3.94.73 \(\int \frac {120 x+44 x^2+4 x^3+(-20 x^2-4 x^3) \log (x)+(72-48 x \log (x)+8 x^2 \log ^2(x)) \log (5+x)}{45+9 x+(-30 x-6 x^2) \log (x)+(5 x^2+x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {4 x^2}{3-x \log (x)}+4 \log ^2(5+x) \]

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Rubi [F]  time = 1.27, 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 {120 x+44 x^2+4 x^3+\left (-20 x^2-4 x^3\right ) \log (x)+\left (72-48 x \log (x)+8 x^2 \log ^2(x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2\right ) \log (x)+\left (5 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(120*x + 44*x^2 + 4*x^3 + (-20*x^2 - 4*x^3)*Log[x] + (72 - 48*x*Log[x] + 8*x^2*Log[x]^2)*Log[5 + x])/(45 +
9*x + (-30*x - 6*x^2)*Log[x] + (5*x^2 + x^3)*Log[x]^2),x]

[Out]

4*Log[5 + x]^2 + 12*Defer[Int][x/(-3 + x*Log[x])^2, x] + 4*Defer[Int][x^2/(-3 + x*Log[x])^2, x] - 4*Defer[Int]
[x/(-3 + x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {120 x+44 x^2+4 x^3+\left (-20 x^2-4 x^3\right ) \log (x)+\left (72-48 x \log (x)+8 x^2 \log ^2(x)\right ) \log (5+x)}{(5+x) (3-x \log (x))^2} \, dx\\ &=\int \left (\frac {120 x}{(5+x) (-3+x \log (x))^2}+\frac {44 x^2}{(5+x) (-3+x \log (x))^2}+\frac {4 x^3}{(5+x) (-3+x \log (x))^2}-\frac {4 x^2 \log (x)}{(-3+x \log (x))^2}+\frac {8 \log (5+x)}{5+x}\right ) \, dx\\ &=4 \int \frac {x^3}{(5+x) (-3+x \log (x))^2} \, dx-4 \int \frac {x^2 \log (x)}{(-3+x \log (x))^2} \, dx+8 \int \frac {\log (5+x)}{5+x} \, dx+44 \int \frac {x^2}{(5+x) (-3+x \log (x))^2} \, dx+120 \int \frac {x}{(5+x) (-3+x \log (x))^2} \, dx\\ &=4 \int \left (\frac {25}{(-3+x \log (x))^2}-\frac {5 x}{(-3+x \log (x))^2}+\frac {x^2}{(-3+x \log (x))^2}-\frac {125}{(5+x) (-3+x \log (x))^2}\right ) \, dx-4 \int \left (\frac {3 x}{(-3+x \log (x))^2}+\frac {x}{-3+x \log (x)}\right ) \, dx+8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,5+x\right )+44 \int \left (-\frac {5}{(-3+x \log (x))^2}+\frac {x}{(-3+x \log (x))^2}+\frac {25}{(5+x) (-3+x \log (x))^2}\right ) \, dx+120 \int \left (\frac {1}{(-3+x \log (x))^2}-\frac {5}{(5+x) (-3+x \log (x))^2}\right ) \, dx\\ &=4 \log ^2(5+x)+4 \int \frac {x^2}{(-3+x \log (x))^2} \, dx-4 \int \frac {x}{-3+x \log (x)} \, dx-12 \int \frac {x}{(-3+x \log (x))^2} \, dx-20 \int \frac {x}{(-3+x \log (x))^2} \, dx+44 \int \frac {x}{(-3+x \log (x))^2} \, dx+100 \int \frac {1}{(-3+x \log (x))^2} \, dx+120 \int \frac {1}{(-3+x \log (x))^2} \, dx-220 \int \frac {1}{(-3+x \log (x))^2} \, dx-500 \int \frac {1}{(5+x) (-3+x \log (x))^2} \, dx-600 \int \frac {1}{(5+x) (-3+x \log (x))^2} \, dx+1100 \int \frac {1}{(5+x) (-3+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} -\frac {4 x^2}{-3+x \log (x)}+4 \log ^2(5+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(120*x + 44*x^2 + 4*x^3 + (-20*x^2 - 4*x^3)*Log[x] + (72 - 48*x*Log[x] + 8*x^2*Log[x]^2)*Log[5 + x])
/(45 + 9*x + (-30*x - 6*x^2)*Log[x] + (5*x^2 + x^3)*Log[x]^2),x]

[Out]

(-4*x^2)/(-3 + x*Log[x]) + 4*Log[5 + x]^2

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fricas [A]  time = 1.32, size = 29, normalized size = 1.26 \begin {gather*} \frac {4 \, {\left ({\left (x \log \relax (x) - 3\right )} \log \left (x + 5\right )^{2} - x^{2}\right )}}{x \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(x)^2-48*x*log(x)+72)*log(5+x)+(-4*x^3-20*x^2)*log(x)+4*x^3+44*x^2+120*x)/((x^3+5*x^2)*lo
g(x)^2+(-6*x^2-30*x)*log(x)+9*x+45),x, algorithm="fricas")

[Out]

4*((x*log(x) - 3)*log(x + 5)^2 - x^2)/(x*log(x) - 3)

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giac [A]  time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} 4 \, \log \left (x + 5\right )^{2} - \frac {4 \, x^{2}}{x \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(x)^2-48*x*log(x)+72)*log(5+x)+(-4*x^3-20*x^2)*log(x)+4*x^3+44*x^2+120*x)/((x^3+5*x^2)*lo
g(x)^2+(-6*x^2-30*x)*log(x)+9*x+45),x, algorithm="giac")

[Out]

4*log(x + 5)^2 - 4*x^2/(x*log(x) - 3)

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maple [A]  time = 0.04, size = 23, normalized size = 1.00

method	result	size
default	\(4 \ln \left (5+x \right )^{2}-\frac {4 x^{2}}{x \ln \relax (x )-3}\)	\(23\)
risch	\(4 \ln \left (5+x \right )^{2}-\frac {4 x^{2}}{x \ln \relax (x )-3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2*ln(x)^2-48*x*ln(x)+72)*ln(5+x)+(-4*x^3-20*x^2)*ln(x)+4*x^3+44*x^2+120*x)/((x^3+5*x^2)*ln(x)^2+(-6*
x^2-30*x)*ln(x)+9*x+45),x,method=_RETURNVERBOSE)

[Out]

4*ln(5+x)^2-4*x^2/(x*ln(x)-3)

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maxima [A]  time = 0.38, size = 29, normalized size = 1.26 \begin {gather*} \frac {4 \, {\left ({\left (x \log \relax (x) - 3\right )} \log \left (x + 5\right )^{2} - x^{2}\right )}}{x \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(x)^2-48*x*log(x)+72)*log(5+x)+(-4*x^3-20*x^2)*log(x)+4*x^3+44*x^2+120*x)/((x^3+5*x^2)*lo
g(x)^2+(-6*x^2-30*x)*log(x)+9*x+45),x, algorithm="maxima")

[Out]

4*((x*log(x) - 3)*log(x + 5)^2 - x^2)/(x*log(x) - 3)

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mupad [B]  time = 7.54, size = 22, normalized size = 0.96 \begin {gather*} 4\,{\ln \left (x+5\right )}^2-\frac {4\,x^2}{x\,\ln \relax (x)-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((120*x - log(x)*(20*x^2 + 4*x^3) + log(x + 5)*(8*x^2*log(x)^2 - 48*x*log(x) + 72) + 44*x^2 + 4*x^3)/(9*x +
log(x)^2*(5*x^2 + x^3) - log(x)*(30*x + 6*x^2) + 45),x)

[Out]

4*log(x + 5)^2 - (4*x^2)/(x*log(x) - 3)

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sympy [A]  time = 0.35, size = 19, normalized size = 0.83 \begin {gather*} - \frac {4 x^{2}}{x \log {\relax (x )} - 3} + 4 \log {\left (x + 5 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2*ln(x)**2-48*x*ln(x)+72)*ln(5+x)+(-4*x**3-20*x**2)*ln(x)+4*x**3+44*x**2+120*x)/((x**3+5*x**2
)*ln(x)**2+(-6*x**2-30*x)*ln(x)+9*x+45),x)

[Out]

-4*x**2/(x*log(x) - 3) + 4*log(x + 5)**2

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