3.96.39 \(\int \frac {-20 e^{\frac {1}{5} (11 x+10 \log (3 x))} x^2 \log ^2(x)+e^{\frac {1}{10} (11 x+10 \log (3 x))} (30+30 x+(60+123 x+33 x^2) \log (x))}{90-120 e^{\frac {1}{10} (11 x+10 \log (3 x))} x \log (x)+40 e^{\frac {1}{5} (11 x+10 \log (3 x))} x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {x (1+x)}{-2 x+\frac {e^{-11 x/10}}{x \log (x)}} \]

________________________________________________________________________________________

Rubi [F]  time = 4.47, 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 {-20 e^{\frac {1}{5} (11 x+10 \log (3 x))} x^2 \log ^2(x)+e^{\frac {1}{10} (11 x+10 \log (3 x))} \left (30+30 x+\left (60+123 x+33 x^2\right ) \log (x)\right )}{90-120 e^{\frac {1}{10} (11 x+10 \log (3 x))} x \log (x)+40 e^{\frac {1}{5} (11 x+10 \log (3 x))} x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20*E^((11*x + 10*Log[3*x])/5)*x^2*Log[x]^2 + E^((11*x + 10*Log[3*x])/10)*(30 + 30*x + (60 + 123*x + 33*x
^2)*Log[x]))/(90 - 120*E^((11*x + 10*Log[3*x])/10)*x*Log[x] + 40*E^((11*x + 10*Log[3*x])/5)*x^2*Log[x]^2),x]

[Out]

-1/2*x - (2 - 1/(E^((11*x)/10)*x^2*Log[x]))^(-1) + Defer[Int][(E^((11*x)/10)*x^2)/(-1 + 2*E^((11*x)/10)*x^2*Lo
g[x])^2, x] + 2*Defer[Int][(E^((11*x)/10)*x^2*Log[x])/(-1 + 2*E^((11*x)/10)*x^2*Log[x])^2, x] + (11*Defer[Int]
[(E^((11*x)/10)*x^3*Log[x])/(-1 + 2*E^((11*x)/10)*x^2*Log[x])^2, x])/10 - Defer[Int][(-1 + 2*E^((11*x)/10)*x^2
*Log[x])^(-1), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{11 x/10} x \left (10 (1+x)+\left (20+41 x+11 x^2\right ) \log (x)-20 e^{11 x/10} x^3 \log ^2(x)\right )}{10 \left (1-2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx\\ &=\frac {1}{10} \int \frac {e^{11 x/10} x \left (10 (1+x)+\left (20+41 x+11 x^2\right ) \log (x)-20 e^{11 x/10} x^3 \log ^2(x)\right )}{\left (1-2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx\\ &=\frac {1}{10} \int \left (\frac {e^{11 x/10} x (1+x) (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}-\frac {10 e^{11 x/10} x^2 \log (x)}{-1+2 e^{11 x/10} x^2 \log (x)}\right ) \, dx\\ &=\frac {1}{10} \int \frac {e^{11 x/10} x (1+x) (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx-\int \frac {e^{11 x/10} x^2 \log (x)}{-1+2 e^{11 x/10} x^2 \log (x)} \, dx\\ &=\frac {1}{10} \int \left (\frac {e^{11 x/10} x (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}+\frac {e^{11 x/10} x^2 (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}\right ) \, dx-\int \left (\frac {1}{2}+\frac {1}{2 \left (-1+2 e^{11 x/10} x^2 \log (x)\right )}\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{10} \int \frac {e^{11 x/10} x (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx+\frac {1}{10} \int \frac {e^{11 x/10} x^2 (10+20 \log (x)+11 x \log (x))}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx-\frac {1}{2} \int \frac {1}{-1+2 e^{11 x/10} x^2 \log (x)} \, dx\\ &=-\frac {x}{2}+\frac {1}{10} \int \left (\frac {10 e^{11 x/10} x^2}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}+\frac {20 e^{11 x/10} x^2 \log (x)}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}+\frac {11 e^{11 x/10} x^3 \log (x)}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2}\right ) \, dx-\frac {1}{2} \int \frac {1}{-1+2 e^{11 x/10} x^2 \log (x)} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,-\frac {e^{-11 x/10}}{2 x^2 \log (x)}\right )\\ &=-\frac {x}{2}-\frac {1}{2-\frac {e^{-11 x/10}}{x^2 \log (x)}}-\frac {1}{2} \int \frac {1}{-1+2 e^{11 x/10} x^2 \log (x)} \, dx+\frac {11}{10} \int \frac {e^{11 x/10} x^3 \log (x)}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx+2 \int \frac {e^{11 x/10} x^2 \log (x)}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx+\int \frac {e^{11 x/10} x^2}{\left (-1+2 e^{11 x/10} x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.90, size = 31, normalized size = 1.19 \begin {gather*} \frac {1}{10} \left (-5 x-\frac {5 (1+x)}{-1+2 e^{11 x/10} x^2 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*E^((11*x + 10*Log[3*x])/5)*x^2*Log[x]^2 + E^((11*x + 10*Log[3*x])/10)*(30 + 30*x + (60 + 123*x
+ 33*x^2)*Log[x]))/(90 - 120*E^((11*x + 10*Log[3*x])/10)*x*Log[x] + 40*E^((11*x + 10*Log[3*x])/5)*x^2*Log[x]^2
),x]

[Out]

(-5*x - (5*(1 + x))/(-1 + 2*E^((11*x)/10)*x^2*Log[x]))/10

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 38, normalized size = 1.46 \begin {gather*} -\frac {2 \, x^{2} e^{\left (\frac {11}{10} \, x + \log \relax (3) + \log \relax (x)\right )} \log \relax (x) + 3}{2 \, {\left (2 \, x e^{\left (\frac {11}{10} \, x + \log \relax (3) + \log \relax (x)\right )} \log \relax (x) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2+((33*x^2+123*x+60)*log(x)+30*x+30)*exp(log(3*x)+11/10*x))/
(40*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2-120*x*log(x)*exp(log(3*x)+11/10*x)+90),x, algorithm="fricas")

[Out]

-1/2*(2*x^2*e^(11/10*x + log(3) + log(x))*log(x) + 3)/(2*x*e^(11/10*x + log(3) + log(x))*log(x) - 3)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 30, normalized size = 1.15 \begin {gather*} -\frac {2 \, x^{3} e^{\left (\frac {11}{10} \, x\right )} \log \relax (x) + 1}{2 \, {\left (2 \, x^{2} e^{\left (\frac {11}{10} \, x\right )} \log \relax (x) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2+((33*x^2+123*x+60)*log(x)+30*x+30)*exp(log(3*x)+11/10*x))/
(40*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2-120*x*log(x)*exp(log(3*x)+11/10*x)+90),x, algorithm="giac")

[Out]

-1/2*(2*x^3*e^(11/10*x)*log(x) + 1)/(2*x^2*e^(11/10*x)*log(x) - 1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 25, normalized size = 0.96

method	result	size
risch	\(-\frac {x}{2}-\frac {3 \left (x +1\right )}{2 \left (6 x^{2} \ln \relax (x ) {\mathrm e}^{\frac {11 x}{10}}-3\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20*x^2*ln(x)^2*exp(ln(3*x)+11/10*x)^2+((33*x^2+123*x+60)*ln(x)+30*x+30)*exp(ln(3*x)+11/10*x))/(40*x^2*ln
(x)^2*exp(ln(3*x)+11/10*x)^2-120*x*ln(x)*exp(ln(3*x)+11/10*x)+90),x,method=_RETURNVERBOSE)

[Out]

-1/2*x-3/2*(x+1)/(6*x^2*ln(x)*exp(11/10*x)-3)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 30, normalized size = 1.15 \begin {gather*} -\frac {2 \, x^{3} e^{\left (\frac {11}{10} \, x\right )} \log \relax (x) + 1}{2 \, {\left (2 \, x^{2} e^{\left (\frac {11}{10} \, x\right )} \log \relax (x) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2+((33*x^2+123*x+60)*log(x)+30*x+30)*exp(log(3*x)+11/10*x))/
(40*x^2*log(x)^2*exp(log(3*x)+11/10*x)^2-120*x*log(x)*exp(log(3*x)+11/10*x)+90),x, algorithm="maxima")

[Out]

-1/2*(2*x^3*e^(11/10*x)*log(x) + 1)/(2*x^2*e^(11/10*x)*log(x) - 1)

________________________________________________________________________________________

mupad [B]  time = 8.40, size = 26, normalized size = 1.00 \begin {gather*} -\frac {x}{2}-\frac {\frac {x}{2}+\frac {1}{2}}{2\,x^2\,{\mathrm {e}}^{\frac {11\,x}{10}}\,\ln \relax (x)-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((11*x)/10 + log(3*x))*(30*x + log(x)*(123*x + 33*x^2 + 60) + 30) - 20*x^2*exp((11*x)/5 + 2*log(3*x))*
log(x)^2)/(40*x^2*exp((11*x)/5 + 2*log(3*x))*log(x)^2 - 120*x*exp((11*x)/10 + log(3*x))*log(x) + 90),x)

[Out]

- x/2 - (x/2 + 1/2)/(2*x^2*exp((11*x)/10)*log(x) - 1)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 24, normalized size = 0.92 \begin {gather*} - \frac {x}{2} + \frac {- x - 1}{4 x^{2} e^{\frac {11 x}{10}} \log {\relax (x )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*x**2*ln(x)**2*exp(ln(3*x)+11/10*x)**2+((33*x**2+123*x+60)*ln(x)+30*x+30)*exp(ln(3*x)+11/10*x))/
(40*x**2*ln(x)**2*exp(ln(3*x)+11/10*x)**2-120*x*ln(x)*exp(ln(3*x)+11/10*x)+90),x)

[Out]

-x/2 + (-x - 1)/(4*x**2*exp(11*x/10)*log(x) - 2)

________________________________________________________________________________________