Integrand size = 63, antiderivative size = 621 \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=-\frac {e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d+a f\right )-4 a b c \left (6 \sqrt {b^2-4 a c} d+13 a f\right )-a b^2 \left (30 c d-\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac {3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {6 c^2 e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:
-1/4*e*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/4*x*(b^2*d-2*a*c*d-a*b *f+c*(-2*a*f+b*d)*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/2*c*e*(2*c*x^2+b )/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/8*x*(3*b^4*d-25*a*b^2*c*d+28*a^2*c^2*d+ a*b^3*f+8*a^2*b*c*f+c*(20*a^2*c*f+a*b^2*f-24*a*b*c*d+3*b^3*d)*x^2)/a^2/(-4 *a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*c^(1/2)*(3*b^4*d+b^3*(3*(-4*a*c+b^2)^(1/2 )*d+a*f)-4*a*b*c*(6*(-4*a*c+b^2)^(1/2)*d+13*a*f)-a*b^2*(30*c*d-f*(-4*a*c+b ^2)^(1/2))+4*a^2*c*(42*c*d+5*f*(-4*a*c+b^2)^(1/2)))*arctan(2^(1/2)*c^(1/2) *x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^(5/2)/(b-(-4*a*c +b^2)^(1/2))^(1/2)+1/16*c^(1/2)*(3*b^3*d-24*a*b*c*d+a*b^2*f+20*a^2*c*f-(-5 2*a^2*b*c*f+168*a^2*c^2*d+a*b^3*f-30*a*b^2*c*d+3*b^4*d)/(-4*a*c+b^2)^(1/2) )*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a *c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)-6*c^2*e*arctanh((2*c*x^2+b)/(-4*a*c +b^2)^(1/2))/(-4*a*c+b^2)^(5/2)
Time = 3.47 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.01 \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\frac {1}{16} \left (\frac {4 a b (e+f x)-4 b d x \left (b+c x^2\right )+8 a c x (d+x (e+f x))}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {6 b^3 d x \left (b+c x^2\right )+2 a b x \left (-25 b c d+b^2 f-24 c^2 d x^2+b c f x^2\right )+8 a^2 c \left (b (3 e+2 f x)+c x \left (7 d+6 e x+5 f x^2\right )\right )}{a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d+a f\right )-4 a b c \left (6 \sqrt {b^2-4 a c} d+13 a f\right )+a b^2 \left (-30 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d-a f\right )+4 a b c \left (-6 \sqrt {b^2-4 a c} d+13 a f\right )+a b^2 \left (30 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (-42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {48 c^2 e \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {48 c^2 e \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}\right ) \] Input:
Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e *x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4,x]
Output:
((4*a*b*(e + f*x) - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)))/(a*(- b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) + 2*a*b*x*(-2 5*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + 8*a^2*c*(b*(3*e + 2*f*x) + c *x*(7*d + 6*e*x + 5*f*x^2)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ( Sqrt[2]*Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6* Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(-30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4* a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c] ]) + (Sqrt[2]*Sqrt[c]*(-3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d - a*f) + 4*a* b*c*(-6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(30*c*d + Sqrt[b^2 - 4*a*c]* f) + 4*a^2*c*(-42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x) /Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c )^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5 /2))/16
Time = 1.13 (sec) , antiderivative size = 605, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2019, 2202, 27, 1432, 1086, 1086, 1083, 219, 1492, 25, 1492, 25, 1480, 218}
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.
| \(\displaystyle \int \frac {x^2 (a f+b d)+a d+a e x+x^4 (b f+c d)+b e x^3+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^3}dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\int \frac {e x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+e \int \frac {x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} e \int \frac {1}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {1}{2} e \left (-\frac {3 c \int \frac {1}{\left (c x^4+b x^2+a\right )^2}dx^2}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {1}{2} e \left (-\frac {3 c \left (-\frac {2 c \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\int -\frac {3 d b^2+a f b+5 c (b d-2 a f) x^2-14 a c d}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 d b^2+a f b+5 c (b d-2 a f) x^2-14 a c d}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 d b^4+a f b^3-27 a c d b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-24 a c d b+20 a^2 c f\right ) x^2+84 a^2 c^2 d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {3 d b^4+a f b^3-27 a c d b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-24 a c d b+20 a^2 c f\right ) x^2+84 a^2 c^2 d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {1}{2} c \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
Input:
Int[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4,x]
Output:
(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((x*(3*b^4*d - 25*a*b^2*c*d + 28*a^2*c^2*d + a*b^3*f + 8*a^2*b*c*f + c*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^2*c*f)*x^2))/(2* a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((Sqrt[c]*(3*b^3*d - 24*a*b*c*d + a *b^2*f + 20*a^2*c*f + (3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + a*b^3*f - 52*a^2*b*c*f)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[ b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^2*c*f - (3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^ 2*d + a*b^3*f - 52*a^2*b*c*f)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x )/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2* a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c)) + (e*(-1/2*(b + 2*c*x^2)/((b^2 - 4*a *c)*(a + b*x^2 + c*x^4)^2) - (3*c*(-((b + 2*c*x^2)/((b^2 - 4*a*c)*(a + b*x ^2 + c*x^4))) + (4*c*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a* c)^(3/2)))/(b^2 - 4*a*c)))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.41 (sec) , antiderivative size = 607, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {\frac {c^{2} \left (20 a^{2} c f +a \,b^{2} f -24 a b c d +3 b^{3} d \right ) x^{7}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c^{3} e \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c \left (28 a^{2} b c f +28 a^{2} c^{2} d +2 a \,b^{3} f -49 a \,b^{2} c d +6 b^{4} d \right ) x^{5}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \,c^{2} e \,x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 a^{3} c^{2} f +5 a^{2} b^{2} c f -4 a^{2} c^{2} b d +a \,b^{4} f -20 a \,b^{3} c d +3 b^{5} d \right ) x^{3}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (5 a c +b^{2}\right ) c e \,x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {\left (16 a^{2} b c f +44 a^{2} c^{2} d -a \,b^{3} f -37 a \,b^{2} c d +5 b^{4} d \right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {b \left (10 a c -b^{2}\right ) e}{64 a^{2} c^{2}-32 a \,b^{2} c +4 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (20 a^{2} c f +a \,b^{2} f -24 a b c d +3 b^{3} d \right ) \textit {\_R}^{2}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {48 c^{2} e \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {16 a^{2} b c f -84 a^{2} c^{2} d -a \,b^{3} f +27 a \,b^{2} c d -3 b^{4} d}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{16}\) | \(607\) |
| default | \(\text {Expression too large to display}\) | \(1311\) |
Input:
int((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/(c*x^4 +b*x^2+a)^4,x,method=_RETURNVERBOSE)
Output:
(1/8*c^2*(20*a^2*c*f+a*b^2*f-24*a*b*c*d+3*b^3*d)/a^2/(16*a^2*c^2-8*a*b^2*c +b^4)*x^7+3*c^3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/8/a^2*c*(28*a^2*b*c*f+2 8*a^2*c^2*d+2*a*b^3*f-49*a*b^2*c*d+6*b^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5 +9/2*b*c^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+1/8*(36*a^3*c^2*f+5*a^2*b^2*c* f-4*a^2*b*c^2*d+a*b^4*f-20*a*b^3*c*d+3*b^5*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^ 4)*x^3+(5*a*c+b^2)*c*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/8*(16*a^2*b*c*f+44 *a^2*c^2*d-a*b^3*f-37*a*b^2*c*d+5*b^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/ 4*b*(10*a*c-b^2)*e/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/16*sum( (c*(20*a^2*c*f+a*b^2*f-24*a*b*c*d+3*b^3*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)* _R^2+48*c^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*_R-(16*a^2*b*c*f-84*a^2*c^2*d-a*b ^3*f+27*a*b^2*c*d-3*b^4*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b) *ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\text {Timed out} \] Input:
integrate((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/ (c*x^4+b*x^2+a)^4,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\text {Timed out} \] Input:
integrate((a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x **6)/(c*x**4+b*x**2+a)**4,x)
Output:
Timed out
\[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\int { \frac {c f x^{6} + c e x^{5} + b e x^{3} + {\left (c d + b f\right )} x^{4} + a e x + {\left (b d + a f\right )} x^{2} + a d}{{\left (c x^{4} + b x^{2} + a\right )}^{4}} \,d x } \] Input:
integrate((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/ (c*x^4+b*x^2+a)^4,x, algorithm="maxima")
Output:
1/8*(24*a^2*c^3*e*x^6 + 36*a^2*b*c^2*e*x^4 + (3*(b^3*c^2 - 8*a*b*c^3)*d + (a*b^2*c^2 + 20*a^2*c^3)*f)*x^7 + ((6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d + 2*(a*b^3*c + 14*a^2*b*c^2)*f)*x^5 + 8*(a^2*b^2*c + 5*a^3*c^2)*e*x^2 + ( (3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d + (a*b^4 + 5*a^2*b^2*c + 36*a^3*c^2)* f)*x^3 - 2*(a^2*b^3 - 10*a^3*b*c)*e + ((5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^ 2)*d - (a^2*b^3 - 16*a^3*b*c)*f)*x)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4 *c^4)*x^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3* c^2 + 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^ 3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) + 1/8*integrate((48*a^2*c^2*e*x + (3*(b^3*c - 8*a*b*c^2)*d + (a*b^2*c + 20*a^2*c^2)*f)*x^2 + 3*(b^4 - 9*a*b ^2*c + 28*a^2*c^2)*d + (a*b^3 - 16*a^2*b*c)*f)/(c*x^4 + b*x^2 + a), x)/(a^ 2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 5280 vs. \(2 (561) = 1122\).
Time = 1.04 (sec) , antiderivative size = 5280, normalized size of antiderivative = 8.50 \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/ (c*x^4+b*x^2+a)^4,x, algorithm="giac")
Output:
-3*(b^2*c^4 - 4*a*c^5 - 2*b*c^5 + c^6)*sqrt(b^2 - 4*a*c)*e*log(x^2 + 1/2*( a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + sqrt((a^2*b^5 - 8*a^3*b^3*c + 16*a^ 4*b*c^2)^2 - 4*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(a^2*b^4*c - 8*a^3*b^2 *c^2 + 16*a^4*c^3)))/(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3))/((b^8 - 16* a*b^6*c - 2*b^7*c + 96*a^2*b^4*c^2 + 24*a*b^5*c^2 + b^6*c^2 - 256*a^3*b^2* c^3 - 96*a^2*b^3*c^3 - 12*a*b^4*c^3 + 256*a^4*c^4 + 128*a^3*b*c^4 + 48*a^2 *b^2*c^4 - 64*a^3*c^5)*c^2) + 3*(b^2*c^4 - 4*a*c^5 - 2*b*c^5 + c^6)*sqrt(b ^2 - 4*a*c)*e*log(x^2 + 1/2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 - sqrt(( a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)^2 - 4*(a^3*b^4 - 8*a^4*b^2*c + 16*a^ 5*c^2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)))/(a^2*b^4*c - 8*a^3*b^2*c ^2 + 16*a^4*c^3))/((b^8 - 16*a*b^6*c - 2*b^7*c + 96*a^2*b^4*c^2 + 24*a*b^5 *c^2 + b^6*c^2 - 256*a^3*b^2*c^3 - 96*a^2*b^3*c^3 - 12*a*b^4*c^3 + 256*a^4 *c^4 + 128*a^3*b*c^4 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*c^2) + 1/32*(3*(sqrt(2 )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 17*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7*c - 2*b^8 *c + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 26*sqrt(2)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 34*a*b^6*c^2 + 2*b^7*c^2 - 368*sqrt(2)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 128*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a^2*b^3*c^3 - 13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 - 2...
Time = 20.45 (sec) , antiderivative size = 8689, normalized size of antiderivative = 13.99 \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx=\text {Too large to display} \] Input:
int((a*d + x^2*(b*d + a*f) + x^4*(c*d + b*f) + a*e*x + b*e*x^3 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4,x)
Output:
((x^2*(5*a*c^2*e + b^2*c*e))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) - (b^3*e - 10* a*b*c*e)/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^5*(28*a^2*c^3*d + 6*b^4*c *d + 2*a*b^3*c*f - 49*a*b^2*c^2*d + 28*a^2*b*c^2*f))/(8*a^2*(b^4 + 16*a^2* c^2 - 8*a*b^2*c)) + (x*(5*b^4*d + 44*a^2*c^2*d - a*b^3*f - 37*a*b^2*c*d + 16*a^2*b*c*f))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^3*e*x^6)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (x^3*(3*b^5*d + 36*a^3*c^2*f + a*b^4*f - 20*a*b ^3*c*d - 4*a^2*b*c^2*d + 5*a^2*b^2*c*f))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^ 2*c)) + (9*b*c^2*e*x^4)/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^7*(20*a^ 2*c^2*f + 3*b^3*c*d - 24*a*b*c^2*d + a*b^2*c*f))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) + symsum(log(root(56371445760*a^11*b^8*c^6*z^4 - 503316480*a^8*b^14*c^3*z ^4 + 47185920*a^7*b^16*c^2*z^4 - 171798691840*a^14*b^2*c^9*z^4 + 193273528 320*a^13*b^4*c^8*z^4 - 128849018880*a^12*b^6*c^7*z^4 - 16911433728*a^10*b^ 10*c^5*z^4 + 3523215360*a^9*b^12*c^4*z^4 - 2621440*a^6*b^18*c*z^4 + 687194 76736*a^15*c^10*z^4 + 65536*a^5*b^20*z^4 - 73728*a^2*b^16*c*d*f*z^2 - 1321 205760*a^9*b^2*c^8*d*f*z^2 + 732168192*a^7*b^6*c^6*d*f*z^2 - 366280704*a^6 *b^8*c^5*d*f*z^2 - 330301440*a^8*b^4*c^7*d*f*z^2 + 96583680*a^5*b^10*c^4*d *f*z^2 - 15175680*a^4*b^12*c^3*d*f*z^2 + 1428480*a^3*b^14*c^2*d*f*z^2 - 44 0401920*a^10*b*c^8*f^2*z^2 + 1761607680*a^10*c^9*d*f*z^2 - 14080*a^3*b^15* c*f^2*z^2 + 6936330240*a^8*b^3*c^8*d^2*z^2 + 2464874496*a^6*b^7*c^6*d^2...
Time = 9.53 (sec) , antiderivative size = 13638, normalized size of antiderivative = 21.96 \[ \int \frac {a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left (a+b x^2+c x^4\right )^4} \, dx =\text {Too large to display} \] Input:
int((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/(c*x^4 +b*x^2+a)^4,x)
Output:
( - 192*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt (2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**5*b *c**2*e - 384*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan ((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))* a**4*b**2*c**2*e*x**2 - 384*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqr t(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)* sqrt(a) + b))*a**4*b*c**3*e*x**4 - 192*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2* sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt (2*sqrt(c)*sqrt(a) + b))*a**3*b**3*c**2*e*x**4 - 384*sqrt(2*sqrt(c)*sqrt(a ) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*s qrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**3*e*x**6 - 192*sqrt(2* sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt (a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**4*e*x**8 - 80*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**5*b*c**2*f - 36*sqrt(a)*sqrt (2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s qrt(2*sqrt(c)*sqrt(a) + b))*a**4*b**3*c*f + 264*sqrt(a)*sqrt(2*sqrt(c)*sqr t(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)* sqrt(a) + b))*a**4*b**2*c**2*d - 160*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*a tan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) ...