3.30.33 \(\int \frac {10 x+160 x^2+3 x^3+(150 x+3 x^2) \log (x)+(10 x^2+10 x \log (x)) \log (x^2+2 x \log (x)+\log ^2(x))}{10 x+10 \log (x)} \, dx\)

Optimal. Leaf size=22 \[ x^2 \left (7+\frac {1}{10} \left (5+x+5 \log \left ((x+\log (x))^2\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.39, 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 {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 x+10 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x + 160*x^2 + 3*x^3 + (150*x + 3*x^2)*Log[x] + (10*x^2 + 10*x*Log[x])*Log[x^2 + 2*x*Log[x] + Log[x]^2]
)/(10*x + 10*Log[x]),x]

[Out]

(15*x^2)/2 + x^3/10 + Defer[Int][x/(x + Log[x]), x] + Defer[Int][x^2/(x + Log[x]), x] + Defer[Int][x*Log[(x +
Log[x])^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{10 (x+\log (x))} \, dx\\ &=\frac {1}{10} \int \frac {10 x+160 x^2+3 x^3+\left (150 x+3 x^2\right ) \log (x)+\left (10 x^2+10 x \log (x)\right ) \log \left (x^2+2 x \log (x)+\log ^2(x)\right )}{x+\log (x)} \, dx\\ &=\frac {1}{10} \int \left (\frac {x \left (10+160 x+3 x^2+150 \log (x)+3 x \log (x)\right )}{x+\log (x)}+10 x \log \left ((x+\log (x))^2\right )\right ) \, dx\\ &=\frac {1}{10} \int \frac {x \left (10+160 x+3 x^2+150 \log (x)+3 x \log (x)\right )}{x+\log (x)} \, dx+\int x \log \left ((x+\log (x))^2\right ) \, dx\\ &=\frac {1}{10} \int \left (3 x (50+x)+\frac {10 x (1+x)}{x+\log (x)}\right ) \, dx+\int x \log \left ((x+\log (x))^2\right ) \, dx\\ &=\frac {3}{10} \int x (50+x) \, dx+\int \frac {x (1+x)}{x+\log (x)} \, dx+\int x \log \left ((x+\log (x))^2\right ) \, dx\\ &=\frac {3}{10} \int \left (50 x+x^2\right ) \, dx+\int \left (\frac {x}{x+\log (x)}+\frac {x^2}{x+\log (x)}\right ) \, dx+\int x \log \left ((x+\log (x))^2\right ) \, dx\\ &=\frac {15 x^2}{2}+\frac {x^3}{10}+\int \frac {x}{x+\log (x)} \, dx+\int \frac {x^2}{x+\log (x)} \, dx+\int x \log \left ((x+\log (x))^2\right ) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 25, normalized size = 1.14 \begin {gather*} \frac {1}{10} \left (75 x^2+x^3+5 x^2 \log \left ((x+\log (x))^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x + 160*x^2 + 3*x^3 + (150*x + 3*x^2)*Log[x] + (10*x^2 + 10*x*Log[x])*Log[x^2 + 2*x*Log[x] + Log
[x]^2])/(10*x + 10*Log[x]),x]

[Out]

(75*x^2 + x^3 + 5*x^2*Log[(x + Log[x])^2])/10

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 30, normalized size = 1.36 \begin {gather*} \frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}\right ) + \frac {15}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x)*log(x)+3*x^3+160*x^2+10*x)/(10*log(
x)+10*x),x, algorithm="fricas")

[Out]

1/10*x^3 + 1/2*x^2*log(x^2 + 2*x*log(x) + log(x)^2) + 15/2*x^2

________________________________________________________________________________________

giac [A]  time = 0.29, size = 30, normalized size = 1.36 \begin {gather*} \frac {1}{10} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}\right ) + \frac {15}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x)*log(x)+3*x^3+160*x^2+10*x)/(10*log(
x)+10*x),x, algorithm="giac")

[Out]

1/10*x^3 + 1/2*x^2*log(x^2 + 2*x*log(x) + log(x)^2) + 15/2*x^2

________________________________________________________________________________________

maple [C]  time = 0.09, size = 94, normalized size = 4.27

method	result	size
risch	\(x^{2} \ln \left (x +\ln \relax (x )\right )+\frac {x^{3}}{10}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )}{4}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )^{2}}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )^{3}}{4}+\frac {15 x^{2}}{2}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x*ln(x)+10*x^2)*ln(ln(x)^2+2*x*ln(x)+x^2)+(3*x^2+150*x)*ln(x)+3*x^3+160*x^2+10*x)/(10*ln(x)+10*x),x,m
ethod=_RETURNVERBOSE)

[Out]

x^2*ln(x+ln(x))+1/10*x^3-1/4*I*Pi*x^2*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)+1/2*I*Pi*x^2*csgn(I*(x+ln(x)))*c
sgn(I*(x+ln(x))^2)^2-1/4*I*Pi*x^2*csgn(I*(x+ln(x))^2)^3+15/2*x^2

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{10} \, x^{3} + x^{2} \log \left (x + \log \relax (x)\right ) + \frac {15}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)+10*x^2)*log(log(x)^2+2*x*log(x)+x^2)+(3*x^2+150*x)*log(x)+3*x^3+160*x^2+10*x)/(10*log(
x)+10*x),x, algorithm="maxima")

[Out]

1/10*x^3 + x^2*log(x + log(x)) + 15/2*x^2

________________________________________________________________________________________

mupad [B]  time = 1.94, size = 30, normalized size = 1.36 \begin {gather*} \frac {15\,x^2}{2}+\frac {x^3}{10}+\frac {x^2\,\ln \left (x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x + log(log(x)^2 + 2*x*log(x) + x^2)*(10*x*log(x) + 10*x^2) + log(x)*(150*x + 3*x^2) + 160*x^2 + 3*x^3
)/(10*x + 10*log(x)),x)

[Out]

(15*x^2)/2 + x^3/10 + (x^2*log(log(x)^2 + 2*x*log(x) + x^2))/2

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 32, normalized size = 1.45 \begin {gather*} \frac {x^{3}}{10} + \frac {x^{2} \log {\left (x^{2} + 2 x \log {\relax (x )} + \log {\relax (x )}^{2} \right )}}{2} + \frac {15 x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*ln(x)+10*x**2)*ln(ln(x)**2+2*x*ln(x)+x**2)+(3*x**2+150*x)*ln(x)+3*x**3+160*x**2+10*x)/(10*ln(
x)+10*x),x)

[Out]

x**3/10 + x**2*log(x**2 + 2*x*log(x) + log(x)**2)/2 + 15*x**2/2

________________________________________________________________________________________