\(\int \frac {x^5 (2 x^5 c_3-3 c_4) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{(x^3+2 x^5 c_3+2 c_4) (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2)} \, dx\) [2760]

Optimal result

Integrand size = 99, antiderivative size = 259 \[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+c_0}\right ) \sqrt {1-c_0}}{\sqrt {-1+c_1}}+\frac {\arctan \left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{1+c_0}\right ) \sqrt {-1-c_0}}{3 \sqrt {1+c_1}}-\frac {4 \arctan \left (\frac {\sqrt {1-2 c_0} \sqrt {-1+2 c_1} \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+2 c_0}\right ) \sqrt {1-2 c_0}}{3 \sqrt {-1+2 c_1}} \] Output:

arctan((1-_C0)^(1/2)*(-1+_C1)^(1/2)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^ 
3+_C4))^(1/2)/(-1+_C0))*(1-_C0)^(1/2)/(-1+_C1)^(1/2)+1/3*arctan((-1-_C0)^( 
1/2)*(1+_C1)^(1/2)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1/2)/(1+ 
_C0))*(-1-_C0)^(1/2)/(1+_C1)^(1/2)-4/3*arctan((1-2*_C0)^(1/2)*(-1+2*_C1)^( 
1/2)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1/2)/(-1+2*_C0))*(1-2* 
_C0)^(1/2)/(-1+2*_C1)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx \] Input:

Integrate[(x^5*(2*x^5*C[3] - 3*C[4])*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^ 
3*C[1] + x^5*C[3] + C[4])])/((x^3 + 2*x^5*C[3] + 2*C[4])*(-x^6 + x^10*C[3] 
^2 + 2*x^5*C[3]*C[4] + C[4]^2)),x]
 

Output:

Integrate[(x^5*(2*x^5*C[3] - 3*C[4])*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^ 
3*C[1] + x^5*C[3] + C[4])])/((x^3 + 2*x^5*C[3] + 2*C[4])*(-x^6 + x^10*C[3] 
^2 + 2*x^5*C[3]*C[4] + C[4]^2)), x]
 

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.

\(\displaystyle \int \frac {x^5 \left (2 c_3 x^5-3 c_4\right ) \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{\left (2 c_3 x^5+x^3+2 c_4\right ) \left (c_3{}^2 x^{10}-x^6+2 c_3 c_4 x^5+c_4{}^2\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {x^5 \left (1-c_3 x^2\right ) \left (2 c_3 x^5-3 c_4\right ) \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{2 c_4 \left (c_3 x^5-x^3+c_4\right ) \left (2 c_3 x^5+x^3+2 c_4\right )}+\frac {x^5 \left (1+c_3 x^2\right ) \left (2 c_3 x^5-3 c_4\right ) \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}}}{2 c_4 \left (c_3 x^5+x^3+c_4\right ) \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^5 \sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \left (3 c_4-2 c_3 x^5\right )}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \left (2 c_3 x^5+x^3+2 c_4\right )}dx\)

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int -\frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \left (\frac {\left (5 x^2 c_3-3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{6 \left (-c_3 x^5+x^3-c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}-\frac {\left (5 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{2 \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4}}+\frac {2 \left (10 c_3 x^2+3\right ) \sqrt {c_3 x^5+c_0 x^3+c_4} x^2}{3 \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}\right )dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\sqrt {\frac {c_3 x^5+c_0 x^3+c_4}{c_3 x^5+c_1 x^3+c_4}} \sqrt {c_3 x^5+c_1 x^3+c_4} \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {c_3 x^5+c_0 x^3+c_4}}{\left (-c_3 x^5+x^3-c_4\right ) \left (c_3 x^5+x^3+c_4\right ) \sqrt {c_3 x^5+c_1 x^3+c_4} \left (2 c_3 x^5+x^3+2 c_4\right )}dx}{\sqrt {c_3 x^5+c_0 x^3+c_4}}\)

Input:

Int[(x^5*(2*x^5*C[3] - 3*C[4])*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^3*C[1] 
 + x^5*C[3] + C[4])])/((x^3 + 2*x^5*C[3] + 2*C[4])*(-x^6 + x^10*C[3]^2 + 2 
*x^5*C[3]*C[4] + C[4]^2)),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {x^{5} \left (2 \textit {\_C3} \,x^{5}-3 \textit {\_C4} \right ) \sqrt {\frac {\textit {\_C3} \,x^{5}+\textit {\_C0} \,x^{3}+\textit {\_C4}}{\textit {\_C3} \,x^{5}+\textit {\_C1} \,x^{3}+\textit {\_C4}}}}{\left (2 \textit {\_C3} \,x^{5}+x^{3}+2 \textit {\_C4} \right ) \left (\textit {\_C3}^{2} x^{10}+2 \textit {\_C3} \textit {\_C4} \,x^{5}-x^{6}+\textit {\_C4}^{2}\right )}d x\]

Input:

int(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1 
/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x)
 

Output:

int(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1 
/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C 
4))^(1/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x, al 
gorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x**5*(2*_C3*x**5-3*_C4)*((_C3*x**5+_C0*x**3+_C4)/(_C3*x**5+_C1*x 
**3+_C4))**(1/2)/(2*_C3*x**5+x**3+2*_C4)/(_C3**2*x**10+2*_C3*_C4*x**5-x**6 
+_C4**2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \text {Unable to generate Latex} \] Input:

integrate(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C 
4))^(1/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x, al 
gorithm="maxima")
 

Output:

integrate((2*_C3*x^5 - 3*_C4)*x^5*sqrt((_C3*x^5 + _C0*x^3 + _C4)/(_C3*x^5 
+ _C1*x^3 + _C4))/((_C3^2*x^10 + 2*_C3*_C4*x^5 - x^6 + _C4^2)*(2*_C3*x^5 + 
 x^3 + 2*_C4)), x)
 

Giac [F]

\[ \text {Unable to generate Latex} \] Input:

integrate(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C 
4))^(1/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x, al 
gorithm="giac")
 

Output:

integrate((2*_C3*x^5 - 3*_C4)*x^5*sqrt((_C3*x^5 + _C0*x^3 + _C4)/(_C3*x^5 
+ _C1*x^3 + _C4))/((_C3^2*x^10 + 2*_C3*_C4*x^5 - x^6 + _C4^2)*(2*_C3*x^5 + 
 x^3 + 2*_C4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\int -\frac {x^5\,\sqrt {\frac {_{\mathrm {C3}}\,x^5+_{\mathrm {C0}}\,x^3+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^5+_{\mathrm {C1}}\,x^3+_{\mathrm {C4}}}}\,\left (3\,_{\mathrm {C4}}-2\,_{\mathrm {C3}}\,x^5\right )}{\left (2\,_{\mathrm {C3}}\,x^5+x^3+2\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^{10}+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^5+{_{\mathrm {C4}}}^2-x^6\right )} \,d x \] Input:

int(-(x^5*((_C4 + _C0*x^3 + _C3*x^5)/(_C4 + _C1*x^3 + _C3*x^5))^(1/2)*(3*_ 
C4 - 2*_C3*x^5))/((2*_C4 + 2*_C3*x^5 + x^3)*(_C4^2 - x^6 + _C3^2*x^10 + 2* 
_C3*_C4*x^5)),x)
 

Output:

int(-(x^5*((_C4 + _C0*x^3 + _C3*x^5)/(_C4 + _C1*x^3 + _C3*x^5))^(1/2)*(3*_ 
C4 - 2*_C3*x^5))/((2*_C4 + 2*_C3*x^5 + x^3)*(_C4^2 - x^6 + _C3^2*x^10 + 2* 
_C3*_C4*x^5)), x)
 

Reduce [F]

\[ \int \frac {x^5 \left (2 x^5 c_3-3 c_4\right ) \sqrt {\frac {x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left (x^3+2 x^5 c_3+2 c_4\right ) \left (-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {x^{5} \left (2 \textit {\_C3} \,x^{5}-3 \textit {\_C4} \right ) \sqrt {\frac {\textit {\_C3} \,x^{5}+\textit {\_C0} \,x^{3}+\textit {\_C4}}{\textit {\_C3} \,x^{5}+\textit {\_C1} \,x^{3}+\textit {\_C4}}}}{\left (2 \textit {\_C3} \,x^{5}+x^{3}+2 \textit {\_C4} \right ) \left (\textit {\_C3}^{2} x^{10}+2 \textit {\_C3} \textit {\_C4} \,x^{5}-x^{6}+\textit {\_C4}^{2}\right )}d x \] Input:

int(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1 
/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x)
 

Output:

int(x^5*(2*_C3*x^5-3*_C4)*((_C3*x^5+_C0*x^3+_C4)/(_C3*x^5+_C1*x^3+_C4))^(1 
/2)/(2*_C3*x^5+x^3+2*_C4)/(_C3^2*x^10+2*_C3*_C4*x^5-x^6+_C4^2),x)