3.62.7 \(\int \frac {100 x \log (x)+100 x \log ^2(x)+(2 x^4+2 e^x x^4) \log ^4(x)}{625+(250 x^2-50 e^x x^2-50 x^3) \log ^2(x)+(25 x^4+e^{2 x} x^4-10 x^5+x^6+e^x (-10 x^4+2 x^5)) \log ^4(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {2}{5-e^x-x+\frac {25}{x^2 \log ^2(x)}} \]

________________________________________________________________________________________

Rubi [F]  time = 14.51, 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 {100 x \log (x)+100 x \log ^2(x)+\left (2 x^4+2 e^x x^4\right ) \log ^4(x)}{625+\left (250 x^2-50 e^x x^2-50 x^3\right ) \log ^2(x)+\left (25 x^4+e^{2 x} x^4-10 x^5+x^6+e^x \left (-10 x^4+2 x^5\right )\right ) \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100*x*Log[x] + 100*x*Log[x]^2 + (2*x^4 + 2*E^x*x^4)*Log[x]^4)/(625 + (250*x^2 - 50*E^x*x^2 - 50*x^3)*Log[
x]^2 + (25*x^4 + E^(2*x)*x^4 - 10*x^5 + x^6 + E^x*(-10*x^4 + 2*x^5))*Log[x]^4),x]

[Out]

100*Defer[Int][(x*Log[x])/(25 - x^2*(-5 + E^x + x)*Log[x]^2)^2, x] + 100*Defer[Int][(x*Log[x]^2)/(25 - x^2*(-5
+ E^x + x)*Log[x]^2)^2, x] + 50*Defer[Int][(x^2*Log[x]^2)/(25 - x^2*(-5 + E^x + x)*Log[x]^2)^2, x] + 12*Defer
[Int][(x^4*Log[x]^4)/(25 - x^2*(-5 + E^x + x)*Log[x]^2)^2, x] - 2*Defer[Int][(x^5*Log[x]^4)/(25 - x^2*(-5 + E^
x + x)*Log[x]^2)^2, x] + 2*Defer[Int][(x^2*Log[x]^2)/(-25 + x^2*(-5 + E^x + x)*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \log (x) \left (50+50 \log (x)+\left (1+e^x\right ) x^3 \log ^3(x)\right )}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx\\ &=2 \int \frac {x \log (x) \left (50+50 \log (x)+\left (1+e^x\right ) x^3 \log ^3(x)\right )}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx\\ &=2 \int \left (\frac {x^2 \log ^2(x)}{-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)}-\frac {x \log (x) \left (-50-50 \log (x)-25 x \log (x)-6 x^3 \log ^3(x)+x^4 \log ^3(x)\right )}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}\right ) \, dx\\ &=2 \int \frac {x^2 \log ^2(x)}{-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)} \, dx-2 \int \frac {x \log (x) \left (-50-50 \log (x)-25 x \log (x)-6 x^3 \log ^3(x)+x^4 \log ^3(x)\right )}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx\\ &=2 \int \frac {x^2 \log ^2(x)}{-25+x^2 \left (-5+e^x+x\right ) \log ^2(x)} \, dx-2 \int \frac {x \log (x) \left (-50-25 (2+x) \log (x)+(-6+x) x^3 \log ^3(x)\right )}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx\\ &=2 \int \frac {x^2 \log ^2(x)}{-25+x^2 \left (-5+e^x+x\right ) \log ^2(x)} \, dx-2 \int \left (-\frac {50 x \log (x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}-\frac {50 x \log ^2(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}-\frac {25 x^2 \log ^2(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}-\frac {6 x^4 \log ^4(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}+\frac {x^5 \log ^4(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^5 \log ^4(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx\right )+2 \int \frac {x^2 \log ^2(x)}{-25+x^2 \left (-5+e^x+x\right ) \log ^2(x)} \, dx+12 \int \frac {x^4 \log ^4(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx+50 \int \frac {x^2 \log ^2(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx+100 \int \frac {x \log (x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx+100 \int \frac {x \log ^2(x)}{\left (-25-5 x^2 \log ^2(x)+e^x x^2 \log ^2(x)+x^3 \log ^2(x)\right )^2} \, dx\\ &=-\left (2 \int \frac {x^5 \log ^4(x)}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx\right )+2 \int \frac {x^2 \log ^2(x)}{-25+x^2 \left (-5+e^x+x\right ) \log ^2(x)} \, dx+12 \int \frac {x^4 \log ^4(x)}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx+50 \int \frac {x^2 \log ^2(x)}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx+100 \int \frac {x \log (x)}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx+100 \int \frac {x \log ^2(x)}{\left (25-x^2 \left (-5+e^x+x\right ) \log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.45, size = 27, normalized size = 1.17 \begin {gather*} -\frac {2 x^2 \log ^2(x)}{-25+x^2 \left (-5+e^x+x\right ) \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*x*Log[x] + 100*x*Log[x]^2 + (2*x^4 + 2*E^x*x^4)*Log[x]^4)/(625 + (250*x^2 - 50*E^x*x^2 - 50*x^3
)*Log[x]^2 + (25*x^4 + E^(2*x)*x^4 - 10*x^5 + x^6 + E^x*(-10*x^4 + 2*x^5))*Log[x]^4),x]

[Out]

(-2*x^2*Log[x]^2)/(-25 + x^2*(-5 + E^x + x)*Log[x]^2)

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 33, normalized size = 1.43 \begin {gather*} -\frac {2 \, x^{2} \log \relax (x)^{2}}{{\left (x^{3} + x^{2} e^{x} - 5 \, x^{2}\right )} \log \relax (x)^{2} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x^4+2*x^4)*log(x)^4+100*x*log(x)^2+100*x*log(x))/((exp(x)^2*x^4+(2*x^5-10*x^4)*exp(x)+x^6
-10*x^5+25*x^4)*log(x)^4+(-50*exp(x)*x^2-50*x^3+250*x^2)*log(x)^2+625),x, algorithm="fricas")

[Out]

-2*x^2*log(x)^2/((x^3 + x^2*e^x - 5*x^2)*log(x)^2 - 25)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x^4+2*x^4)*log(x)^4+100*x*log(x)^2+100*x*log(x))/((exp(x)^2*x^4+(2*x^5-10*x^4)*exp(x)+x^6
-10*x^5+25*x^4)*log(x)^4+(-50*exp(x)*x^2-50*x^3+250*x^2)*log(x)^2+625),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 51, normalized size = 2.22

method	result	size
risch	\(-\frac {2}{{\mathrm e}^{x}+x -5}-\frac {50}{\left ({\mathrm e}^{x}+x -5\right ) \left (x^{2} {\mathrm e}^{x} \ln \relax (x )^{2}+x^{3} \ln \relax (x )^{2}-5 x^{2} \ln \relax (x )^{2}-25\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(x)*x^4+2*x^4)*ln(x)^4+100*x*ln(x)^2+100*x*ln(x))/((exp(x)^2*x^4+(2*x^5-10*x^4)*exp(x)+x^6-10*x^5+2
5*x^4)*ln(x)^4+(-50*exp(x)*x^2-50*x^3+250*x^2)*ln(x)^2+625),x,method=_RETURNVERBOSE)

[Out]

-2/(exp(x)+x-5)-50/(exp(x)+x-5)/(x^2*exp(x)*ln(x)^2+x^3*ln(x)^2-5*x^2*ln(x)^2-25)

________________________________________________________________________________________

maxima [B]  time = 0.55, size = 37, normalized size = 1.61 \begin {gather*} -\frac {2 \, x^{2} \log \relax (x)^{2}}{x^{2} e^{x} \log \relax (x)^{2} + {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (x)^{2} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x^4+2*x^4)*log(x)^4+100*x*log(x)^2+100*x*log(x))/((exp(x)^2*x^4+(2*x^5-10*x^4)*exp(x)+x^6
-10*x^5+25*x^4)*log(x)^4+(-50*exp(x)*x^2-50*x^3+250*x^2)*log(x)^2+625),x, algorithm="maxima")

[Out]

-2*x^2*log(x)^2/(x^2*e^x*log(x)^2 + (x^3 - 5*x^2)*log(x)^2 - 25)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (2\,x^4\,{\mathrm {e}}^x+2\,x^4\right )\,{\ln \relax (x)}^4+100\,x\,{\ln \relax (x)}^2+100\,x\,\ln \relax (x)}{\left (x^4\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (10\,x^4-2\,x^5\right )+25\,x^4-10\,x^5+x^6\right )\,{\ln \relax (x)}^4+\left (250\,x^2-50\,x^2\,{\mathrm {e}}^x-50\,x^3\right )\,{\ln \relax (x)}^2+625} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x*log(x)^2 + log(x)^4*(2*x^4*exp(x) + 2*x^4) + 100*x*log(x))/(log(x)^4*(x^4*exp(2*x) - exp(x)*(10*x^4
- 2*x^5) + 25*x^4 - 10*x^5 + x^6) - log(x)^2*(50*x^2*exp(x) - 250*x^2 + 50*x^3) + 625),x)

[Out]

int((100*x*log(x)^2 + log(x)^4*(2*x^4*exp(x) + 2*x^4) + 100*x*log(x))/(log(x)^4*(x^4*exp(2*x) - exp(x)*(10*x^4
- 2*x^5) + 25*x^4 - 10*x^5 + x^6) - log(x)^2*(50*x^2*exp(x) - 250*x^2 + 50*x^3) + 625), x)

________________________________________________________________________________________

sympy [B]  time = 0.36, size = 42, normalized size = 1.83 \begin {gather*} - \frac {2 x^{2} \log {\relax (x )}^{2}}{x^{3} \log {\relax (x )}^{2} + x^{2} e^{x} \log {\relax (x )}^{2} - 5 x^{2} \log {\relax (x )}^{2} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x**4+2*x**4)*ln(x)**4+100*x*ln(x)**2+100*x*ln(x))/((exp(x)**2*x**4+(2*x**5-10*x**4)*exp(x
)+x**6-10*x**5+25*x**4)*ln(x)**4+(-50*exp(x)*x**2-50*x**3+250*x**2)*ln(x)**2+625),x)

[Out]

-2*x**2*log(x)**2/(x**3*log(x)**2 + x**2*exp(x)*log(x)**2 - 5*x**2*log(x)**2 - 25)

________________________________________________________________________________________