Integrand size = 40, antiderivative size = 384 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}+\frac {(b d-a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {f \log \left (a+b x^4\right )}{4 b} \]
g*x/b+1/2*h*x^2/b+1/3*i*x^3/b+1/4*f*ln(b*x^4+a)/b+1/2*(-a*h+b*d)*arctan(x^ 2*b^(1/2)/a^(1/2))/b^(3/2)/a^(1/2)-1/8*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/ 2)+x^2*b^(1/2))*(-(-a*i+b*e)*a^(1/2)+(-a*g+b*c)*b^(1/2))/a^(3/4)/b^(7/4)*2 ^(1/2)+1/8*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-(-a*i+b*e)* a^(1/2)+(-a*g+b*c)*b^(1/2))/a^(3/4)/b^(7/4)*2^(1/2)+1/4*arctan(-1+b^(1/4)* x*2^(1/2)/a^(1/4))*((-a*i+b*e)*a^(1/2)+(-a*g+b*c)*b^(1/2))/a^(3/4)/b^(7/4) *2^(1/2)+1/4*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*((-a*i+b*e)*a^(1/2)+(-a*g +b*c)*b^(1/2))/a^(3/4)/b^(7/4)*2^(1/2)
Time = 0.26 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+\frac {6 \left (-\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d-\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h+\sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {6 \left (\sqrt {2} b^{3/2} c-2 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e-\sqrt {2} a \sqrt {b} g+2 a^{5/4} \sqrt [4]{b} h-\sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+\frac {3 \sqrt {2} \left (b^{3/2} c-\sqrt {a} b e-a \sqrt {b} g+a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{3/4}}+6 b^{3/4} f \log \left (a+b x^4\right )}{24 b^{7/4}} \]
(24*b^(3/4)*g*x + 12*b^(3/4)*h*x^2 + 8*b^(3/4)*i*x^3 + (6*(-(Sqrt[2]*b^(3/ 2)*c) - 2*a^(1/4)*b^(5/4)*d - Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^ (1/4)])/a^(3/4) + (6*(Sqrt[2]*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sq rt[a]*b*e - Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - Sqrt[2]*a^(3/2)*i) *ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^( 1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*S qrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^ 2])/a^(3/4) + 6*b^(3/4)*f*Log[a + b*x^4])/(24*b^(7/4))
Time = 0.69 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2424, 2009}
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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (\frac {c+e x^2+g x^4+i x^6}{a+b x^4}+\frac {x \left (d+f x^2+h x^4\right )}{a+b x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (b c-a g)+\sqrt {a} (b e-a i)\right )}{2 \sqrt {2} a^{3/4} b^{7/4}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (b c-a g)-\sqrt {a} (b e-a i)\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (b d-a h)}{2 \sqrt {a} b^{3/2}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {g x}{b}+\frac {h x^2}{2 b}+\frac {i x^3}{3 b}\) |
(g*x)/b + (h*x^2)/(2*b) + (i*x^3)/(3*b) + ((b*d - a*h)*ArcTan[(Sqrt[b]*x^2 )/Sqrt[a]])/(2*Sqrt[a]*b^(3/2)) - ((Sqrt[b]*(b*c - a*g) + Sqrt[a]*(b*e - a *i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*(b*c - a*g) + Sqrt[a]*(b*e - a*i))*ArcTan[1 + (Sqrt[2]*b^(1/4) *x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(7/4)) - ((Sqrt[b]*(b*c - a*g) - Sqrt[a ]*(b*e - a*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4* Sqrt[2]*a^(3/4)*b^(7/4)) + ((Sqrt[b]*(b*c - a*g) - Sqrt[a]*(b*e - a*i))*Lo g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b ^(7/4)) + (f*Log[a + b*x^4])/(4*b)
3.2.90.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 *((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.23
| method | result | size |
| risch | \(\frac {i \,x^{3}}{3 b}+\frac {h \,x^{2}}{2 b}+\frac {g x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c -a g +\left (-a h +b d \right ) \textit {\_R} +\left (-a i +b e \right ) \textit {\_R}^{2}+\textit {\_R}^{3} b f \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(88\) |
| default | \(\frac {\frac {1}{3} i \,x^{3}+\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (-a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a h +b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (-a i +b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {f \ln \left (b \,x^{4}+a \right )}{4}}{b}\) | \(283\) |
1/3*i*x^3/b+1/2*h*x^2/b+g*x/b+1/4/b^2*sum((b*c-a*g+(-a*h+b*d)*_R+(-a*i+b*e )*_R^2+_R^3*b*f)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {2 \, i x^{3} + 3 \, h x^{2} + 6 \, g x}{6 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f + b^{2} c - \sqrt {a} b^{\frac {3}{2}} e - a b g + a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - b^{2} c + \sqrt {a} b^{\frac {3}{2}} e + a b g - a^{\frac {3}{2}} \sqrt {b} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i - 2 \, \sqrt {a} b^{2} d + 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} i + 2 \, \sqrt {a} b^{2} d - 2 \, a^{\frac {3}{2}} b h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]
1/6*(2*i*x^3 + 3*h*x^2 + 6*g*x)/b + 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)* f + b^2*c - sqrt(a)*b^(3/2)*e - a*b*g + a^(3/2)*sqrt(b)*i)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4)) + sqrt(2)*(sqrt( 2)*a^(3/4)*b^(5/4)*f - b^2*c + sqrt(a)*b^(3/2)*e + a*b*g - a^(3/2)*sqrt(b) *i)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(5/4 )) + 2*(sqrt(2)*a^(1/4)*b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*e - sqrt(2)*a^ (5/4)*b^(5/4)*g - sqrt(2)*a^(7/4)*b^(3/4)*i - 2*sqrt(a)*b^2*d + 2*a^(3/2)* b*h)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt( a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(5/4)) + 2*(sqrt(2)*a^(1/4)* b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*e - sqrt(2)*a^(5/4)*b^(5/4)*g - sqrt(2 )*a^(7/4)*b^(3/4)*i + 2*sqrt(a)*b^2*d - 2*a^(3/2)*b*h)*arctan(1/2*sqrt(2)* (2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sq rt(sqrt(a)*sqrt(b))*b^(5/4)))/b
Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\frac {f \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {2 \, b^{2} i x^{3} + 3 \, b^{2} h x^{2} + 6 \, b^{2} g x}{6 \, b^{3}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{3} d + \sqrt {2} \sqrt {a b} a b^{2} h - \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e + \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{4}} \]
1/4*f*log(abs(b*x^4 + a))/b + 1/6*(2*b^2*i*x^3 + 3*b^2*h*x^2 + 6*b^2*g*x)/ b^3 - 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^3*d + sqrt(2)*sqrt(a*b)*a*b^2*h - ( a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e + (a*b^3)^( 3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b ^4) - 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^3*d + sqrt(2)*sqrt(a*b)*a*b^2*h - ( a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e + (a*b^3)^( 3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b ^4) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c - (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^( 3/4)*b*e + (a*b^3)^(3/4)*a*i)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b)) /(a*b^4) - 1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c - (a*b^3)^(1/4)*a*b^2*g - (a*b ^3)^(3/4)*b*e + (a*b^3)^(3/4)*a*i)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt( a/b))/(a*b^4)
Time = 9.42 (sec) , antiderivative size = 3798, normalized size of antiderivative = 9.89 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx=\text {Too large to display} \]
symsum(log((a^4*i^3 - a*b^3*e^3 + b^4*c*d^2 - b^4*c^2*e + a^2*b^2*c*h^2 - a^2*b^2*e*g^2 + a^2*b^2*f^2*g + 3*a^2*b^2*e^2*i - a*b^3*c*f^2 - a*b^3*d^2* g + a*b^3*c^2*i - 3*a^3*b*e*i^2 - a^3*b*g*h^2 + a^3*b*g^2*i - 2*a^2*b^2*c* g*i - 2*a^2*b^2*d*f*i + 2*a^2*b^2*d*g*h - 2*a^2*b^2*e*f*h - 2*a*b^3*c*d*h + 2*a*b^3*c*e*g + 2*a*b^3*d*e*f + 2*a^3*b*f*h*i)/b^2 + root(256*a^3*b^7*z^ 4 - 256*a^3*b^6*f*z^3 + 64*a^4*b^4*g*i*z^2 - 64*a^3*b^5*e*g*z^2 - 64*a^3*b ^5*d*h*z^2 - 64*a^3*b^5*c*i*z^2 + 64*a^2*b^6*c*e*z^2 + 32*a^4*b^4*h^2*z^2 + 96*a^3*b^5*f^2*z^2 + 32*a^2*b^6*d^2*z^2 - 32*a^4*b^3*f*g*i*z + 32*a^4*b^ 3*e*h*i*z + 32*a^3*b^4*e*f*g*z + 32*a^3*b^4*d*f*h*z - 32*a^3*b^4*d*e*i*z - 32*a^3*b^4*c*g*h*z + 32*a^3*b^4*c*f*i*z - 32*a^2*b^5*c*e*f*z + 32*a^2*b^5 *c*d*g*z - 16*a^5*b^2*h*i^2*z + 16*a^4*b^3*g^2*h*z - 16*a^4*b^3*f*h^2*z + 16*a^4*b^3*d*i^2*z - 16*a^3*b^4*e^2*h*z - 16*a^3*b^4*d*g^2*z + 16*a^2*b^5* c^2*h*z - 16*a^2*b^5*d^2*f*z + 16*a^2*b^5*d*e^2*z - 16*a*b^6*c^2*d*z - 16* a^3*b^4*f^3*z - 8*a^4*b^2*e*f*h*i + 8*a^4*b^2*d*g*h*i - 8*a^3*b^3*d*e*g*h + 8*a^3*b^3*d*e*f*i + 8*a^3*b^3*c*f*g*h + 8*a^3*b^3*c*e*g*i - 8*a^3*b^3*c* d*h*i - 8*a^2*b^4*c*d*f*g + 8*a^2*b^4*c*d*e*h + 4*a^4*b^2*f^2*g*i - 4*a^4* b^2*f*g^2*h - 4*a^4*b^2*e*g^2*i + 4*a^4*b^2*e*g*h^2 + 4*a^4*b^2*c*h^2*i - 4*a^3*b^3*d^2*g*i - 4*a^4*b^2*d*f*i^2 - 4*a^4*b^2*c*g*i^2 + 4*a^3*b^3*e^2* f*h - 4*a^3*b^3*e*f^2*g - 4*a^3*b^3*d*f^2*h - 4*a^3*b^3*c*f^2*i + 4*a^3*b^ 3*d*f*g^2 - 4*a^2*b^4*c^2*f*h - 4*a^2*b^4*c^2*e*i - 4*a^3*b^3*c*e*h^2 +...