3.69.100 \(\int \frac {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+(-10 x^2+18 x^3+4 x^4+(30 x^2+6 x^3) \log (x)) \log (5+x)}{45+9 x+(-30 x-6 x^2+(-30-6 x) \log (x)) \log (5+x)+(5 x^2+x^3+(10 x+2 x^2) \log (x)+(5+x) \log ^2(x)) \log ^2(5+x)} \, dx\)

Optimal. Leaf size=20 \[ 27+\frac {2 x^3}{-3+(x+\log (x)) \log (5+x)} \]

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Rubi [F]  time = 14.52, 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 {-90 x^2-18 x^3-2 x^4-2 x^3 \log (x)+\left (-10 x^2+18 x^3+4 x^4+\left (30 x^2+6 x^3\right ) \log (x)\right ) \log (5+x)}{45+9 x+\left (-30 x-6 x^2+(-30-6 x) \log (x)\right ) \log (5+x)+\left (5 x^2+x^3+\left (10 x+2 x^2\right ) \log (x)+(5+x) \log ^2(x)\right ) \log ^2(5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-90*x^2 - 18*x^3 - 2*x^4 - 2*x^3*Log[x] + (-10*x^2 + 18*x^3 + 4*x^4 + (30*x^2 + 6*x^3)*Log[x])*Log[5 + x]
)/(45 + 9*x + (-30*x - 6*x^2 + (-30 - 6*x)*Log[x])*Log[5 + x] + (5*x^2 + x^3 + (10*x + 2*x^2)*Log[x] + (5 + x)
*Log[x]^2)*Log[5 + x]^2),x]

[Out]

-1250*Defer[Int][1/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] + 250*Defer[Int][x/((x + Log[x
])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 56*Defer[Int][x^2/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x
]*Log[5 + x])^2), x] + 4*Defer[Int][x^3/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 2*Defer
[Int][x^4/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] + 6250*Defer[Int][1/((5 + x)*(x + Log[x
])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] + 500*Defer[Int][Log[x]/((x + Log[x])*(-3 + x*Log[5 + x] + L
og[x]*Log[5 + x])^2), x] - 100*Defer[Int][(x*Log[x])/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2),
x] + 20*Defer[Int][(x^2*Log[x])/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 4*Defer[Int][(
x^3*Log[x])/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 2500*Defer[Int][Log[x]/((5 + x)*(x
+ Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 50*Defer[Int][Log[x]^2/((x + Log[x])*(-3 + x*Log[5
+ x] + Log[x]*Log[5 + x])^2), x] + 10*Defer[Int][(x*Log[x]^2)/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5
+ x])^2), x] - 2*Defer[Int][(x^2*Log[x]^2)/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] + 250*
Defer[Int][Log[x]^2/((5 + x)*(x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])^2), x] - 2*Defer[Int][x^2/((
x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])), x] + 4*Defer[Int][x^3/((x + Log[x])*(-3 + x*Log[5 + x] +
Log[x]*Log[5 + x])), x] + 6*Defer[Int][(x^2*Log[x])/((x + Log[x])*(-3 + x*Log[5 + x] + Log[x]*Log[5 + x])), x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2 \left (-45-9 x-x^2+\left (-5+9 x+2 x^2\right ) \log (5+x)+\log (x) (-x+3 (5+x) \log (5+x))\right )}{(5+x) (3-(x+\log (x)) \log (5+x))^2} \, dx\\ &=2 \int \frac {x^2 \left (-45-9 x-x^2+\left (-5+9 x+2 x^2\right ) \log (5+x)+\log (x) (-x+3 (5+x) \log (5+x))\right )}{(5+x) (3-(x+\log (x)) \log (5+x))^2} \, dx\\ &=2 \int \left (-\frac {x^2 \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x^2 (-1+2 x+3 \log (x))}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))}\right ) \, dx\\ &=-\left (2 \int \frac {x^2 \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx\right )+2 \int \frac {x^2 (-1+2 x+3 \log (x))}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx\\ &=-\left (2 \int \left (-\frac {5 \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {25 \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}\right ) \, dx\right )+2 \int \left (-\frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))}+\frac {2 x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))}+\frac {3 x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)\right )}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx\right )-2 \int \frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+4 \int \frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+6 \int \frac {x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+10 \int \frac {15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-50 \int \frac {15+18 x+3 x^2+x^3+2 x^2 \log (x)+x \log ^2(x)}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx\\ &=-\left (2 \int \frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx\right )-2 \int \left (\frac {15 x}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {18 x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {3 x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x^4}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {2 x^3 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x^2 \log ^2(x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}\right ) \, dx+4 \int \frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+6 \int \frac {x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+10 \int \left (\frac {15}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {18 x}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {3 x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {2 x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x \log ^2(x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}\right ) \, dx-50 \int \left (\frac {15}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {18 x}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {3 x^2}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x^3}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {2 x^2 \log (x)}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}+\frac {x \log ^2(x)}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^4}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx\right )-2 \int \frac {x^2 \log ^2(x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-2 \int \frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx-4 \int \frac {x^3 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+4 \int \frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx-6 \int \frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+6 \int \frac {x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))} \, dx+10 \int \frac {x^3}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+10 \int \frac {x \log ^2(x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+20 \int \frac {x^2 \log (x)}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-30 \int \frac {x}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+30 \int \frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-36 \int \frac {x^2}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-50 \int \frac {x^3}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-50 \int \frac {x \log ^2(x)}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-100 \int \frac {x^2 \log (x)}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+150 \int \frac {1}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-150 \int \frac {x^2}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx+180 \int \frac {x}{(x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-750 \int \frac {1}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx-900 \int \frac {x}{(5+x) (x+\log (x)) (-3+x \log (5+x)+\log (x) \log (5+x))^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.38, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 x^3}{-3+(x+\log (x)) \log (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90*x^2 - 18*x^3 - 2*x^4 - 2*x^3*Log[x] + (-10*x^2 + 18*x^3 + 4*x^4 + (30*x^2 + 6*x^3)*Log[x])*Log[
5 + x])/(45 + 9*x + (-30*x - 6*x^2 + (-30 - 6*x)*Log[x])*Log[5 + x] + (5*x^2 + x^3 + (10*x + 2*x^2)*Log[x] + (
5 + x)*Log[x]^2)*Log[5 + x]^2),x]

[Out]

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

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fricas [A]  time = 0.78, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, x^{3}}{{\left (x + \log \relax (x)\right )} \log \left (x + 5\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log(x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(
x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*log(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="fr
icas")

[Out]

2*x^3/((x + log(x))*log(x + 5) - 3)

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giac [A]  time = 0.30, size = 22, normalized size = 1.10 \begin {gather*} \frac {2 \, x^{3}}{x \log \left (x + 5\right ) + \log \left (x + 5\right ) \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log(x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(
x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*log(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="gi
ac")

[Out]

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

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maple [A]  time = 0.03, size = 23, normalized size = 1.15

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((6*x^3+30*x^2)*ln(x)+4*x^4+18*x^3-10*x^2)*ln(5+x)-2*x^3*ln(x)-2*x^4-18*x^3-90*x^2)/(((5+x)*ln(x)^2+(2*x^
2+10*x)*ln(x)+x^3+5*x^2)*ln(5+x)^2+((-6*x-30)*ln(x)-6*x^2-30*x)*ln(5+x)+9*x+45),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 18, normalized size = 0.90 \begin {gather*} \frac {2 \, x^{3}}{{\left (x + \log \relax (x)\right )} \log \left (x + 5\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^3+30*x^2)*log(x)+4*x^4+18*x^3-10*x^2)*log(5+x)-2*x^3*log(x)-2*x^4-18*x^3-90*x^2)/(((5+x)*log(
x)^2+(2*x^2+10*x)*log(x)+x^3+5*x^2)*log(5+x)^2+((-6*x-30)*log(x)-6*x^2-30*x)*log(5+x)+9*x+45),x, algorithm="ma
xima")

[Out]

2*x^3/((x + log(x))*log(x + 5) - 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {2\,x^3\,\ln \relax (x)-\ln \left (x+5\right )\,\left (\ln \relax (x)\,\left (6\,x^3+30\,x^2\right )-10\,x^2+18\,x^3+4\,x^4\right )+90\,x^2+18\,x^3+2\,x^4}{\left ({\ln \relax (x)}^2\,\left (x+5\right )+\ln \relax (x)\,\left (2\,x^2+10\,x\right )+5\,x^2+x^3\right )\,{\ln \left (x+5\right )}^2+\left (-30\,x-\ln \relax (x)\,\left (6\,x+30\right )-6\,x^2\right )\,\ln \left (x+5\right )+9\,x+45} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^3*log(x) - log(x + 5)*(log(x)*(30*x^2 + 6*x^3) - 10*x^2 + 18*x^3 + 4*x^4) + 90*x^2 + 18*x^3 + 2*x^4)
/(9*x - log(x + 5)*(30*x + log(x)*(6*x + 30) + 6*x^2) + log(x + 5)^2*(log(x)^2*(x + 5) + log(x)*(10*x + 2*x^2)
+ 5*x^2 + x^3) + 45),x)

[Out]

int(-(2*x^3*log(x) - log(x + 5)*(log(x)*(30*x^2 + 6*x^3) - 10*x^2 + 18*x^3 + 4*x^4) + 90*x^2 + 18*x^3 + 2*x^4)
/(9*x - log(x + 5)*(30*x + log(x)*(6*x + 30) + 6*x^2) + log(x + 5)^2*(log(x)^2*(x + 5) + log(x)*(10*x + 2*x^2)
+ 5*x^2 + x^3) + 45), x)

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sympy [A]  time = 0.37, size = 15, normalized size = 0.75 \begin {gather*} \frac {2 x^{3}}{\left (x + \log {\relax (x )}\right ) \log {\left (x + 5 \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x**3+30*x**2)*ln(x)+4*x**4+18*x**3-10*x**2)*ln(5+x)-2*x**3*ln(x)-2*x**4-18*x**3-90*x**2)/(((5+x
)*ln(x)**2+(2*x**2+10*x)*ln(x)+x**3+5*x**2)*ln(5+x)**2+((-6*x-30)*ln(x)-6*x**2-30*x)*ln(5+x)+9*x+45),x)

[Out]

2*x**3/((x + log(x))*log(x + 5) - 3)

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