\(\int \frac {x^8}{a+f x^2+c x^4} \, dx\) [783]

Optimal result

Integrand size = 18, antiderivative size = 355 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=-\frac {\left (a c-f^2\right ) x}{c^3}-\frac {f x^3}{3 c^2}+\frac {x^5}{5 c}-\frac {\left (2 a c f-f^3+\sqrt {a} \sqrt {c} \left (a c-f^2\right )\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{7/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\left (2 a c f-f^3+\sqrt {a} \sqrt {c} \left (a c-f^2\right )\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{7/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\left (a^{3/2} c^{3/2}-2 a c f-\sqrt {a} \sqrt {c} f^2+f^3\right ) \text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 c^{7/2} \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:

-(a*c-f^2)*x/c^3-1/3*f*x^3/c^2+1/5*x^5/c-1/2*(2*a*c*f-f^3+a^(1/2)*c^(1/2)* 
(a*c-f^2))*arctan(((2*a^(1/2)*c^(1/2)-f)^(1/2)-2*c^(1/2)*x)/(2*a^(1/2)*c^( 
1/2)+f)^(1/2))/c^(7/2)/(2*a^(1/2)*c^(1/2)+f)^(1/2)+1/2*(2*a*c*f-f^3+a^(1/2 
)*c^(1/2)*(a*c-f^2))*arctan(((2*a^(1/2)*c^(1/2)-f)^(1/2)+2*c^(1/2)*x)/(2*a 
^(1/2)*c^(1/2)+f)^(1/2))/c^(7/2)/(2*a^(1/2)*c^(1/2)+f)^(1/2)+1/2*(a^(3/2)* 
c^(3/2)-2*a*c*f-a^(1/2)*c^(1/2)*f^2+f^3)*arctanh((2*a^(1/2)*c^(1/2)-f)^(1/ 
2)*x/(a^(1/2)+c^(1/2)*x^2))/c^(7/2)/(2*a^(1/2)*c^(1/2)-f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.79 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=\frac {\left (-a c+f^2\right ) x}{c^3}-\frac {f x^3}{3 c^2}+\frac {x^5}{5 c}+\frac {\left (2 a^2 c^2+f^3 \left (f-\sqrt {-4 a c+f^2}\right )+2 a c f \left (-2 f+\sqrt {-4 a c+f^2}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{7/2} \sqrt {-4 a c+f^2} \sqrt {f-\sqrt {-4 a c+f^2}}}-\frac {\left (2 a^2 c^2+f^3 \left (f+\sqrt {-4 a c+f^2}\right )-2 a c f \left (2 f+\sqrt {-4 a c+f^2}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{7/2} \sqrt {-4 a c+f^2} \sqrt {f+\sqrt {-4 a c+f^2}}} \] Input:

Integrate[x^8/(a + f*x^2 + c*x^4),x]
 

Output:

((-(a*c) + f^2)*x)/c^3 - (f*x^3)/(3*c^2) + x^5/(5*c) + ((2*a^2*c^2 + f^3*( 
f - Sqrt[-4*a*c + f^2]) + 2*a*c*f*(-2*f + Sqrt[-4*a*c + f^2]))*ArcTan[(Sqr 
t[2]*Sqrt[c]*x)/Sqrt[f - Sqrt[-4*a*c + f^2]]])/(Sqrt[2]*c^(7/2)*Sqrt[-4*a* 
c + f^2]*Sqrt[f - Sqrt[-4*a*c + f^2]]) - ((2*a^2*c^2 + f^3*(f + Sqrt[-4*a* 
c + f^2]) - 2*a*c*f*(2*f + Sqrt[-4*a*c + f^2]))*ArcTan[(Sqrt[2]*Sqrt[c]*x) 
/Sqrt[f + Sqrt[-4*a*c + f^2]]])/(Sqrt[2]*c^(7/2)*Sqrt[-4*a*c + f^2]*Sqrt[f 
 + Sqrt[-4*a*c + f^2]])
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.50, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1442, 27, 1602, 27, 1602, 25, 1483, 27, 1142, 25, 27, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {x^8}{a+c x^4+f x^2} \, dx\)

\(\Big \downarrow \) 1442

\(\displaystyle \frac {x^5}{5 c}-\frac {\int \frac {5 x^4 \left (f x^2+a\right )}{c x^4+f x^2+a}dx}{5 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^5}{5 c}-\frac {\int \frac {x^4 \left (f x^2+a\right )}{c x^4+f x^2+a}dx}{c}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\int \frac {3 x^2 \left (a f-\left (a c-f^2\right ) x^2\right )}{c x^4+f x^2+a}dx}{3 c}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\int \frac {x^2 \left (a f-\left (a c-f^2\right ) x^2\right )}{c x^4+f x^2+a}dx}{c}}{c}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {-\frac {\int -\frac {f \left (2 a c-f^2\right ) x^2+a \left (a c-f^2\right )}{c x^4+f x^2+a}dx}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\int \frac {f \left (2 a c-f^2\right ) x^2+a \left (a c-f^2\right )}{c x^4+f x^2+a}dx}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\int \frac {\sqrt {a} \left (\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )-\sqrt {c} \left (\sqrt {a} \left (a c-f^2\right )-\frac {f \left (2 a c-f^2\right )}{\sqrt {c}}\right ) x\right )}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {a} \left (\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )+\sqrt {c} \left (\sqrt {a} \left (a c-f^2\right )-\frac {f \left (2 a c-f^2\right )}{\sqrt {c}}\right ) x\right )}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\int \frac {\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )-\left (f^3-\sqrt {a} \sqrt {c} f^2-2 a c f+a^{3/2} c^{3/2}\right ) x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )+\left (f^3-\sqrt {a} \sqrt {c} f^2-2 a c f+a^{3/2} c^{3/2}\right ) x}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int -\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x^5}{5 c}-\frac {\frac {f x^3}{3 c}-\frac {\frac {\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2+2 a c f-f^3\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {1}{2} \left (a^{3/2} c^{3/2}-\sqrt {a} \sqrt {c} f^2-2 a c f+f^3\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}-\frac {x \left (a c-f^2\right )}{c}}{c}}{c}\)

Input:

Int[x^8/(a + f*x^2 + c*x^4),x]
 

Output:

x^5/(5*c) - ((f*x^3)/(3*c) - (-(((a*c - f^2)*x)/c) + (((Sqrt[2*Sqrt[a]*Sqr 
t[c] - f]*(a^(3/2)*c^(3/2) + 2*a*c*f - Sqrt[a]*Sqrt[c]*f^2 - f^3)*ArcTan[( 
Sqrt[c]*(-(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqrt[c]) + 2*x))/Sqrt[2*Sqrt[a]*Sqr 
t[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f] - ((a^(3/2)*c^(3/2) - 2*a*c*f - Sq 
rt[a]*Sqrt[c]*f^2 + f^3)*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqr 
t[c]*x^2])/2)/(2*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]) + ((Sqrt[2*Sqrt[a]*S 
qrt[c] - f]*(a^(3/2)*c^(3/2) + 2*a*c*f - Sqrt[a]*Sqrt[c]*f^2 - f^3)*ArcTan 
[(Sqrt[c]*(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqrt[c] + 2*x))/Sqrt[2*Sqrt[a]*Sqrt 
[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f] + ((a^(3/2)*c^(3/2) - 2*a*c*f - Sqr 
t[a]*Sqrt[c]*f^2 + f^3)*Log[Sqrt[a] + Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt 
[c]*x^2])/2)/(2*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]))/c)/c)/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.28

method	result	size
risch	\(\frac {x^{5}}{5 c}-\frac {f \,x^{3}}{3 c^{2}}-\frac {a x}{c^{2}}+\frac {x \,f^{2}}{c^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +f \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (f \left (2 a c -f^{2}\right ) \textit {\_R}^{2}+c \,a^{2}-a \,f^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} f}}{2 c^{3}}\)	\(101\)
default	\(-\frac {-\frac {1}{5} c^{2} x^{5}+\frac {1}{3} f \,x^{3} c +a c x -x \,f^{2}}{c^{3}}+\frac {\frac {\left (2 a c f \sqrt {-4 a c +f^{2}}-f^{3} \sqrt {-4 a c +f^{2}}-2 a^{2} c^{2}+4 a c \,f^{2}-f^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\left (2 a c f \sqrt {-4 a c +f^{2}}-f^{3} \sqrt {-4 a c +f^{2}}+2 a^{2} c^{2}-4 a c \,f^{2}+f^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}}{c^{2}}\)	\(257\)

Input:

int(x^8/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/5*x^5/c-1/3*f*x^3/c^2-1/c^2*a*x+1/c^3*x*f^2+1/2/c^3*sum((f*(2*a*c-f^2)*_ 
R^2+c*a^2-a*f^2)/(2*_R^3*c+_R*f)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*f+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2133 vs. \(2 (270) = 540\).

Time = 0.16 (sec) , antiderivative size = 2133, normalized size of antiderivative = 6.01 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

integrate(x^8/(c*x^4+f*x^2+a),x, algorithm="fricas")
 

Output:

1/30*(6*c^2*x^5 - 10*c*f*x^3 - 15*sqrt(1/2)*c^3*sqrt(-(7*a^3*c^3*f - 14*a^ 
2*c^2*f^3 + 7*a*c*f^5 - f^7 + (4*a*c^8 - c^7*f^2)*sqrt(-(a^6*c^6 - 12*a^5* 
c^5*f^2 + 46*a^4*c^4*f^4 - 62*a^3*c^3*f^6 + 37*a^2*c^2*f^8 - 10*a*c*f^10 + 
 f^12)/(4*a*c^15 - c^14*f^2)))/(4*a*c^8 - c^7*f^2))*log(-2*(a^6*c^3 - 6*a^ 
5*c^2*f^2 + 5*a^4*c*f^4 - a^3*f^6)*x + sqrt(1/2)*(4*a^5*c^5 - 29*a^4*c^4*f 
^2 + 51*a^3*c^3*f^4 - 35*a^2*c^2*f^6 + 10*a*c*f^8 - f^10 + (12*a^2*c^9*f - 
 7*a*c^8*f^3 + c^7*f^5)*sqrt(-(a^6*c^6 - 12*a^5*c^5*f^2 + 46*a^4*c^4*f^4 - 
 62*a^3*c^3*f^6 + 37*a^2*c^2*f^8 - 10*a*c*f^10 + f^12)/(4*a*c^15 - c^14*f^ 
2)))*sqrt(-(7*a^3*c^3*f - 14*a^2*c^2*f^3 + 7*a*c*f^5 - f^7 + (4*a*c^8 - c^ 
7*f^2)*sqrt(-(a^6*c^6 - 12*a^5*c^5*f^2 + 46*a^4*c^4*f^4 - 62*a^3*c^3*f^6 + 
 37*a^2*c^2*f^8 - 10*a*c*f^10 + f^12)/(4*a*c^15 - c^14*f^2)))/(4*a*c^8 - c 
^7*f^2))) + 15*sqrt(1/2)*c^3*sqrt(-(7*a^3*c^3*f - 14*a^2*c^2*f^3 + 7*a*c*f 
^5 - f^7 + (4*a*c^8 - c^7*f^2)*sqrt(-(a^6*c^6 - 12*a^5*c^5*f^2 + 46*a^4*c^ 
4*f^4 - 62*a^3*c^3*f^6 + 37*a^2*c^2*f^8 - 10*a*c*f^10 + f^12)/(4*a*c^15 - 
c^14*f^2)))/(4*a*c^8 - c^7*f^2))*log(-2*(a^6*c^3 - 6*a^5*c^2*f^2 + 5*a^4*c 
*f^4 - a^3*f^6)*x - sqrt(1/2)*(4*a^5*c^5 - 29*a^4*c^4*f^2 + 51*a^3*c^3*f^4 
 - 35*a^2*c^2*f^6 + 10*a*c*f^8 - f^10 + (12*a^2*c^9*f - 7*a*c^8*f^3 + c^7* 
f^5)*sqrt(-(a^6*c^6 - 12*a^5*c^5*f^2 + 46*a^4*c^4*f^4 - 62*a^3*c^3*f^6 + 3 
7*a^2*c^2*f^8 - 10*a*c*f^10 + f^12)/(4*a*c^15 - c^14*f^2)))*sqrt(-(7*a^3*c 
^3*f - 14*a^2*c^2*f^3 + 7*a*c*f^5 - f^7 + (4*a*c^8 - c^7*f^2)*sqrt(-(a^...
 

Sympy [A] (verification not implemented)

Time = 9.74 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.74 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=x \left (- \frac {a}{c^{2}} + \frac {f^{2}}{c^{3}}\right ) + \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{9} - 128 a c^{8} f^{2} + 16 c^{7} f^{4}\right ) + t^{2} \cdot \left (112 a^{4} c^{4} f - 252 a^{3} c^{3} f^{3} + 168 a^{2} c^{2} f^{5} - 44 a c f^{7} + 4 f^{9}\right ) + a^{7}, \left ( t \mapsto t \log {\left (x + \frac {- 96 t^{3} a^{2} c^{9} f + 56 t^{3} a c^{8} f^{3} - 8 t^{3} c^{7} f^{5} + 4 t a^{5} c^{5} - 50 t a^{4} c^{4} f^{2} + 100 t a^{3} c^{3} f^{4} - 70 t a^{2} c^{2} f^{6} + 20 t a c f^{8} - 2 t f^{10}}{a^{6} c^{3} - 6 a^{5} c^{2} f^{2} + 5 a^{4} c f^{4} - a^{3} f^{6}} \right )} \right )\right )} + \frac {x^{5}}{5 c} - \frac {f x^{3}}{3 c^{2}} \] Input:

integrate(x**8/(c*x**4+f*x**2+a),x)
 

Output:

x*(-a/c**2 + f**2/c**3) + RootSum(_t**4*(256*a**2*c**9 - 128*a*c**8*f**2 + 
 16*c**7*f**4) + _t**2*(112*a**4*c**4*f - 252*a**3*c**3*f**3 + 168*a**2*c* 
*2*f**5 - 44*a*c*f**7 + 4*f**9) + a**7, Lambda(_t, _t*log(x + (-96*_t**3*a 
**2*c**9*f + 56*_t**3*a*c**8*f**3 - 8*_t**3*c**7*f**5 + 4*_t*a**5*c**5 - 5 
0*_t*a**4*c**4*f**2 + 100*_t*a**3*c**3*f**4 - 70*_t*a**2*c**2*f**6 + 20*_t 
*a*c*f**8 - 2*_t*f**10)/(a**6*c**3 - 6*a**5*c**2*f**2 + 5*a**4*c*f**4 - a* 
*3*f**6)))) + x**5/(5*c) - f*x**3/(3*c**2)
 

Maxima [F]

\[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=\int { \frac {x^{8}}{c x^{4} + f x^{2} + a} \,d x } \] Input:

integrate(x^8/(c*x^4+f*x^2+a),x, algorithm="maxima")
 

Output:

1/15*(3*c^2*x^5 - 5*c*f*x^3 - 15*(a*c - f^2)*x)/c^3 - integrate(-(a^2*c - 
a*f^2 + (2*a*c*f - f^3)*x^2)/(c*x^4 + f*x^2 + a), x)/c^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3064 vs. \(2 (270) = 540\).

Time = 0.65 (sec) , antiderivative size = 3064, normalized size of antiderivative = 8.63 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

integrate(x^8/(c*x^4+f*x^2+a),x, algorithm="giac")
 

Output:

1/8*(16*a^3*c^7*f - 36*a^2*c^6*f^3 + 16*a*c^5*f^5 - 2*c^4*f^7 - 8*sqrt(2)* 
sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^3*c^5*f + 2*sqrt(2)* 
sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^6*f - 4*sqrt(2)* 
sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^5*f^2 + 18*sqrt( 
2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4*f^3 - 4*sqr 
t(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^5*f^3 + 8*sqr 
t(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^4*f^4 - 8*sqr 
t(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f^5 + sqrt( 
2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^4*f^5 - 2*sqrt(2) 
*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^3*f^6 + sqrt(2)*sqr 
t(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^7 - 4*(4*a*c - f^2) 
*a^2*c^6*f + 8*(4*a*c - f^2)*a*c^5*f^3 - 2*(4*a*c - f^2)*c^4*f^5 - (64*a^3 
*c^5*f - 64*a^2*c^4*f^3 + 20*a*c^3*f^5 - 2*c^2*f^7 - 32*sqrt(2)*sqrt(-4*a* 
c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^3*c^3*f + 8*sqrt(2)*sqrt(-4*a* 
c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4*f - 16*sqrt(2)*sqrt(-4*a 
*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^3*f^2 + 32*sqrt(2)*sqrt(- 
4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^2*f^3 - 6*sqrt(2)*sqrt 
(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f^3 + 12*sqrt(2)*sqr 
t(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^2*f^4 - 10*sqrt(2)*sq 
rt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f^5 + sqrt(2)*sqr...
 

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 5317, normalized size of antiderivative = 14.98 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

int(x^8/(a + c*x^4 + f*x^2),x)
 

Output:

atan(((((16*a^3*c^6 + 4*a*c^4*f^4 - 20*a^2*c^5*f^2)/c^5 - (2*x*(4*c^7*f^3 
- 16*a*c^8*f)*(-(f^9 + f^6*(-(4*a*c - f^2)^3)^(1/2) + 28*a^4*c^4*f + 42*a^ 
2*c^2*f^5 - 63*a^3*c^3*f^3 - a^3*c^3*(-(4*a*c - f^2)^3)^(1/2) - 11*a*c*f^7 
 + 6*a^2*c^2*f^2*(-(4*a*c - f^2)^3)^(1/2) - 5*a*c*f^4*(-(4*a*c - f^2)^3)^( 
1/2))/(8*(16*a^2*c^9 + c^7*f^4 - 8*a*c^8*f^2)))^(1/2))/c^5)*(-(f^9 + f^6*( 
-(4*a*c - f^2)^3)^(1/2) + 28*a^4*c^4*f + 42*a^2*c^2*f^5 - 63*a^3*c^3*f^3 - 
 a^3*c^3*(-(4*a*c - f^2)^3)^(1/2) - 11*a*c*f^7 + 6*a^2*c^2*f^2*(-(4*a*c - 
f^2)^3)^(1/2) - 5*a*c*f^4*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^9 + c^7*f 
^4 - 8*a*c^8*f^2)))^(1/2) - (2*x*(f^8 + 2*a^4*c^4 + 20*a^2*c^2*f^4 - 16*a^ 
3*c^3*f^2 - 8*a*c*f^6))/c^5)*(-(f^9 + f^6*(-(4*a*c - f^2)^3)^(1/2) + 28*a^ 
4*c^4*f + 42*a^2*c^2*f^5 - 63*a^3*c^3*f^3 - a^3*c^3*(-(4*a*c - f^2)^3)^(1/ 
2) - 11*a*c*f^7 + 6*a^2*c^2*f^2*(-(4*a*c - f^2)^3)^(1/2) - 5*a*c*f^4*(-(4* 
a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^9 + c^7*f^4 - 8*a*c^8*f^2)))^(1/2)*1i - 
(((16*a^3*c^6 + 4*a*c^4*f^4 - 20*a^2*c^5*f^2)/c^5 + (2*x*(4*c^7*f^3 - 16*a 
*c^8*f)*(-(f^9 + f^6*(-(4*a*c - f^2)^3)^(1/2) + 28*a^4*c^4*f + 42*a^2*c^2* 
f^5 - 63*a^3*c^3*f^3 - a^3*c^3*(-(4*a*c - f^2)^3)^(1/2) - 11*a*c*f^7 + 6*a 
^2*c^2*f^2*(-(4*a*c - f^2)^3)^(1/2) - 5*a*c*f^4*(-(4*a*c - f^2)^3)^(1/2))/ 
(8*(16*a^2*c^9 + c^7*f^4 - 8*a*c^8*f^2)))^(1/2))/c^5)*(-(f^9 + f^6*(-(4*a* 
c - f^2)^3)^(1/2) + 28*a^4*c^4*f + 42*a^2*c^2*f^5 - 63*a^3*c^3*f^3 - a^3*c 
^3*(-(4*a*c - f^2)^3)^(1/2) - 11*a*c*f^7 + 6*a^2*c^2*f^2*(-(4*a*c - f^2...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.79 \[ \int \frac {x^8}{a+f x^2+c x^4} \, dx =\text {Too large to display} \] Input:

int(x^8/(c*x^4+f*x^2+a),x)
 

Output:

( - 90*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - 
f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c**2*f + 30*sqrt(a)*sqrt( 
2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sq 
rt(2*sqrt(c)*sqrt(a) + f))*c*f**3 - 60*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f) 
*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + 
 f))*a**2*c**2 + 120*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt 
(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c*f**2 - 30 
*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2 
*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f**4 + 90*sqrt(a)*sqrt(2*sqrt(c)* 
sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt( 
c)*sqrt(a) + f))*a*c**2*f - 30*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((s 
qrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*c*f 
**3 + 60*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) 
- f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a**2*c**2 - 120*sqrt(c)*s 
qrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x 
)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c*f**2 + 30*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a 
) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqr 
t(a) + f))*f**4 + 45*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqr 
t(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c**2*f - 15*sqrt(a)*sqrt(2 
*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + ...