left (x+\ln \left (x\right )\right )} \]

[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)))