Integrand size = 121, antiderivative size = 27 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=-4+x+\frac {x}{2 \sqrt [3]{e}-\frac {(5+x) \log \left (x^2\right )}{x}} \end {dmath*}
Time = 0.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=x+\frac {x^2}{2 \sqrt [3]{e} x-(5+x) \log \left (x^2\right )} \end {dmath*}
Integrate[(10*x + 2*x^2 + 2*E^(1/3)*x^2 + 4*E^(2/3)*x^2 + (-10*x - x^2 + E ^(1/3)*(-20*x - 4*x^2))*Log[x^2] + (25 + 10*x + x^2)*Log[x^2]^2)/(4*E^(2/3 )*x^2 + E^(1/3)*(-20*x - 4*x^2)*Log[x^2] + (25 + 10*x + x^2)*Log[x^2]^2),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.
\(\displaystyle \int \frac {4 e^{2/3} x^2+2 \sqrt [3]{e} x^2+2 x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\left (-x^2+\sqrt [3]{e} \left (-4 x^2-20 x\right )-10 x\right ) \log \left (x^2\right )+10 x}{4 e^{2/3} x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\sqrt [3]{e} \left (-4 x^2-20 x\right ) \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {4 e^{2/3} x^2+\left (2+2 \sqrt [3]{e}\right ) x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\left (-x^2+\sqrt [3]{e} \left (-4 x^2-20 x\right )-10 x\right ) \log \left (x^2\right )+10 x}{4 e^{2/3} x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\sqrt [3]{e} \left (-4 x^2-20 x\right ) \log \left (x^2\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (2+2 \sqrt [3]{e}+4 e^{2/3}\right ) x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\left (-x^2+\sqrt [3]{e} \left (-4 x^2-20 x\right )-10 x\right ) \log \left (x^2\right )+10 x}{4 e^{2/3} x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\sqrt [3]{e} \left (-4 x^2-20 x\right ) \log \left (x^2\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (2+2 \sqrt [3]{e}+4 e^{2/3}\right ) x^2+\left (x^2+10 x+25\right ) \log ^2\left (x^2\right )+\left (-x^2+\sqrt [3]{e} \left (-4 x^2-20 x\right )-10 x\right ) \log \left (x^2\right )+10 x}{\left (x \left (-\log \left (x^2\right )\right )-5 \log \left (x^2\right )+2 \sqrt [3]{e} x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x (x+10)}{(x+5) \left (x \left (-\log \left (x^2\right )\right )-5 \log \left (x^2\right )+2 \sqrt [3]{e} x\right )}+\frac {2 x \left (x^2+5 \left (2-\sqrt [3]{e}\right ) x+25\right )}{(x+5) \left (x \left (-\log \left (x^2\right )\right )-5 \log \left (x^2\right )+2 \sqrt [3]{e} x\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 50 \sqrt [3]{e} \int \frac {1}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+10 \left (1-\sqrt [3]{e}\right ) \int \frac {x}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+2 \int \frac {x^2}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx-250 \sqrt [3]{e} \int \frac {1}{(x+5) \left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+5 \int \frac {1}{-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )}dx+\int \frac {x}{-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )}dx-25 \int \frac {1}{(x+5) \left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )}dx+x\) |
Int[(10*x + 2*x^2 + 2*E^(1/3)*x^2 + 4*E^(2/3)*x^2 + (-10*x - x^2 + E^(1/3) *(-20*x - 4*x^2))*Log[x^2] + (25 + 10*x + x^2)*Log[x^2]^2)/(4*E^(2/3)*x^2 + E^(1/3)*(-20*x - 4*x^2)*Log[x^2] + (25 + 10*x + x^2)*Log[x^2]^2),x]
3.10.62.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Time = 3.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
risch | \(x +\frac {x^{2}}{2 x \,{\mathrm e}^{\frac {1}{3}}-x \ln \left (x^{2}\right )-5 \ln \left (x^{2}\right )}\) | \(28\) |
norman | \(\frac {25 \ln \left (x^{2}\right )-10 x \,{\mathrm e}^{\frac {1}{3}}+\left (2 \,{\mathrm e}^{\frac {1}{3}}+1\right ) x^{2}-x^{2} \ln \left (x^{2}\right )}{2 x \,{\mathrm e}^{\frac {1}{3}}-x \ln \left (x^{2}\right )-5 \ln \left (x^{2}\right )}\) | \(54\) |
parallelrisch | \(-\frac {-4 x^{2} {\mathrm e}^{\frac {1}{3}}+2 x^{2} \ln \left (x^{2}\right )+20 x \,{\mathrm e}^{\frac {1}{3}}-2 x^{2}-50 \ln \left (x^{2}\right )}{2 \left (2 x \,{\mathrm e}^{\frac {1}{3}}-x \ln \left (x^{2}\right )-5 \ln \left (x^{2}\right )\right )}\) | \(57\) |
int(((x^2+10*x+25)*ln(x^2)^2+((-4*x^2-20*x)*exp(1/3)-x^2-10*x)*ln(x^2)+4*x ^2*exp(1/3)^2+2*x^2*exp(1/3)+2*x^2+10*x)/((x^2+10*x+25)*ln(x^2)^2+(-4*x^2- 20*x)*exp(1/3)*ln(x^2)+4*x^2*exp(1/3)^2),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {2 \, x^{2} e^{\frac {1}{3}} + x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x^{2}\right )}{2 \, x e^{\frac {1}{3}} - {\left (x + 5\right )} \log \left (x^{2}\right )} \end {dmath*}
integrate(((x^2+10*x+25)*log(x^2)^2+((-4*x^2-20*x)*exp(1/3)-x^2-10*x)*log( x^2)+4*x^2*exp(1/3)^2+2*x^2*exp(1/3)+2*x^2+10*x)/((x^2+10*x+25)*log(x^2)^2 +(-4*x^2-20*x)*exp(1/3)*log(x^2)+4*x^2*exp(1/3)^2),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=- \frac {x^{2}}{- 2 x e^{\frac {1}{3}} + \left (x + 5\right ) \log {\left (x^{2} \right )}} + x \end {dmath*}
integrate(((x**2+10*x+25)*ln(x**2)**2+((-4*x**2-20*x)*exp(1/3)-x**2-10*x)* ln(x**2)+4*x**2*exp(1/3)**2+2*x**2*exp(1/3)+2*x**2+10*x)/((x**2+10*x+25)*l n(x**2)**2+(-4*x**2-20*x)*exp(1/3)*ln(x**2)+4*x**2*exp(1/3)**2),x)
Exception generated. \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: RuntimeError} \end {dmath*}
integrate(((x^2+10*x+25)*log(x^2)^2+((-4*x^2-20*x)*exp(1/3)-x^2-10*x)*log( x^2)+4*x^2*exp(1/3)^2+2*x^2*exp(1/3)+2*x^2+10*x)/((x^2+10*x+25)*log(x^2)^2 +(-4*x^2-20*x)*exp(1/3)*log(x^2)+4*x^2*exp(1/3)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {2 \, x^{2} e^{\frac {1}{3}} - x^{2} \log \left (x^{2}\right ) + x^{2} - 5 \, x \log \left (x^{2}\right )}{2 \, x e^{\frac {1}{3}} - x \log \left (x^{2}\right ) - 5 \, \log \left (x^{2}\right )} \end {dmath*}
integrate(((x^2+10*x+25)*log(x^2)^2+((-4*x^2-20*x)*exp(1/3)-x^2-10*x)*log( x^2)+4*x^2*exp(1/3)^2+2*x^2*exp(1/3)+2*x^2+10*x)/((x^2+10*x+25)*log(x^2)^2 +(-4*x^2-20*x)*exp(1/3)*log(x^2)+4*x^2*exp(1/3)^2),x, algorithm=\
Time = 15.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \begin {dmath*} \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {25\,\ln \left (x^2\right )\,{\mathrm {e}}^{1/3}+10\,x\,\ln \left (x^2\right )-10\,x\,{\mathrm {e}}^{2/3}+2\,x^2\,\ln \left (x^2\right )-4\,x^2\,{\mathrm {e}}^{1/3}-2\,x^2+5\,x\,\ln \left (x^2\right )\,{\mathrm {e}}^{1/3}}{2\,\left (5\,\ln \left (x^2\right )+x\,\ln \left (x^2\right )-2\,x\,{\mathrm {e}}^{1/3}\right )} \end {dmath*}
int((10*x + log(x^2)^2*(10*x + x^2 + 25) + 2*x^2*exp(1/3) + 4*x^2*exp(2/3) + 2*x^2 - log(x^2)*(10*x + exp(1/3)*(20*x + 4*x^2) + x^2))/(log(x^2)^2*(1 0*x + x^2 + 25) + 4*x^2*exp(2/3) - log(x^2)*exp(1/3)*(20*x + 4*x^2)),x)