Integrand size = 38, antiderivative size = 313 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {(b c-a f) x}{b^2}+\frac {(b d-a g) x^2}{2 b^2}+\frac {(b e-a h) x^3}{3 b^2}+\frac {f x^4}{4 b}+\frac {g x^5}{5 b}+\frac {h x^6}{6 b}+\frac {\sqrt [3]{a} \left (b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f-a^{4/3} g\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}-\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {\sqrt [3]{a} \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}-\frac {a (b e-a h) \log \left (a+b x^3\right )}{3 b^3} \]
(-a*f+b*c)*x/b^2+1/2*(-a*g+b*d)*x^2/b^2+1/3*(-a*h+b*e)*x^3/b^2+1/4*f*x^4/b +1/5*g*x^5/b+1/6*h*x^6/b-1/3*a^(1/3)*(b^(1/3)*(-a*f+b*c)-a^(1/3)*(-a*g+b*d ))*ln(a^(1/3)+b^(1/3)*x)/b^(8/3)+1/6*a^(1/3)*(b^(1/3)*(-a*f+b*c)-a^(1/3)*( -a*g+b*d))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(8/3)-1/3*a*(-a*h+b *e)*ln(b*x^3+a)/b^3+1/3*a^(1/3)*(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^(4/3) *g)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(8/3)*3^(1/2)
Time = 0.22 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {60 b (b c-a f) x+30 b (b d-a g) x^2+20 b (b e-a h) x^3+15 b^2 f x^4+12 b^2 g x^5+10 b^2 h x^6-20 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (-b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-20 \sqrt [3]{a} \sqrt [3]{b} \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+10 \sqrt [3]{a} \sqrt [3]{b} \left (b^{4/3} c-\sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 a (-b e+a h) \log \left (a+b x^3\right )}{60 b^3} \]
(60*b*(b*c - a*f)*x + 30*b*(b*d - a*g)*x^2 + 20*b*(b*e - a*h)*x^3 + 15*b^2 *f*x^4 + 12*b^2*g*x^5 + 10*b^2*h*x^6 - 20*Sqrt[3]*a^(1/3)*b^(1/3)*(-(b^(4/ 3)*c) - a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a ^(1/3))/Sqrt[3]] - 20*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3) *f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + 10*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*a*(-(b*e) + a*h)*Log[a + b*x^3])/(60*b^3)
Time = 1.05 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\int \frac {6 x^3 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{6 b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 \left (b g x^4+b f x^3+(b e-a h) x^2+b d x+b c\right )}{b x^3+a}dx}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\int \frac {5 x^3 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{5 b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^3 \left (b^2 f x^3+b (b e-a h) x^2+b (b d-a g) x+b^2 c\right )}{b x^3+a}dx}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 x^3 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{4 b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^3 \left ((b e-a h) x^2 b^2+(b c-a f) b^2+(b d-a g) x b^2\right )}{b x^3+a}dx}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {\frac {\frac {\int \left (b (b e-a h) x^2+b (b d-a g) x+b (b c-a f)-\frac {a b (b e-a h) x^2+a b (b d-a g) x+a b (b c-a f)}{b x^3+a}\right )dx}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt [3]{a} \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt {3}}+\frac {1}{6} \sqrt [3]{a} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )+b x (b c-a f)+\frac {1}{2} b x^2 (b d-a g)+\frac {1}{3} b x^3 (b e-a h)-\frac {1}{3} a (b e-a h) \log \left (a+b x^3\right )}{b}+\frac {1}{4} b f x^4}{b}+\frac {g x^5}{5}}{b}+\frac {h x^6}{6 b}\) |
(h*x^6)/(6*b) + ((g*x^5)/5 + ((b*f*x^4)/4 + (b*(b*c - a*f)*x + (b*(b*d - a *g)*x^2)/2 + (b*(b*e - a*h)*x^3)/3 + (a^(1/3)*b^(1/3)*(b^(4/3)*c + a^(1/3) *b*d - a*b^(1/3)*f - a^(4/3)*g)*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*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(1/3) + b^(1/3)*x])/3 + (a^(1/3)*b^(1/3)*(b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/6 - (a*(b*e - a*h)*Log[a + b*x^3])/3)/b)/b)/b
3.5.4.3.1 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 = 138, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {h \,x^{6}}{6 b}+\frac {g \,x^{5}}{5 b}+\frac {f \,x^{4}}{4 b}-\frac {a h \,x^{3}}{3 b^{2}}+\frac {e \,x^{3}}{3 b}-\frac {a g \,x^{2}}{2 b^{2}}+\frac {d \,x^{2}}{2 b}-\frac {a f x}{b^{2}}+\frac {c x}{b}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\left (a h -b e \right ) \textit {\_R}^{2}+\left (a g -b d \right ) \textit {\_R} +a f -b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{3}}\) | \(138\) |
| default | \(-\frac {-\frac {1}{6} b h \,x^{6}-\frac {1}{5} b g \,x^{5}-\frac {1}{4} b f \,x^{4}+\frac {1}{3} a h \,x^{3}-\frac {1}{3} b e \,x^{3}+\frac {1}{2} a g \,x^{2}-\frac {1}{2} b d \,x^{2}+a f x -b c x}{b^{2}}+\frac {\left (\left (a f -b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a g -b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a h -b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b}\right ) a}{b^{2}}\) | \(291\) |
1/6*h*x^6/b+1/5*g*x^5/b+1/4*f*x^4/b-1/3/b^2*a*h*x^3+1/3*e*x^3/b-1/2/b^2*a* g*x^2+1/2*d*x^2/b-1/b^2*a*f*x+c*x/b+1/3/b^3*a*sum(((a*h-b*e)*_R^2+(a*g-b*d )*_R+a*f-b*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 15451, normalized size of antiderivative = 49.36 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=\frac {10 \, b h x^{6} + 12 \, b g x^{5} + 15 \, b f x^{4} + 20 \, {\left (b e - a h\right )} x^{3} + 30 \, {\left (b d - a g\right )} x^{2} + 60 \, {\left (b c - a f\right )} x}{60 \, b^{2}} - \frac {\sqrt {3} {\left (a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3}} - \frac {{\left (2 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b c + a^{2} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b c - a^{2} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/60*(10*b*h*x^6 + 12*b*g*x^5 + 15*b*f*x^4 + 20*(b*e - a*h)*x^3 + 30*(b*d - a*g)*x^2 + 60*(b*c - a*f)*x)/b^2 - 1/3*sqrt(3)*(a*b^2*d*(a/b)^(2/3) - a^ 2*b*g*(a/b)^(2/3) + a*b^2*c*(a/b)^(1/3) - a^2*b*f*(a/b)^(1/3))*arctan(1/3* sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3) - 1/6*(2*a*b*e*(a/b)^(2/3 ) - 2*a^2*h*(a/b)^(2/3) + a*b*d*(a/b)^(1/3) - a^2*g*(a/b)^(1/3) - a*b*c + a^2*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) - 1/3*(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) + a*b*c - a^2*f)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.12 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx=-\frac {{\left (a b e - a^{2} h\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b f - \left (-a b^{2}\right )^{\frac {2}{3}} b d + \left (-a b^{2}\right )^{\frac {2}{3}} a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b f + \left (-a b^{2}\right )^{\frac {2}{3}} b d - \left (-a b^{2}\right )^{\frac {2}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {10 \, b^{5} h x^{6} + 12 \, b^{5} g x^{5} + 15 \, b^{5} f x^{4} + 20 \, b^{5} e x^{3} - 20 \, a b^{4} h x^{3} + 30 \, b^{5} d x^{2} - 30 \, a b^{4} g x^{2} + 60 \, b^{5} c x - 60 \, a b^{4} f x}{60 \, b^{6}} + \frac {{\left (a b^{12} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{11} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{12} c - a^{2} b^{11} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{13}} \]
-1/3*(a*b*e - a^2*h)*log(abs(b*x^3 + a))/b^3 - 1/3*sqrt(3)*((-a*b^2)^(1/3) *b^2*c - (-a*b^2)^(1/3)*a*b*f - (-a*b^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*a rctan(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*c - (-a*b^2)^(1/3)*a*b*f + (-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a* g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 1/60*(10*b^5*h*x^6 + 12* b^5*g*x^5 + 15*b^5*f*x^4 + 20*b^5*e*x^3 - 20*a*b^4*h*x^3 + 30*b^5*d*x^2 - 30*a*b^4*g*x^2 + 60*b^5*c*x - 60*a*b^4*f*x)/b^6 + 1/3*(a*b^12*d*(-a/b)^(1/ 3) - a^2*b^11*g*(-a/b)^(1/3) + a*b^12*c - a^2*b^11*f)*(-a/b)^(1/3)*log(abs (x - (-a/b)^(1/3)))/(a*b^13)
Time = 0.15 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.95 \[ \int \frac {x^3 \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 {d}{2\,b}-\frac {a\,g}{2\,b^2}\right )+x^3\,\left (\frac {e}{3\,b}-\frac {a\,h}{3\,b^2}\right )+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^9\,z^3+27\,a\,b^7\,e\,z^2-27\,a^2\,b^6\,h\,z^2+9\,a\,b^6\,c\,d\,z-18\,a^3\,b^4\,e\,h\,z+9\,a^3\,b^4\,f\,g\,z-9\,a^2\,b^5\,d\,f\,z-9\,a^2\,b^5\,c\,g\,z+9\,a^4\,b^3\,h^2\,z+9\,a^2\,b^5\,e^2\,z-3\,a^5\,b\,f\,g\,h+3\,a^4\,b^2\,e\,f\,g+3\,a^4\,b^2\,d\,f\,h+3\,a^4\,b^2\,c\,g\,h-3\,a^3\,b^3\,d\,e\,f-3\,a^3\,b^3\,c\,e\,g-3\,a^3\,b^3\,c\,d\,h+3\,a^2\,b^4\,c\,d\,e+3\,a^5\,b\,e\,h^2-3\,a^4\,b^2\,e^2\,h-3\,a^4\,b^2\,d\,g^2+3\,a^3\,b^3\,d^2\,g+3\,a^3\,b^3\,c\,f^2-3\,a^2\,b^4\,c^2\,f+a^3\,b^3\,e^3+a^5\,b\,g^3+a\,b^5\,c^3-a^4\,b^2\,f^3-a^2\,b^4\,d^3-a^6\,h^3,z,k\right )\,\left (\frac {6\,a^2\,b^4\,e-6\,a^3\,b^3\,h}{b^4}+\frac {x\,\left (3\,a^2\,b^3\,f-3\,a\,b^4\,c\right )}{b^3}+\mathrm {root}\left (27\,b^9\,z^3+27\,a\,b^7\,e\,z^2-27\,a^2\,b^6\,h\,z^2+9\,a\,b^6\,c\,d\,z-18\,a^3\,b^4\,e\,h\,z+9\,a^3\,b^4\,f\,g\,z-9\,a^2\,b^5\,d\,f\,z-9\,a^2\,b^5\,c\,g\,z+9\,a^4\,b^3\,h^2\,z+9\,a^2\,b^5\,e^2\,z-3\,a^5\,b\,f\,g\,h+3\,a^4\,b^2\,e\,f\,g+3\,a^4\,b^2\,d\,f\,h+3\,a^4\,b^2\,c\,g\,h-3\,a^3\,b^3\,d\,e\,f-3\,a^3\,b^3\,c\,e\,g-3\,a^3\,b^3\,c\,d\,h+3\,a^2\,b^4\,c\,d\,e+3\,a^5\,b\,e\,h^2-3\,a^4\,b^2\,e^2\,h-3\,a^4\,b^2\,d\,g^2+3\,a^3\,b^3\,d^2\,g+3\,a^3\,b^3\,c\,f^2-3\,a^2\,b^4\,c^2\,f+a^3\,b^3\,e^3+a^5\,b\,g^3+a\,b^5\,c^3-a^4\,b^2\,f^3-a^2\,b^4\,d^3-a^6\,h^3,z,k\right )\,a\,b^2\,9\right )+\frac {a^5\,h^2+a^3\,b^2\,e^2-2\,a^4\,b\,e\,h+a^4\,b\,f\,g+a^2\,b^3\,c\,d-a^3\,b^2\,c\,g-a^3\,b^2\,d\,f}{b^4}+\frac {x\,\left (a^4\,g^2+a^2\,b^2\,d^2-a^4\,f\,h+a^3\,b\,c\,h-2\,a^3\,b\,d\,g+a^3\,b\,e\,f-a^2\,b^2\,c\,e\right )}{b^3}\right )\,\mathrm {root}\left (27\,b^9\,z^3+27\,a\,b^7\,e\,z^2-27\,a^2\,b^6\,h\,z^2+9\,a\,b^6\,c\,d\,z-18\,a^3\,b^4\,e\,h\,z+9\,a^3\,b^4\,f\,g\,z-9\,a^2\,b^5\,d\,f\,z-9\,a^2\,b^5\,c\,g\,z+9\,a^4\,b^3\,h^2\,z+9\,a^2\,b^5\,e^2\,z-3\,a^5\,b\,f\,g\,h+3\,a^4\,b^2\,e\,f\,g+3\,a^4\,b^2\,d\,f\,h+3\,a^4\,b^2\,c\,g\,h-3\,a^3\,b^3\,d\,e\,f-3\,a^3\,b^3\,c\,e\,g-3\,a^3\,b^3\,c\,d\,h+3\,a^2\,b^4\,c\,d\,e+3\,a^5\,b\,e\,h^2-3\,a^4\,b^2\,e^2\,h-3\,a^4\,b^2\,d\,g^2+3\,a^3\,b^3\,d^2\,g+3\,a^3\,b^3\,c\,f^2-3\,a^2\,b^4\,c^2\,f+a^3\,b^3\,e^3+a^5\,b\,g^3+a\,b^5\,c^3-a^4\,b^2\,f^3-a^2\,b^4\,d^3-a^6\,h^3,z,k\right )\right )+x\,\left (\frac {c}{b}-\frac {a\,f}{b^2}\right )+\frac {f\,x^4}{4\,b}+\frac {g\,x^5}{5\,b}+\frac {h\,x^6}{6\,b} \]
x^2*(d/(2*b) - (a*g)/(2*b^2)) + x^3*(e/(3*b) - (a*h)/(3*b^2)) + symsum(log (root(27*b^9*z^3 + 27*a*b^7*e*z^2 - 27*a^2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18* a^3*b^4*e*h*z + 9*a^3*b^4*f*g*z - 9*a^2*b^5*d*f*z - 9*a^2*b^5*c*g*z + 9*a^ 4*b^3*h^2*z + 9*a^2*b^5*e^2*z - 3*a^5*b*f*g*h + 3*a^4*b^2*e*f*g + 3*a^4*b^ 2*d*f*h + 3*a^4*b^2*c*g*h - 3*a^3*b^3*d*e*f - 3*a^3*b^3*c*e*g - 3*a^3*b^3* c*d*h + 3*a^2*b^4*c*d*e + 3*a^5*b*e*h^2 - 3*a^4*b^2*e^2*h - 3*a^4*b^2*d*g^ 2 + 3*a^3*b^3*d^2*g + 3*a^3*b^3*c*f^2 - 3*a^2*b^4*c^2*f + a^3*b^3*e^3 + a^ 5*b*g^3 + a*b^5*c^3 - a^4*b^2*f^3 - a^2*b^4*d^3 - a^6*h^3, z, k)*((6*a^2*b ^4*e - 6*a^3*b^3*h)/b^4 + (x*(3*a^2*b^3*f - 3*a*b^4*c))/b^3 + 9*root(27*b^ 9*z^3 + 27*a*b^7*e*z^2 - 27*a^2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18*a^3*b^4*e*h *z + 9*a^3*b^4*f*g*z - 9*a^2*b^5*d*f*z - 9*a^2*b^5*c*g*z + 9*a^4*b^3*h^2*z + 9*a^2*b^5*e^2*z - 3*a^5*b*f*g*h + 3*a^4*b^2*e*f*g + 3*a^4*b^2*d*f*h + 3 *a^4*b^2*c*g*h - 3*a^3*b^3*d*e*f - 3*a^3*b^3*c*e*g - 3*a^3*b^3*c*d*h + 3*a ^2*b^4*c*d*e + 3*a^5*b*e*h^2 - 3*a^4*b^2*e^2*h - 3*a^4*b^2*d*g^2 + 3*a^3*b ^3*d^2*g + 3*a^3*b^3*c*f^2 - 3*a^2*b^4*c^2*f + a^3*b^3*e^3 + a^5*b*g^3 + a *b^5*c^3 - a^4*b^2*f^3 - a^2*b^4*d^3 - a^6*h^3, z, k)*a*b^2) + (a^5*h^2 + a^3*b^2*e^2 - 2*a^4*b*e*h + a^4*b*f*g + a^2*b^3*c*d - a^3*b^2*c*g - a^3*b^ 2*d*f)/b^4 + (x*(a^4*g^2 + a^2*b^2*d^2 - a^4*f*h + a^3*b*c*h - 2*a^3*b*d*g + a^3*b*e*f - a^2*b^2*c*e))/b^3)*root(27*b^9*z^3 + 27*a*b^7*e*z^2 - 27*a^ 2*b^6*h*z^2 + 9*a*b^6*c*d*z - 18*a^3*b^4*e*h*z + 9*a^3*b^4*f*g*z - 9*a^...