\(\int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+(-10 x-x^2+\sqrt [3]{e} (-20 x-4 x^2)) \log (x^2)+(25+10 x+x^2) \log ^2(x^2)}{4 e^{2/3} x^2+\sqrt [3]{e} (-20 x-4 x^2) \log (x^2)+(25+10 x+x^2) \log ^2(x^2)} \, dx\) [962]

Optimal result

Integrand size = 121, antiderivative size = 27 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=-4+x+\frac {x}{2 \sqrt [3]{e}-\frac {(5+x) \log \left (x^2\right )}{x}} \] Output:

x+x/(2*exp(1/3)-ln(x^2)/x*(5+x))-4
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=x+\frac {x^2}{2 \sqrt [3]{e} x-(5+x) \log \left (x^2\right )} \] Input:

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 
]
 

Output:

x + x^2/(2*E^(1/3)*x - (5 + x)*Log[x^2])
 

Rubi [F]

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

\(\displaystyle 50 \sqrt [3]{e} \int \frac {1}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+10 \left (1-\sqrt [3]{e}\right ) \int \frac {x}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+2 \int \frac {x^2}{\left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx-250 \sqrt [3]{e} \int \frac {1}{(x+5) \left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )^2}dx+5 \int \frac {1}{-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )}dx+\int \frac {x}{-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )}dx-25 \int \frac {1}{(x+5) \left (-\log \left (x^2\right ) x+2 \sqrt [3]{e} x-5 \log \left (x^2\right )\right )}dx+x\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.77 (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	\(-\frac {-4 x^{2} {\mathrm e}^{\frac {1}{3}}+2 x^{2} \ln \left (x^{2}\right )+20 x \,{\mathrm e}^{\frac {1}{3}}-2 x^{2}-50 \ln \left (x^{2}\right )}{2 \left (2 x \,{\mathrm e}^{\frac {1}{3}}-x \ln \left (x^{2}\right )-5 \ln \left (x^{2}\right )\right )}\)	\(57\)

Input:

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)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {2 \, x^{2} e^{\frac {1}{3}} + x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x^{2}\right )}{2 \, x e^{\frac {1}{3}} - {\left (x + 5\right )} \log \left (x^{2}\right )} \] Input:

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="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=- \frac {x^{2}}{- 2 x e^{\frac {1}{3}} + \left (x + 5\right ) \log {\left (x^{2} \right )}} + x \] Input:

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)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

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="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {2 \, x^{2} e^{\frac {1}{3}} - x^{2} \log \left (x^{2}\right ) + x^{2} - 5 \, x \log \left (x^{2}\right )}{2 \, x e^{\frac {1}{3}} - x \log \left (x^{2}\right ) - 5 \, \log \left (x^{2}\right )} \] Input:

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="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 7.90 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {25\,\ln \left (x^2\right )\,{\mathrm {e}}^{1/3}+10\,x\,\ln \left (x^2\right )-10\,x\,{\mathrm {e}}^{2/3}+2\,x^2\,\ln \left (x^2\right )-4\,x^2\,{\mathrm {e}}^{1/3}-2\,x^2+5\,x\,\ln \left (x^2\right )\,{\mathrm {e}}^{1/3}}{2\,\left (5\,\ln \left (x^2\right )+x\,\ln \left (x^2\right )-2\,x\,{\mathrm {e}}^{1/3}\right )} \] Input:

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)
 

Output:

(25*log(x^2)*exp(1/3) + 10*x*log(x^2) - 10*x*exp(2/3) + 2*x^2*log(x^2) - 4 
*x^2*exp(1/3) - 2*x^2 + 5*x*log(x^2)*exp(1/3))/(2*(5*log(x^2) + x*log(x^2) 
 - 2*x*exp(1/3)))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {10 x+2 x^2+2 \sqrt [3]{e} x^2+4 e^{2/3} x^2+\left (-10 x-x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right )\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )}{4 e^{2/3} x^2+\sqrt [3]{e} \left (-20 x-4 x^2\right ) \log \left (x^2\right )+\left (25+10 x+x^2\right ) \log ^2\left (x^2\right )} \, dx=\frac {x \left (2 e^{\frac {1}{3}} x -\mathrm {log}\left (x^{2}\right ) x -5 \,\mathrm {log}\left (x^{2}\right )+x \right )}{2 e^{\frac {1}{3}} x -\mathrm {log}\left (x^{2}\right ) x -5 \,\mathrm {log}\left (x^{2}\right )} \] Input:

int(((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)
 

Output:

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