Integrand size = 123, antiderivative size = 26 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=2 \left (x+\frac {8-2 x+\log (x)}{12+\left (\sqrt [4]{e}+x\right ) \log (x)}\right ) \] Output:
2*(8-2*x+ln(x))/((x+exp(1/4))*ln(x)+12)+2*x
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\frac {2 \left (8+10 x+\left (1+\sqrt [4]{e} x+x^2\right ) \log (x)\right )}{12+\left (\sqrt [4]{e}+x\right ) \log (x)} \] Input:
Integrate[(24 + 224*x + 4*x^2 + E^(1/4)*(-16 + 4*x) + (-16*x + 44*E^(1/4)* x + 48*x^2)*Log[x] + (-2*x + 2*Sqrt[E]*x + 4*E^(1/4)*x^2 + 2*x^3)*Log[x]^2 )/(144*x + (24*E^(1/4)*x + 24*x^2)*Log[x] + (Sqrt[E]*x + 2*E^(1/4)*x^2 + x ^3)*Log[x]^2),x]
Output:
(2*(8 + 10*x + (1 + E^(1/4)*x + x^2)*Log[x]))/(12 + (E^(1/4) + x)*Log[x])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {4 x^2+\left (48 x^2+44 \sqrt [4]{e} x-16 x\right ) \log (x)+\left (2 x^3+4 \sqrt [4]{e} x^2+2 \sqrt {e} x-2 x\right ) \log ^2(x)+224 x+\sqrt [4]{e} (4 x-16)+24}{\left (24 x^2+24 \sqrt [4]{e} x\right ) \log (x)+\left (x^3+2 \sqrt [4]{e} x^2+\sqrt {e} x\right ) \log ^2(x)+144 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 x^2+\left (48 x^2+44 \sqrt [4]{e} x-16 x\right ) \log (x)+\left (2 x^3+4 \sqrt [4]{e} x^2+2 \sqrt {e} x-2 x\right ) \log ^2(x)+224 x+\sqrt [4]{e} (4 x-16)+24}{x \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (x^2+2 \sqrt [4]{e} x+\sqrt {e}-1\right )}{\left (x+\sqrt [4]{e}\right )^2}+\frac {4 \left (x^4-\left (16-3 \sqrt [4]{e}\right ) x^3+3 \left (18-8 \sqrt [4]{e}+\sqrt {e}\right ) x^2-\left (72-60 \sqrt [4]{e}+12 \sqrt {e}-e^{3/4}\right ) x+2 \left (3-2 \sqrt [4]{e}\right ) \sqrt {e}\right )}{x \left (x+\sqrt [4]{e}\right )^2 \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}+\frac {4 \left (-\left (\left (4+\sqrt [4]{e}\right ) x\right )-\sqrt {e}-4 \sqrt [4]{e}+12\right )}{\left (x+\sqrt [4]{e}\right )^2 \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (16-\sqrt [4]{e}\right ) \int \frac {1}{\left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx+8 \left (3-2 \sqrt [4]{e}\right ) \int \frac {1}{x \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx+4 \int \frac {x}{\left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx-288 \int \frac {1}{\left (x+\sqrt [4]{e}\right )^2 \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx+48 \left (4+\sqrt [4]{e}\right ) \int \frac {1}{\left (x+\sqrt [4]{e}\right ) \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )^2}dx+48 \int \frac {1}{\left (x+\sqrt [4]{e}\right )^2 \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )}dx-4 \left (4+\sqrt [4]{e}\right ) \int \frac {1}{\left (x+\sqrt [4]{e}\right ) \left (x \log (x)+\sqrt [4]{e} \log (x)+12\right )}dx+2 x+\frac {2}{x+\sqrt [4]{e}}\) |
Input:
Int[(24 + 224*x + 4*x^2 + E^(1/4)*(-16 + 4*x) + (-16*x + 44*E^(1/4)*x + 48 *x^2)*Log[x] + (-2*x + 2*Sqrt[E]*x + 4*E^(1/4)*x^2 + 2*x^3)*Log[x]^2)/(144 *x + (24*E^(1/4)*x + 24*x^2)*Log[x] + (Sqrt[E]*x + 2*E^(1/4)*x^2 + x^3)*Lo g[x]^2),x]
Output:
$Aborted
Time = 3.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
| method | result | size |
| default | \(\frac {2 \left (-{\mathrm e}^{\frac {1}{2}}+1\right ) \ln \left (x \right )+2 x^{2} \ln \left (x \right )+20 x +16-24 \,{\mathrm e}^{\frac {1}{4}}}{\ln \left (x \right ) {\mathrm e}^{\frac {1}{4}}+x \ln \left (x \right )+12}\) | \(42\) |
| norman | \(\frac {\left (-2 \,{\mathrm e}^{\frac {1}{2}}+2\right ) \ln \left (x \right )+20 x +2 x^{2} \ln \left (x \right )+16-24 \,{\mathrm e}^{\frac {1}{4}}}{\ln \left (x \right ) {\mathrm e}^{\frac {1}{4}}+x \ln \left (x \right )+12}\) | \(42\) |
| parallelrisch | \(-\frac {2 \ln \left (x \right ) {\mathrm e}^{\frac {1}{2}}-16-2 x^{2} \ln \left (x \right )+24 \,{\mathrm e}^{\frac {1}{4}}-20 x -2 \ln \left (x \right )}{\ln \left (x \right ) {\mathrm e}^{\frac {1}{4}}+x \ln \left (x \right )+12}\) | \(44\) |
| risch | \(\frac {2 x \,{\mathrm e}^{\frac {1}{4}}+2 x^{2}+2}{x +{\mathrm e}^{\frac {1}{4}}}-\frac {4 \left (x \,{\mathrm e}^{\frac {1}{4}}+x^{2}-4 \,{\mathrm e}^{\frac {1}{4}}-4 x +6\right )}{\left (x +{\mathrm e}^{\frac {1}{4}}\right ) \left (\ln \left (x \right ) {\mathrm e}^{\frac {1}{4}}+x \ln \left (x \right )+12\right )}\) | \(56\) |
Input:
int(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*ln(x)^2+(44*x*exp(1/4)+48*x ^2-16*x)*ln(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2*x^2*exp( 1/4)+x^3)*ln(x)^2+(24*x*exp(1/4)+24*x^2)*ln(x)+144*x),x,method=_RETURNVERB OSE)
Output:
2*((-exp(1/4)^2+1)*ln(x)+x^2*ln(x)+10*x+8-12*exp(1/4))/(ln(x)*exp(1/4)+x*l n(x)+12)
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\frac {2 \, {\left ({\left (x^{2} + x e^{\frac {1}{4}} + 1\right )} \log \left (x\right ) + 10 \, x + 8\right )}}{{\left (x + e^{\frac {1}{4}}\right )} \log \left (x\right ) + 12} \] Input:
integrate(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*log(x)^2+(44*x*exp(1/ 4)+48*x^2-16*x)*log(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2* x^2*exp(1/4)+x^3)*log(x)^2+(24*x*exp(1/4)+24*x^2)*log(x)+144*x),x, algorit hm="fricas")
Output:
2*((x^2 + x*e^(1/4) + 1)*log(x) + 10*x + 8)/((x + e^(1/4))*log(x) + 12)
Timed out. \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\text {Timed out} \] Input:
integrate(((2*x*exp(1/4)**2+4*x**2*exp(1/4)+2*x**3-2*x)*ln(x)**2+(44*x*exp (1/4)+48*x**2-16*x)*ln(x)+(4*x-16)*exp(1/4)+4*x**2+224*x+24)/((x*exp(1/4)* *2+2*x**2*exp(1/4)+x**3)*ln(x)**2+(24*x*exp(1/4)+24*x**2)*ln(x)+144*x),x)
Output:
Timed out
Time = 1.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\frac {2 \, {\left ({\left (x^{2} + x e^{\frac {1}{4}} + 1\right )} \log \left (x\right ) + 10 \, x + 8\right )}}{{\left (x + e^{\frac {1}{4}}\right )} \log \left (x\right ) + 12} \] Input:
integrate(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*log(x)^2+(44*x*exp(1/ 4)+48*x^2-16*x)*log(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2* x^2*exp(1/4)+x^3)*log(x)^2+(24*x*exp(1/4)+24*x^2)*log(x)+144*x),x, algorit hm="maxima")
Output:
2*((x^2 + x*e^(1/4) + 1)*log(x) + 10*x + 8)/((x + e^(1/4))*log(x) + 12)
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\frac {2 \, {\left (x^{2} \log \left (x\right ) + x e^{\frac {1}{4}} \log \left (x\right ) + 10 \, x + \log \left (x\right ) + 8\right )}}{x \log \left (x\right ) + e^{\frac {1}{4}} \log \left (x\right ) + 12} \] Input:
integrate(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*log(x)^2+(44*x*exp(1/ 4)+48*x^2-16*x)*log(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2* x^2*exp(1/4)+x^3)*log(x)^2+(24*x*exp(1/4)+24*x^2)*log(x)+144*x),x, algorit hm="giac")
Output:
2*(x^2*log(x) + x*e^(1/4)*log(x) + 10*x + log(x) + 8)/(x*log(x) + e^(1/4)* log(x) + 12)
Time = 2.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\frac {2\,\left (10\,x+\ln \left (x\right )+x^2\,\ln \left (x\right )+x\,{\mathrm {e}}^{1/4}\,\ln \left (x\right )+8\right )}{{\mathrm {e}}^{1/4}\,\ln \left (x\right )+x\,\ln \left (x\right )+12} \] Input:
int((224*x + 4*x^2 + log(x)^2*(2*x*exp(1/2) - 2*x + 4*x^2*exp(1/4) + 2*x^3 ) + log(x)*(44*x*exp(1/4) - 16*x + 48*x^2) + exp(1/4)*(4*x - 16) + 24)/(14 4*x + log(x)*(24*x*exp(1/4) + 24*x^2) + log(x)^2*(x*exp(1/2) + 2*x^2*exp(1 /4) + x^3)),x)
Output:
(2*(10*x + log(x) + x^2*log(x) + x*exp(1/4)*log(x) + 8))/(exp(1/4)*log(x) + x*log(x) + 12)
\[ \int \frac {24+224 x+4 x^2+\sqrt [4]{e} (-16+4 x)+\left (-16 x+44 \sqrt [4]{e} x+48 x^2\right ) \log (x)+\left (-2 x+2 \sqrt {e} x+4 \sqrt [4]{e} x^2+2 x^3\right ) \log ^2(x)}{144 x+\left (24 \sqrt [4]{e} x+24 x^2\right ) \log (x)+\left (\sqrt {e} x+2 \sqrt [4]{e} x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {\left (2 x \left ({\mathrm e}^{\frac {1}{4}}\right )^{2}+4 x^{2} {\mathrm e}^{\frac {1}{4}}+2 x^{3}-2 x \right ) \mathrm {log}\left (x \right )^{2}+\left (44 x \,{\mathrm e}^{\frac {1}{4}}+48 x^{2}-16 x \right ) \mathrm {log}\left (x \right )+\left (4 x -16\right ) {\mathrm e}^{\frac {1}{4}}+4 x^{2}+224 x +24}{\left (x \left ({\mathrm e}^{\frac {1}{4}}\right )^{2}+2 x^{2} {\mathrm e}^{\frac {1}{4}}+x^{3}\right ) \mathrm {log}\left (x \right )^{2}+\left (24 x \,{\mathrm e}^{\frac {1}{4}}+24 x^{2}\right ) \mathrm {log}\left (x \right )+144 x}d x \] Input:
int(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*log(x)^2+(44*x*exp(1/4)+48* x^2-16*x)*log(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2*x^2*ex p(1/4)+x^3)*log(x)^2+(24*x*exp(1/4)+24*x^2)*log(x)+144*x),x)
Output:
int(((2*x*exp(1/4)^2+4*x^2*exp(1/4)+2*x^3-2*x)*log(x)^2+(44*x*exp(1/4)+48* x^2-16*x)*log(x)+(4*x-16)*exp(1/4)+4*x^2+224*x+24)/((x*exp(1/4)^2+2*x^2*ex p(1/4)+x^3)*log(x)^2+(24*x*exp(1/4)+24*x^2)*log(x)+144*x),x)