Integrand size = 30, antiderivative size = 646 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {b e-2 a g+(2 c e-b g) x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 (2 c e-b g) \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d+a f\right )-4 a b c \left (6 \sqrt {b^2-4 a c} d+13 a f\right )-a b^2 \left (30 c d-\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac {3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {3 c (2 c e-b g) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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+1/4*(-b*e+2*a*g-(-b*g+2*c*e)*x^2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/4 *(-b*g+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)-3*c*(-b*g+2*c*e)*arcta nh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)+1/16*arctan(x*2^(1/2 )*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^4*d+b^3*(a*f+3*d*(-4* a*c+b^2)^(1/2))-4*a*b*c*(13*a*f+6*d*(-4*a*c+b^2)^(1/2))-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)))/a^2/(-4*a*c+b^2 )^(5/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/16*arctan(x*2^(1/2)*c^(1/2) /(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3*d-24*a*b*c*d+a*b^2*f+20*a^2* c*f+(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)/(-4*a*c+b^2) ^(1/2))/a^2/(-4*a*c+b^2)^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 2.36 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.02 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{16} \left (\frac {-8 a^2 g-4 b d x \left (b+c x^2\right )+8 a c x (d+x (e+f x))+4 a b (e+x (f-g x))}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 \left (3 b^3 d x \left (b+c x^2\right )+a b x \left (-25 b c d+b^2 f-24 c^2 d x^2+b c f x^2\right )+a^2 \left (-6 b^2 g+4 c^2 x \left (7 d+6 e x+5 f x^2\right )+4 b c \left (3 e+2 f x-3 g x^2\right )\right )\right )}{a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d+a f\right )-4 a b c \left (6 \sqrt {b^2-4 a c} d+13 a f\right )+a b^2 \left (-30 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^4 d+b^3 \left (3 \sqrt {b^2-4 a c} d-a f\right )+4 a b c \left (-6 \sqrt {b^2-4 a c} d+13 a f\right )+a b^2 \left (30 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (-42 c d+5 \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {24 c (-2 c e+b g) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {24 c (-2 c e+b g) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}\right ) \]
((-8*a^2*g - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)) + 4*a*b*(e + x*(f - g*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*(3*b^3*d*x*(b + c*x^2) + a*b*x*(-25*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + a^2*(-6* b^2*g + 4*c^2*x*(7*d + 6*e*x + 5*f*x^2) + 4*b*c*(3*e + 2*f*x - 3*g*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*Sq rt[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]]) - (24*c*(-2*c*e + b*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + (24*c*( -2*c*e + b*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/1 6
Time = 1.02 (sec) , antiderivative size = 627, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2202, 1492, 25, 1492, 25, 1480, 218, 1576, 1159, 1086, 1083, 219}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^3}dx+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\int -\frac {3 d b^2+a f b+5 c (b d-2 a f) x^2-14 a c d}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 d b^2+a f b+5 c (b d-2 a f) x^2-14 a c d}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 d b^4+a f b^3-27 a c d b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-24 a c d b+20 a^2 c f\right ) x^2+84 a^2 c^2 d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {3 d b^4+a f b^3-27 a c d b^2-16 a^2 c f b+c \left (3 d b^3+a f b^2-24 a c d b+20 a^2 c f\right ) x^2+84 a^2 c^2 d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {1}{2} c \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^3}dx+\frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int \frac {g x^2+e}{\left (c x^4+b x^2+a\right )^3}dx^2+\frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3 (2 c e-b g) \int \frac {1}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 \left (b^2-4 a c\right )}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3 (2 c e-b g) \left (-\frac {2 c \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3 (2 c e-b g) \left (\frac {4 c \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt {b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {3 (2 c e-b g) \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
(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)) + (-1/2*(b*e - 2*a*g + (2*c*e - b*g) *x^2)/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*(2*c*e - b*g)*(-((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)/Sq rt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c)))/2
3.1.54.3.1 Defintions of rubi rules used
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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
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[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
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 = 0.58 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(\frac {\frac {c^{2} \left (20 a^{2} c f +a \,b^{2} f -24 a b c d +3 b^{3} d \right ) x^{7}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (b g -2 e c \right ) c^{2} x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (28 a^{2} b c f +28 a^{2} c^{2} d +2 a \,b^{3} f -49 a \,b^{2} c d +6 d \,b^{4}\right ) x^{5}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 b \left (b g -2 e c \right ) c \,x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 a^{3} c^{2} f +5 a^{2} b^{2} c f -4 a^{2} b \,c^{2} d +a \,b^{4} f -20 a \,b^{3} c d +3 b^{5} d \right ) x^{3}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) \left (b g -2 e c \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (16 a^{2} b c f +44 a^{2} c^{2} d -a \,b^{3} f -37 a \,b^{2} c d +5 d \,b^{4}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {8 a^{2} c g +a \,b^{2} g -10 a b c e +b^{3} e}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (20 a^{2} c f +a \,b^{2} f -24 a b c d +3 b^{3} d \right ) \textit {\_R}^{2}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {24 \left (b g -2 e c \right ) c \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {16 a^{2} b c f -84 a^{2} c^{2} d -a \,b^{3} f +27 a \,b^{2} c d -3 d \,b^{4}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}\right )}{16}\) | \(644\) |
| default | \(\text {Expression too large to display}\) | \(1638\) |
(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^7-3/2*(b*g-2*c*e)*c^2/(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+28*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^5-9/4*b*(b*g-2*c*e)*c/(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^3-1/2*(5*a*c+b^2)*(b*g-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*(8*a^2*c*g+a*b^2*g-10*a*b*c*e+b^3*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^2-24*(b*g-2*c*e)*c/ (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=R ootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/8*((3*(b^3*c^2 - 8*a*b*c^3)*d + (a*b^2*c^2 + 20*a^2*c^3)*f)*x^7 + 12*(2* a^2*c^3*e - a^2*b*c^2*g)*x^6 + ((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 + 18*(2*a^2*b*c^2*e - a^2*b^2*c*g)*x^4 + ((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 + 4*(2*(a^2*b^2*c + 5*a^3*c^2)*e - (a^2*b^3 + 5*a^3*b*c)*g)*x^2 - 2*(a^2*b^3 - 10*a^3*b*c)*e - 2*(a^3*b^2 + 8*a^4*c)*g + ((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 (((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 + 24*(2*a^2*c^2*e - a^2*b*c *g)*x)/(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. 5619 vs. \(2 (586) = 1172\).
Time = 2.55 (sec) , antiderivative size = 5619, normalized size of antiderivative = 8.70 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
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 - 232*a^2*b^4*c^3 + 30*a*b^5*c^3 + 448*sqrt(2)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a^4*c^4 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b* c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 736*a^3*b^2 *c^4 - 176*a^2*b^3*c^4 - 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c ^5 - 896*a^4*c^5 + 352*a^3*b*c^5 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*b^7 - 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* a*c)*c)*b^6*c + 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*a^2*b^3*c^2 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*a*b^4*c^2 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* b^5*c^2 - 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^ 3*b*c^3 - 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2 *b^2*c^3 - 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)...
Time = 10.42 (sec) , antiderivative size = 13431, normalized size of antiderivative = 20.79 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
symsum(log((x*(13824*a^4*c^8*e^3 - 54*b^7*c^5*d^2*e + 27*b^8*c^4*d^2*g - 1 728*a^4*b^3*c^5*g^3 - 20160*a^4*c^8*d*e*f + 972*a*b^5*c^6*d^2*e + 24192*a^ 3*b*c^8*d^2*e - 486*a*b^6*c^5*d^2*g + 6240*a^4*b*c^7*e*f^2 - 20736*a^4*b*c ^7*e^2*g - 7344*a^2*b^3*c^7*d^2*e + 3672*a^2*b^4*c^6*d^2*g - 6*a^2*b^5*c^5 *e*f^2 - 12096*a^3*b^2*c^7*d^2*g + 192*a^3*b^3*c^6*e*f^2 + 10368*a^4*b^2*c ^6*e*g^2 + 3*a^2*b^6*c^4*f^2*g - 96*a^3*b^4*c^5*f^2*g - 3120*a^4*b^2*c^6*f ^2*g - 36*a*b^6*c^5*d*e*f + 18*a*b^7*c^4*d*f*g + 10080*a^4*b*c^7*d*f*g + 9 00*a^2*b^4*c^6*d*e*f - 4896*a^3*b^2*c^7*d*e*f - 450*a^2*b^5*c^5*d*f*g + 24 48*a^3*b^3*c^6*d*f*g))/(64*(a^4*b^12 + 4096*a^10*c^6 - 24*a^5*b^10*c + 240 *a^6*b^8*c^2 - 1280*a^7*b^6*c^3 + 3840*a^8*b^4*c^4 - 6144*a^9*b^2*c^5)) - 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 + 193273528320*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 + 3 523215360*a^9*b^12*c^4*z^4 - 2621440*a^6*b^18*c*z^4 + 68719476736*a^15*c^1 0*z^4 + 65536*a^5*b^20*z^4 - 73728*a^2*b^16*c*d*f*z^2 + 1509949440*a^9*b^3 *c^7*e*g*z^2 - 1321205760*a^9*b^2*c^8*d*f*z^2 - 754974720*a^8*b^5*c^6*e*g* z^2 + 732168192*a^7*b^6*c^6*d*f*z^2 - 366280704*a^6*b^8*c^5*d*f*z^2 - 3303 01440*a^8*b^4*c^7*d*f*z^2 + 188743680*a^7*b^7*c^5*e*g*z^2 + 96583680*a^5*b ^10*c^4*d*f*z^2 - 23592960*a^6*b^9*c^4*e*g*z^2 + 1179648*a^5*b^11*c^3*e*g* z^2 - 15175680*a^4*b^12*c^3*d*f*z^2 + 1428480*a^3*b^14*c^2*d*f*z^2 - 12...