3.90.9 \(\int \frac {75+225 x+(150 x-30 e x+30 x^2-150 x \log (5)) \log (\frac {3}{x})+(30 x+e^2 x+10 x^2+x^3+e (-10 x-2 x^2)+(-50 x+10 e x-10 x^2) \log (5)+25 x \log ^2(5)) \log ^2(\frac {3}{x})}{225 x+(150 x-30 e x+30 x^2-150 x \log (5)) \log (\frac {3}{x})+(25 x+e^2 x+10 x^2+x^3+e (-10 x-2 x^2)+(-50 x+10 e x-10 x^2) \log (5)+25 x \log ^2(5)) \log ^2(\frac {3}{x})} \, dx\)

Optimal. Leaf size=29 \[ x+\frac {5}{-5+e-x-5 \left (-\log (5)+\frac {3}{\log \left (\frac {3}{x}\right )}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 3.77, 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 {75+225 x+\left (150 x-30 e x+30 x^2-150 x \log (5)\right ) \log \left (\frac {3}{x}\right )+\left (30 x+e^2 x+10 x^2+x^3+e \left (-10 x-2 x^2\right )+\left (-50 x+10 e x-10 x^2\right ) \log (5)+25 x \log ^2(5)\right ) \log ^2\left (\frac {3}{x}\right )}{225 x+\left (150 x-30 e x+30 x^2-150 x \log (5)\right ) \log \left (\frac {3}{x}\right )+\left (25 x+e^2 x+10 x^2+x^3+e \left (-10 x-2 x^2\right )+\left (-50 x+10 e x-10 x^2\right ) \log (5)+25 x \log ^2(5)\right ) \log ^2\left (\frac {3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(75 + 225*x + (150*x - 30*E*x + 30*x^2 - 150*x*Log[5])*Log[3/x] + (30*x + E^2*x + 10*x^2 + x^3 + E*(-10*x
- 2*x^2) + (-50*x + 10*E*x - 10*x^2)*Log[5] + 25*x*Log[5]^2)*Log[3/x]^2)/(225*x + (150*x - 30*E*x + 30*x^2 - 1
50*x*Log[5])*Log[3/x] + (25*x + E^2*x + 10*x^2 + x^3 + E*(-10*x - 2*x^2) + (-50*x + 10*E*x - 10*x^2)*Log[5] +
25*x*Log[5]^2)*Log[3/x]^2),x]

[Out]

x - 5/(5 - E + x - 5*Log[5]) + 75*Defer[Int][1/(x*(15 + (5 - E + x - 5*Log[5])*Log[3/x])^2), x] + 1125*Defer[I
nt][1/((5 - E + x - 5*Log[5])^2*(15 - (-5 + E - x + 5*Log[5])*Log[3/x])^2), x] + 150*Defer[Int][1/((5 - E + x
- 5*Log[5])^2*(-15 + (-5 + E - x + 5*Log[5])*Log[3/x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {75 (1+3 x)+30 x (5-e+x-5 \log (5)) \log \left (\frac {3}{x}\right )+x \left (e^2+x^2-2 e (5+x-5 \log (5))-10 x (-1+\log (5))+5 \left (6-10 \log (5)+5 \log ^2(5)\right )\right ) \log ^2\left (\frac {3}{x}\right )}{x \left (15+(5-e+x-5 \log (5)) \log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=\int \left (\frac {30-10 e+e^2+x^2+2 x (5-e-5 \log (5))-50 \log (5)+10 e \log (5)+25 \log ^2(5)}{(5-e+x-5 \log (5))^2}+\frac {150}{(5-e+x-5 \log (5))^2 \left (-15-x \log \left (\frac {3}{x}\right )-5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )}+\frac {75 \left (x^2+(-5+e+5 \log (5))^2+x (25-2 e-5 \log (25))\right )}{x (5-e+x-5 \log (5))^2 \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2}\right ) \, dx\\ &=75 \int \frac {x^2+(-5+e+5 \log (5))^2+x (25-2 e-5 \log (25))}{x (5-e+x-5 \log (5))^2 \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2} \, dx+150 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (-15-x \log \left (\frac {3}{x}\right )-5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )} \, dx+\int \frac {30-10 e+e^2+x^2+2 x (5-e-5 \log (5))-50 \log (5)+10 e \log (5)+25 \log ^2(5)}{(5-e+x-5 \log (5))^2} \, dx\\ &=75 \int \frac {x^2+(-5+e+5 \log (5))^2+x (25-2 e-5 \log (25))}{x (5-e+x-5 \log (5))^2 \left (15+(5-e+x-5 \log (5)) \log \left (\frac {3}{x}\right )\right )^2} \, dx+150 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (-15+(-5+e-x+5 \log (5)) \log \left (\frac {3}{x}\right )\right )} \, dx+\int \left (1+\frac {5}{(-5+e-x+5 \log (5))^2}\right ) \, dx\\ &=x-\frac {5}{5-e+x-5 \log (5)}+75 \int \left (\frac {1}{x \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2}+\frac {15}{(5-e+x-5 \log (5))^2 \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2}\right ) \, dx+150 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (-15+(-5+e-x+5 \log (5)) \log \left (\frac {3}{x}\right )\right )} \, dx\\ &=x-\frac {5}{5-e+x-5 \log (5)}+75 \int \frac {1}{x \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2} \, dx+150 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (-15+(-5+e-x+5 \log (5)) \log \left (\frac {3}{x}\right )\right )} \, dx+1125 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (15+x \log \left (\frac {3}{x}\right )+5 \left (1-\frac {e}{5}-\log (5)\right ) \log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=x-\frac {5}{5-e+x-5 \log (5)}+75 \int \frac {1}{x \left (15+(5-e+x-5 \log (5)) \log \left (\frac {3}{x}\right )\right )^2} \, dx+150 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (-15+(-5+e-x+5 \log (5)) \log \left (\frac {3}{x}\right )\right )} \, dx+1125 \int \frac {1}{(5-e+x-5 \log (5))^2 \left (15-(-5+e-x+5 \log (5)) \log \left (\frac {3}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 48, normalized size = 1.66 \begin {gather*} \frac {15 x+\left (-5+x^2-x (-5+e+5 \log (5))\right ) \log \left (\frac {3}{x}\right )}{15+(5-e+x-5 \log (5)) \log \left (\frac {3}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75 + 225*x + (150*x - 30*E*x + 30*x^2 - 150*x*Log[5])*Log[3/x] + (30*x + E^2*x + 10*x^2 + x^3 + E*(
-10*x - 2*x^2) + (-50*x + 10*E*x - 10*x^2)*Log[5] + 25*x*Log[5]^2)*Log[3/x]^2)/(225*x + (150*x - 30*E*x + 30*x
^2 - 150*x*Log[5])*Log[3/x] + (25*x + E^2*x + 10*x^2 + x^3 + E*(-10*x - 2*x^2) + (-50*x + 10*E*x - 10*x^2)*Log
[5] + 25*x*Log[5]^2)*Log[3/x]^2),x]

[Out]

(15*x + (-5 + x^2 - x*(-5 + E + 5*Log[5]))*Log[3/x])/(15 + (5 - E + x - 5*Log[5])*Log[3/x])

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 52, normalized size = 1.79 \begin {gather*} \frac {{\left (x^{2} - x e - 5 \, x \log \relax (5) + 5 \, x - 5\right )} \log \left (\frac {3}{x}\right ) + 15 \, x}{{\left (x - e - 5 \, \log \relax (5) + 5\right )} \log \left (\frac {3}{x}\right ) + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+30*x)*lo
g(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*log(3/x)+225*x+75)/((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)
*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+25*x)*log(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*l
og(3/x)+225*x),x, algorithm="fricas")

[Out]

((x^2 - x*e - 5*x*log(5) + 5*x - 5)*log(3/x) + 15*x)/((x - e - 5*log(5) + 5)*log(3/x) + 15)

________________________________________________________________________________________

giac [B]  time = 0.92, size = 271, normalized size = 9.34 \begin {gather*} \frac {e \log \left (\frac {3}{x}\right ) - \frac {10 \, e \log \relax (5) \log \left (\frac {3}{x}\right )}{x} + 5 \, \log \relax (5) \log \left (\frac {3}{x}\right ) - \frac {25 \, \log \relax (5)^{2} \log \left (\frac {3}{x}\right )}{x} - \frac {e^{2} \log \left (\frac {3}{x}\right )}{x} + \frac {10 \, e \log \left (\frac {3}{x}\right )}{x} + \frac {50 \, \log \relax (5) \log \left (\frac {3}{x}\right )}{x} + \frac {15 \, e}{x} + \frac {75 \, \log \relax (5)}{x} - \frac {30 \, \log \left (\frac {3}{x}\right )}{x} - \frac {75}{x} - \frac {75}{x^{2}} - 5 \, \log \left (\frac {3}{x}\right )}{\frac {e \log \left (\frac {3}{x}\right )}{x} + \frac {5 \, \log \relax (5) \log \left (\frac {3}{x}\right )}{x} - \frac {10 \, e \log \relax (5) \log \left (\frac {3}{x}\right )}{x^{2}} - \frac {25 \, \log \relax (5)^{2} \log \left (\frac {3}{x}\right )}{x^{2}} - \frac {5 \, \log \left (\frac {3}{x}\right )}{x} - \frac {e^{2} \log \left (\frac {3}{x}\right )}{x^{2}} + \frac {10 \, e \log \left (\frac {3}{x}\right )}{x^{2}} + \frac {50 \, \log \relax (5) \log \left (\frac {3}{x}\right )}{x^{2}} + \frac {15 \, e}{x^{2}} + \frac {75 \, \log \relax (5)}{x^{2}} - \frac {25 \, \log \left (\frac {3}{x}\right )}{x^{2}} - \frac {75}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+30*x)*lo
g(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*log(3/x)+225*x+75)/((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)
*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+25*x)*log(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*l
og(3/x)+225*x),x, algorithm="giac")

[Out]

(e*log(3/x) - 10*e*log(5)*log(3/x)/x + 5*log(5)*log(3/x) - 25*log(5)^2*log(3/x)/x - e^2*log(3/x)/x + 10*e*log(
3/x)/x + 50*log(5)*log(3/x)/x + 15*e/x + 75*log(5)/x - 30*log(3/x)/x - 75/x - 75/x^2 - 5*log(3/x))/(e*log(3/x)
/x + 5*log(5)*log(3/x)/x - 10*e*log(5)*log(3/x)/x^2 - 25*log(5)^2*log(3/x)/x^2 - 5*log(3/x)/x - e^2*log(3/x)/x
^2 + 10*e*log(3/x)/x^2 + 50*log(5)*log(3/x)/x^2 + 15*e/x^2 + 75*log(5)/x^2 - 25*log(3/x)/x^2 - 75/x^2)

________________________________________________________________________________________

maple [B]  time = 0.28, size = 90, normalized size = 3.10

method	result	size
risch	\(\frac {5 x \ln \relax (5)+x \,{\mathrm e}-x^{2}-5 x +5}{5 \ln \relax (5)+{\mathrm e}-x -5}+\frac {75}{\left (5 \ln \relax (5)+{\mathrm e}-x -5\right ) \left (5 \ln \left (\frac {3}{x}\right ) \ln \relax (5)+{\mathrm e} \ln \left (\frac {3}{x}\right )-x \ln \left (\frac {3}{x}\right )-5 \ln \left (\frac {3}{x}\right )-15\right )}\)	\(90\)
norman	\(\frac {\left (25 \ln \relax (5)^{2}+10 \,{\mathrm e} \ln \relax (5)+{\mathrm e}^{2}-50 \ln \relax (5)-10 \,{\mathrm e}+30\right ) \ln \left (\frac {3}{x}\right )+75-15 x -x^{2} \ln \left (\frac {3}{x}\right )-15 \,{\mathrm e}-75 \ln \relax (5)}{5 \ln \left (\frac {3}{x}\right ) \ln \relax (5)+{\mathrm e} \ln \left (\frac {3}{x}\right )-x \ln \left (\frac {3}{x}\right )-5 \ln \left (\frac {3}{x}\right )-15}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x*ln(5)^2+(10*x*exp(1)-10*x^2-50*x)*ln(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+30*x)*ln(3/x)^2+
(-150*x*ln(5)-30*x*exp(1)+30*x^2+150*x)*ln(3/x)+225*x+75)/((25*x*ln(5)^2+(10*x*exp(1)-10*x^2-50*x)*ln(5)+x*exp
(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+25*x)*ln(3/x)^2+(-150*x*ln(5)-30*x*exp(1)+30*x^2+150*x)*ln(3/x)+225*x),x
,method=_RETURNVERBOSE)

[Out]

(5*x*ln(5)+x*exp(1)-x^2-5*x+5)/(5*ln(5)+exp(1)-x-5)+75/(5*ln(5)+exp(1)-x-5)/(5*ln(3/x)*ln(5)+exp(1)*ln(3/x)-x*
ln(3/x)-5*ln(3/x)-15)

________________________________________________________________________________________

maxima [B]  time = 0.52, size = 87, normalized size = 3.00 \begin {gather*} \frac {x^{2} \log \relax (3) - {\left (5 \, {\left (\log \relax (5) - 1\right )} \log \relax (3) + e \log \relax (3) - 15\right )} x - {\left (x^{2} - x {\left (e + 5 \, \log \relax (5) - 5\right )} - 5\right )} \log \relax (x) - 5 \, \log \relax (3)}{x \log \relax (3) - 5 \, {\left (\log \relax (5) - 1\right )} \log \relax (3) - e \log \relax (3) - {\left (x - e - 5 \, \log \relax (5) + 5\right )} \log \relax (x) + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+30*x)*lo
g(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*log(3/x)+225*x+75)/((25*x*log(5)^2+(10*x*exp(1)-10*x^2-50*x)
*log(5)+x*exp(1)^2+(-2*x^2-10*x)*exp(1)+x^3+10*x^2+25*x)*log(3/x)^2+(-150*x*log(5)-30*x*exp(1)+30*x^2+150*x)*l
og(3/x)+225*x),x, algorithm="maxima")

[Out]

(x^2*log(3) - (5*(log(5) - 1)*log(3) + e*log(3) - 15)*x - (x^2 - x*(e + 5*log(5) - 5) - 5)*log(x) - 5*log(3))/
(x*log(3) - 5*(log(5) - 1)*log(3) - e*log(3) - (x - e - 5*log(5) + 5)*log(x) + 15)

________________________________________________________________________________________

mupad [B]  time = 13.77, size = 94, normalized size = 3.24 \begin {gather*} \frac {15\,x-5\,\ln \left (\frac {3}{x}\right )+5\,x\,\ln \left (\frac {3}{x}\right )+x^2\,\ln \left (\frac {3}{x}\right )-x\,\mathrm {e}\,\ln \left (\frac {3}{x}\right )-5\,x\,\ln \relax (5)\,\ln \left (\frac {3}{x}\right )}{5\,\ln \left (\frac {3}{x}\right )-\mathrm {e}\,\ln \left (\frac {3}{x}\right )-5\,\ln \relax (5)\,\ln \left (\frac {3}{x}\right )+x\,\ln \left (\frac {3}{x}\right )+15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((225*x + log(3/x)^2*(30*x - exp(1)*(10*x + 2*x^2) + x*exp(2) - log(5)*(50*x - 10*x*exp(1) + 10*x^2) + 25*x
*log(5)^2 + 10*x^2 + x^3) + log(3/x)*(150*x - 30*x*exp(1) - 150*x*log(5) + 30*x^2) + 75)/(225*x + log(3/x)^2*(
25*x - exp(1)*(10*x + 2*x^2) + x*exp(2) - log(5)*(50*x - 10*x*exp(1) + 10*x^2) + 25*x*log(5)^2 + 10*x^2 + x^3)
+ log(3/x)*(150*x - 30*x*exp(1) - 150*x*log(5) + 30*x^2)),x)

[Out]

(15*x - 5*log(3/x) + 5*x*log(3/x) + x^2*log(3/x) - x*exp(1)*log(3/x) - 5*x*log(5)*log(3/x))/(5*log(3/x) - exp(
1)*log(3/x) - 5*log(5)*log(3/x) + x*log(3/x) + 15)

________________________________________________________________________________________

sympy [B]  time = 0.61, size = 87, normalized size = 3.00 \begin {gather*} x + \frac {75}{15 x + \left (x^{2} - 10 x \log {\relax (5 )} - 2 e x + 10 x - 50 \log {\relax (5 )} - 10 e + e^{2} + 25 + 10 e \log {\relax (5 )} + 25 \log {\relax (5 )}^{2}\right ) \log {\left (\frac {3}{x} \right )} - 75 \log {\relax (5 )} - 15 e + 75} - \frac {5}{x - 5 \log {\relax (5 )} - e + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x*ln(5)**2+(10*x*exp(1)-10*x**2-50*x)*ln(5)+x*exp(1)**2+(-2*x**2-10*x)*exp(1)+x**3+10*x**2+30*x
)*ln(3/x)**2+(-150*x*ln(5)-30*x*exp(1)+30*x**2+150*x)*ln(3/x)+225*x+75)/((25*x*ln(5)**2+(10*x*exp(1)-10*x**2-5
0*x)*ln(5)+x*exp(1)**2+(-2*x**2-10*x)*exp(1)+x**3+10*x**2+25*x)*ln(3/x)**2+(-150*x*ln(5)-30*x*exp(1)+30*x**2+1
50*x)*ln(3/x)+225*x),x)

[Out]

x + 75/(15*x + (x**2 - 10*x*log(5) - 2*E*x + 10*x - 50*log(5) - 10*E + exp(2) + 25 + 10*E*log(5) + 25*log(5)**
2)*log(3/x) - 75*log(5) - 15*E + 75) - 5/(x - 5*log(5) - E + 5)

________________________________________________________________________________________