Integrand size = 38, antiderivative size = 294 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {(b d-a g) x}{b^2}+\frac {(b e-a h) x^2}{2 b^2}+\frac {f x^3}{3 b}+\frac {g x^4}{4 b}+\frac {h x^5}{5 b}+\frac {\sqrt [3]{a} \left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 b^2} \]
(-a*g+b*d)*x/b^2+1/2*(-a*h+b*e)*x^2/b^2+1/3*f*x^3/b+1/4*g*x^4/b+1/5*h*x^5/ b-1/3*a^(1/3)*(b^(1/3)*(-a*g+b*d)-a^(1/3)*(-a*h+b*e))*ln(a^(1/3)+b^(1/3)*x )/b^(8/3)+1/6*a^(1/3)*(b^(1/3)*(-a*g+b*d)-a^(1/3)*(-a*h+b*e))*ln(a^(2/3)-a ^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(8/3)+1/3*(-a*f+b*c)*ln(b*x^3+a)/b^2+1/3*a ^(1/3)*(b^(4/3)*d+a^(1/3)*b*e-a*b^(1/3)*g-a^(4/3)*h)*arctan(1/3*(a^(1/3)-2 *b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(8/3)*3^(1/2)
Time = 0.19 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {60 b^{2/3} (b d-a g) x+30 b^{2/3} (b e-a h) x^2+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5-20 \sqrt {3} \sqrt [3]{a} \left (-b^{4/3} d-\sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+20 \sqrt [3]{a} \left (-b^{4/3} d+\sqrt [3]{a} b e+a \sqrt [3]{b} g-a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+10 \sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 b^{2/3} (b c-a f) \log \left (a+b x^3\right )}{60 b^{8/3}} \]
(60*b^(2/3)*(b*d - a*g)*x + 30*b^(2/3)*(b*e - a*h)*x^2 + 20*b^(5/3)*f*x^3 + 15*b^(5/3)*g*x^4 + 12*b^(5/3)*h*x^5 - 20*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3)) /Sqrt[3]] + 20*a^(1/3)*(-(b^(4/3)*d) + a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3) *h)*Log[a^(1/3) + b^(1/3)*x] + 10*a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^( 1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*b^ (2/3)*(b*c - a*f)*Log[a + b*x^3])/(60*b^(8/3))
Time = 1.04 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2375, 27, 2375, 27, 2375, 27, 2426, 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.
| \(\displaystyle \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int \frac {5 x^2 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{5 b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\int \frac {4 x^2 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{4 b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 x^2 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{3 b}+\frac {1}{3} b f x^3}{b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^2 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{b}+\frac {1}{3} b f x^3}{b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {\frac {\frac {\int \left (b (b d-a g)+b (b e-a h) x-\frac {-b^2 (b c-a f) x^2+a b (b e-a h) x+a b (b d-a g)}{b x^3+a}\right )dx}{b}+\frac {1}{3} b f x^3}{b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt [3]{a} \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3}}+\frac {1}{6} \sqrt [3]{a} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )+\frac {1}{3} b (b c-a f) \log \left (a+b x^3\right )-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )+b x (b d-a g)+\frac {1}{2} b x^2 (b e-a h)}{b}+\frac {1}{3} b f x^3}{b}+\frac {g x^4}{4}}{b}+\frac {h x^5}{5 b}\) |
(h*x^5)/(5*b) + ((g*x^4)/4 + ((b*f*x^3)/3 + (b*(b*d - a*g)*x + (b*(b*e - a *h)*x^2)/2 + (a^(1/3)*b^(1/3)*(b^(4/3)*d + a^(1/3)*b*e - a*b^(1/3)*g - a^( 4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/Sqrt[3] - (a^(1 /3)*b^(1/3)*(b^(1/3)*(b*d - a*g) - a^(1/3)*(b*e - a*h))*Log[a^(1/3) + b^(1 /3)*x])/3 + (a^(1/3)*b^(1/3)*(b^(1/3)*(b*d - a*g) - a^(1/3)*(b*e - a*h))*L og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6 + (b*(b*c - a*f)*Log[a + b*x^3])/3)/b)/b)/b
3.5.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(\frac {h \,x^{5}}{5 b}+\frac {g \,x^{4}}{4 b}+\frac {f \,x^{3}}{3 b}-\frac {a h \,x^{2}}{2 b^{2}}+\frac {e \,x^{2}}{2 b}-\frac {a g x}{b^{2}}+\frac {d x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (b \left (-a f +b c \right ) \textit {\_R}^{2}+a \left (a h -b e \right ) \textit {\_R} +a^{2} g -a b d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{3}}\) | \(123\) |
| default | \(-\frac {-\frac {1}{5} b h \,x^{5}-\frac {1}{4} b g \,x^{4}-\frac {1}{3} f \,x^{3} b +\frac {1}{2} a h \,x^{2}-\frac {1}{2} b e \,x^{2}+a g x -b d x}{b^{2}}+\frac {\left (a^{2} g -a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h -a e b \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a f b +b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{b^{2}}\) | \(285\) |
1/5*h*x^5/b+1/4*g*x^4/b+1/3*f*x^3/b-1/2/b^2*a*h*x^2+1/2*e*x^2/b-1/b^2*a*g* x+d*x/b+1/3/b^3*sum((b*(-a*f+b*c)*_R^2+a*(a*h-b*e)*_R+a^2*g-a*b*d)/_R^2*ln (x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 14746, normalized size of antiderivative = 50.16 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} + \frac {12 \, b h x^{5} + 15 \, b g x^{4} + 20 \, b f x^{3} + 30 \, {\left (b e - a h\right )} x^{2} + 60 \, {\left (b d - a g\right )} x}{60 \, b^{2}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/3*sqrt(3)*(a*b*e*(a/b)^(2/3) - a^2*h*(a/b)^(2/3) + a*b*d*(a/b)^(1/3) - a^2*g*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a* b^2) + 1/60*(12*b*h*x^5 + 15*b*g*x^4 + 20*b*f*x^3 + 30*(b*e - a*h)*x^2 + 6 0*(b*d - a*g)*x)/b^2 + 1/6*(2*b^2*c*(a/b)^(2/3) - 2*a*b*f*(a/b)^(2/3) - a* b*e*(a/b)^(1/3) + a^2*h*(a/b)^(1/3) + a*b*d - a^2*g)*log(x^2 - x*(a/b)^(1/ 3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1/3*(b^2*c*(a/b)^(2/3) - a*b*f*(a/b) ^(2/3) + a*b*e*(a/b)^(1/3) - a^2*h*(a/b)^(1/3) - a*b*d + a^2*g)*log(x + (a /b)^(1/3))/(b^3*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a b g - \left (-a b^{2}\right )^{\frac {2}{3}} b e + \left (-a b^{2}\right )^{\frac {2}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a b g + \left (-a b^{2}\right )^{\frac {2}{3}} b e - \left (-a b^{2}\right )^{\frac {2}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {12 \, b^{4} h x^{5} + 15 \, b^{4} g x^{4} + 20 \, b^{4} f x^{3} + 30 \, b^{4} e x^{2} - 30 \, a b^{3} h x^{2} + 60 \, b^{4} d x - 60 \, a b^{3} g x}{60 \, b^{5}} + \frac {{\left (a b^{10} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{9} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{10} d - a^{2} b^{9} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{11}} \]
1/3*(b*c - a*f)*log(abs(b*x^3 + a))/b^2 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2* d - (-a*b^2)^(1/3)*a*b*g - (-a*b^2)^(2/3)*b*e + (-a*b^2)^(2/3)*a*h)*arctan (1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/6*((-a*b^2)^(1/3)* b^2*d - (-a*b^2)^(1/3)*a*b*g + (-a*b^2)^(2/3)*b*e - (-a*b^2)^(2/3)*a*h)*lo g(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 1/60*(12*b^4*h*x^5 + 15*b^4*g *x^4 + 20*b^4*f*x^3 + 30*b^4*e*x^2 - 30*a*b^3*h*x^2 + 60*b^4*d*x - 60*a*b^ 3*g*x)/b^5 + 1/3*(a*b^10*e*(-a/b)^(1/3) - a^2*b^9*h*(-a/b)^(1/3) + a*b^10* d - a^2*b^9*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^11)
Time = 9.25 (sec) , antiderivative size = 1170, normalized size of antiderivative = 3.98 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=x^2\,\left (\frac {e}{2\,b}-\frac {a\,h}{2\,b^2}\right )+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\,\left (\frac {6\,a^2\,b^3\,f-6\,a\,b^4\,c}{b^3}+\frac {x\,\left (3\,a^2\,b^3\,g-3\,a\,b^4\,d\right )}{b^3}+\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\,a\,b^2\,9\right )+\frac {a\,b^3\,c^2+a^3\,b\,f^2+a^4\,g\,h-a^3\,b\,d\,h-a^3\,b\,e\,g-2\,a^2\,b^2\,c\,f+a^2\,b^2\,d\,e}{b^3}+\frac {x\,\left (a^4\,h^2+a^2\,b^2\,e^2+a\,b^3\,c\,d-2\,a^3\,b\,e\,h+a^3\,b\,f\,g-a^2\,b^2\,c\,g-a^2\,b^2\,d\,f\right )}{b^3}\right )\,\mathrm {root}\left (27\,b^8\,z^3+27\,a\,b^6\,f\,z^2-27\,b^7\,c\,z^2-18\,a\,b^5\,c\,f\,z+9\,a\,b^5\,d\,e\,z+9\,a^3\,b^3\,g\,h\,z-9\,a^2\,b^4\,e\,g\,z-9\,a^2\,b^4\,d\,h\,z+9\,a^2\,b^4\,f^2\,z+9\,b^6\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3,z,k\right )\right )+x\,\left (\frac {d}{b}-\frac {a\,g}{b^2}\right )+\frac {f\,x^3}{3\,b}+\frac {g\,x^4}{4\,b}+\frac {h\,x^5}{5\,b} \]
x^2*(e/(2*b) - (a*h)/(2*b^2)) + symsum(log(root(27*b^8*z^3 + 27*a*b^6*f*z^ 2 - 27*b^7*c*z^2 - 18*a*b^5*c*f*z + 9*a*b^5*d*e*z + 9*a^3*b^3*g*h*z - 9*a^ 2*b^4*e*g*z - 9*a^2*b^4*d*h*z + 9*a^2*b^4*f^2*z + 9*b^6*c^2*z + 3*a^4*b*f* g*h - 3*a*b^4*c*d*e - 3*a^3*b^2*e*f*g - 3*a^3*b^2*d*f*h - 3*a^3*b^2*c*g*h + 3*a^2*b^3*d*e*f + 3*a^2*b^3*c*e*g + 3*a^2*b^3*c*d*h - 3*a^4*b*e*h^2 + 3* a*b^4*c^2*f + 3*a^3*b^2*e^2*h + 3*a^3*b^2*d*g^2 - 3*a^2*b^3*d^2*g - 3*a^2* b^3*c*f^2 + a^3*b^2*f^3 + a*b^4*d^3 + a^5*h^3 - a^2*b^3*e^3 - a^4*b*g^3 - b^5*c^3, z, k)*((6*a^2*b^3*f - 6*a*b^4*c)/b^3 + (x*(3*a^2*b^3*g - 3*a*b^4* d))/b^3 + 9*root(27*b^8*z^3 + 27*a*b^6*f*z^2 - 27*b^7*c*z^2 - 18*a*b^5*c*f *z + 9*a*b^5*d*e*z + 9*a^3*b^3*g*h*z - 9*a^2*b^4*e*g*z - 9*a^2*b^4*d*h*z + 9*a^2*b^4*f^2*z + 9*b^6*c^2*z + 3*a^4*b*f*g*h - 3*a*b^4*c*d*e - 3*a^3*b^2 *e*f*g - 3*a^3*b^2*d*f*h - 3*a^3*b^2*c*g*h + 3*a^2*b^3*d*e*f + 3*a^2*b^3*c *e*g + 3*a^2*b^3*c*d*h - 3*a^4*b*e*h^2 + 3*a*b^4*c^2*f + 3*a^3*b^2*e^2*h + 3*a^3*b^2*d*g^2 - 3*a^2*b^3*d^2*g - 3*a^2*b^3*c*f^2 + a^3*b^2*f^3 + a*b^4 *d^3 + a^5*h^3 - a^2*b^3*e^3 - a^4*b*g^3 - b^5*c^3, z, k)*a*b^2) + (a*b^3* c^2 + a^3*b*f^2 + a^4*g*h - a^3*b*d*h - a^3*b*e*g - 2*a^2*b^2*c*f + a^2*b^ 2*d*e)/b^3 + (x*(a^4*h^2 + a^2*b^2*e^2 + a*b^3*c*d - 2*a^3*b*e*h + a^3*b*f *g - a^2*b^2*c*g - a^2*b^2*d*f))/b^3)*root(27*b^8*z^3 + 27*a*b^6*f*z^2 - 2 7*b^7*c*z^2 - 18*a*b^5*c*f*z + 9*a*b^5*d*e*z + 9*a^3*b^3*g*h*z - 9*a^2*b^4 *e*g*z - 9*a^2*b^4*d*h*z + 9*a^2*b^4*f^2*z + 9*b^6*c^2*z + 3*a^4*b*f*g*...