\(\int \frac {(a+b x+c x^2)^4}{d+e x^3} \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 645 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=-\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}-\frac {\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{8/3}}+\frac {\left (\sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{8/3}}-\frac {\left (\sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{8/3}}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3} \]

[Out]

-2*(-6*a^2*b*c*e-2*a*b^3*e+2*a*c^3*d+3*b^2*c^2*d)*x/e^2-1/2*(-6*a^2*c^2*e-12*a*b^2*c*e-b^4*e+4*b*c^3*d)*x^2/e^
2-1/3*c*(-12*a*b*c*e-4*b^3*e+c^3*d)*x^3/e^2+1/2*c^2*(2*a*c+3*b^2)*x^4/e+4/5*b*c^3*x^5/e+1/6*c^4*x^6/e+1/3*(e^(
1/3)*(a^4*e^2-12*a^2*b*c*d*e-4*a*b^3*d*e+4*a*c^3*d^2+6*b^2*c^2*d^2)+d^(1/3)*(b^4*d*e+12*a*b^2*c*d*e+6*a^2*c^2*
d*e-4*b*(a^3*e^2+c^3*d^2)))*ln(d^(1/3)+e^(1/3)*x)/d^(2/3)/e^(8/3)-1/6*(e^(1/3)*(a^4*e^2-12*a^2*b*c*d*e-4*a*b^3
*d*e+4*a*c^3*d^2+6*b^2*c^2*d^2)+d^(1/3)*(b^4*d*e+12*a*b^2*c*d*e+6*a^2*c^2*d*e-4*b*(a^3*e^2+c^3*d^2)))*ln(d^(2/
3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(2/3)/e^(8/3)+1/3*(c^4*d^2-12*a*b*c^2*d*e+6*a^2*b^2*e^2-4*c*e*(-a^3*e+b^3*
d))*ln(e*x^3+d)/e^3-1/3*(b*d^(1/3)+a*e^(1/3))*(4*c^3*d^2+6*c^2*(b*d^(5/3)*e^(1/3)-a*d^(4/3)*e^(2/3))-12*a*b*c*
d*e-e*(b^3*d+3*a*b^2*d^(2/3)*e^(1/3)-3*a^2*b*d^(1/3)*e^(2/3)-a^3*e))*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*
3^(1/2))/d^(2/3)/e^(8/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=-\frac {x^2 \left (-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right )}{2 e^2}-\frac {2 x \left (-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right )}{e^2}-\frac {\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (-e \left (a^3 (-e)-3 a^2 b \sqrt [3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e+4 c^3 d^2\right )}{\sqrt {3} d^{2/3} e^{8/3}}+\frac {\log \left (d+e x^3\right ) \left (-4 c e \left (b^3 d-a^3 e\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{3 e^3}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^4 e^2-12 a^2 b c d e+\frac {\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )}{\sqrt [3]{e}}-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{3 d^{2/3} e^{8/3}}-\frac {c x^3 \left (-12 a b c e-4 b^3 e+c^3 d\right )}{3 e^2}+\frac {c^2 x^4 \left (2 a c+3 b^2\right )}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e} \]

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x^3),x]

[Out]

(-2*(3*b^2*c^2*d + 2*a*c^3*d - 2*a*b^3*e - 6*a^2*b*c*e)*x)/e^2 - ((4*b*c^3*d - b^4*e - 12*a*b^2*c*e - 6*a^2*c^
2*e)*x^2)/(2*e^2) - (c*(c^3*d - 4*b^3*e - 12*a*b*c*e)*x^3)/(3*e^2) + (c^2*(3*b^2 + 2*a*c)*x^4)/(2*e) + (4*b*c^
3*x^5)/(5*e) + (c^4*x^6)/(6*e) - ((b*d^(1/3) + a*e^(1/3))*(4*c^3*d^2 + 6*c^2*(b*d^(5/3)*e^(1/3) - a*d^(4/3)*e^
(2/3)) - 12*a*b*c*d*e - e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^(2/3) - a^3*e))*ArcTan[(d^(1/3)
- 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(8/3)) + ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b
^3*d*e - 12*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^
2)))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(8/3)) - ((6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 12*a^2*b*c*
d*e + a^4*e^2 + (d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^2)))/e^(1/3))*Log[d^
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) + ((c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 - 4
*c*e*(b^3*d - a^3*e))*Log[d + e*x^3])/(3*e^3)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right )}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x}{e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^2}{e^2}+\frac {2 c^2 \left (3 b^2+2 a c\right ) x^3}{e}+\frac {4 b c^3 x^4}{e}+\frac {c^4 x^5}{e}+\frac {6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2-\left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right ) x+\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) x^2}{e^2 \left (d+e x^3\right )}\right ) \, dx \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}+\frac {\int \frac {6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2-\left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right ) x+\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) x^2}{d+e x^3} \, dx}{e^2} \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}+\frac {\int \frac {6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\right )\right ) x}{d+e x^3} \, dx}{e^2}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \int \frac {x^2}{d+e x^3} \, dx}{e^2} \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac {\int \frac {\sqrt [3]{d} \left (2 \sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\right )\right )\right )+\sqrt [3]{e} \left (-\sqrt [3]{e} \left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2\right )+\sqrt [3]{d} \left (-b^4 d e-12 a b^2 c d e-6 a^2 c^2 d e+4 b \left (c^3 d^2+a^3 e^2\right )\right )\right ) x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} e^{7/3}}+\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e^2} \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}+\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac {\left (\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right )\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d} e^{7/3}}-\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{7/3}} \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}+\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3}+\frac {\left (\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{8/3}} \\ & = -\frac {2 \left (3 b^2 c^2 d+2 a c^3 d-2 a b^3 e-6 a^2 b c e\right ) x}{e^2}-\frac {\left (4 b c^3 d-b^4 e-12 a b^2 c e-6 a^2 c^2 e\right ) x^2}{2 e^2}-\frac {c \left (c^3 d-4 b^3 e-12 a b c e\right ) x^3}{3 e^2}+\frac {c^2 \left (3 b^2+2 a c\right ) x^4}{2 e}+\frac {4 b c^3 x^5}{5 e}+\frac {c^4 x^6}{6 e}-\frac {\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (4 c^3 d^2+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e-e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{8/3}}+\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (6 b^2 c^2 d^2+4 a c^3 d^2-4 a b^3 d e-12 a^2 b c d e+a^4 e^2+\frac {\sqrt [3]{d} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e-4 b \left (c^3 d^2+a^3 e^2\right )\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2-4 c e \left (b^3 d-a^3 e\right )\right ) \log \left (d+e x^3\right )}{3 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=\frac {60 e^{2/3} \left (-3 b^2 c^2 d-2 a c^3 d+2 a b^3 e+6 a^2 b c e\right ) x+15 e^{2/3} \left (-4 b c^3 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^2+10 c e^{2/3} \left (-c^3 d+4 b^3 e+12 a b c e\right ) x^3+15 c^2 \left (3 b^2+2 a c\right ) e^{5/3} x^4+24 b c^3 e^{5/3} x^5+5 c^4 e^{5/3} x^6+\frac {10 \sqrt {3} \left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \left (-4 c^3 d^2+c^2 \left (-6 b d^{5/3} \sqrt [3]{e}+6 a d^{4/3} e^{2/3}\right )+12 a b c d e+e \left (b^3 d+3 a b^2 d^{2/3} \sqrt [3]{e}-3 a^2 b \sqrt [3]{d} e^{2/3}-a^3 e\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {10 \left (4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e+6 a^2 c^2 d^{4/3} e-4 a b^3 d e^{4/3}+a^4 e^{7/3}+6 b^2 \left (c^2 d^2 \sqrt [3]{e}+2 a c d^{4/3} e\right )-4 b \left (c^3 d^{7/3}+3 a^2 c d e^{4/3}+a^3 \sqrt [3]{d} e^2\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {5 \left (4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e+6 a^2 c^2 d^{4/3} e-4 a b^3 d e^{4/3}+a^4 e^{7/3}+6 b^2 \left (c^2 d^2 \sqrt [3]{e}+2 a c d^{4/3} e\right )-4 b \left (c^3 d^{7/3}+3 a^2 c d e^{4/3}+a^3 \sqrt [3]{d} e^2\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}+\frac {10 \left (c^4 d^2-12 a b c^2 d e+6 a^2 b^2 e^2+4 c e \left (-b^3 d+a^3 e\right )\right ) \log \left (d+e x^3\right )}{\sqrt [3]{e}}}{30 e^{8/3}} \]

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x^3),x]

[Out]

(60*e^(2/3)*(-3*b^2*c^2*d - 2*a*c^3*d + 2*a*b^3*e + 6*a^2*b*c*e)*x + 15*e^(2/3)*(-4*b*c^3*d + b^4*e + 12*a*b^2
*c*e + 6*a^2*c^2*e)*x^2 + 10*c*e^(2/3)*(-(c^3*d) + 4*b^3*e + 12*a*b*c*e)*x^3 + 15*c^2*(3*b^2 + 2*a*c)*e^(5/3)*
x^4 + 24*b*c^3*e^(5/3)*x^5 + 5*c^4*e^(5/3)*x^6 + (10*Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*(-4*c^3*d^2 + c^2*(-6*b*d
^(5/3)*e^(1/3) + 6*a*d^(4/3)*e^(2/3)) + 12*a*b*c*d*e + e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^
(2/3) - a^3*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (10*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*
e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4/3)*e) - 4*b*(c
^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (5*(4*a*c^3*d^2*e^(1/3)
+ b^4*d^(4/3)*e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4
/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^
2])/d^(2/3) + (10*(c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 + 4*c*e*(-(b^3*d) + a^3*e))*Log[d + e*x^3])/e^(1/3
))/(30*e^(8/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.69 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {c^{4} x^{6}}{6 e}+\frac {4 b \,c^{3} x^{5}}{5 e}+\frac {a \,c^{3} x^{4}}{e}+\frac {3 b^{2} c^{2} x^{4}}{2 e}+\frac {4 a b \,c^{2} x^{3}}{e}+\frac {4 b^{3} c \,x^{3}}{3 e}-\frac {c^{4} d \,x^{3}}{3 e^{2}}+\frac {3 a^{2} c^{2} x^{2}}{e}+\frac {6 a \,b^{2} c \,x^{2}}{e}+\frac {b^{4} x^{2}}{2 e}-\frac {2 b \,c^{3} d \,x^{2}}{e^{2}}+\frac {12 a^{2} b c x}{e}+\frac {4 a \,b^{3} x}{e}-\frac {4 a \,c^{3} d x}{e^{2}}-\frac {6 x \,b^{2} c^{2} d}{e^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e +d \right )}{\sum }\frac {\left (\left (4 a^{3} c \,e^{2}+6 a^{2} b^{2} e^{2}-12 a b \,c^{2} d e -4 b^{3} c d e +c^{4} d^{2}\right ) \textit {\_R}^{2}+\textit {\_R} \left (4 a^{3} b \,e^{2}-6 a^{2} c^{2} d e -12 a \,b^{2} c d e -b^{4} d e +4 d^{2} c^{3} b \right )+a^{4} e^{2}-12 a^{2} b c d e -4 a \,b^{3} d e +4 a \,c^{3} d^{2}+6 b^{2} c^{2} d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{3}}\)	\(350\)
default	\(\frac {\frac {1}{6} c^{4} x^{6} e +\frac {4}{5} b \,c^{3} x^{5} e +a \,c^{3} e \,x^{4}+\frac {3}{2} b^{2} c^{2} e \,x^{4}+4 a b \,c^{2} e \,x^{3}+\frac {4}{3} b^{3} c e \,x^{3}-\frac {1}{3} c^{4} d \,x^{3}+3 a^{2} c^{2} e \,x^{2}+6 a \,b^{2} c e \,x^{2}+\frac {1}{2} b^{4} e \,x^{2}-2 b \,c^{3} d \,x^{2}+12 a^{2} b c e x +4 a \,b^{3} e x -4 a \,c^{3} d x -6 x \,b^{2} c^{2} d}{e^{2}}+\frac {\left (a^{4} e^{2}-12 a^{2} b c d e -4 a \,b^{3} d e +4 a \,c^{3} d^{2}+6 b^{2} c^{2} d^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )+\left (4 a^{3} b \,e^{2}-6 a^{2} c^{2} d e -12 a \,b^{2} c d e -b^{4} d e +4 d^{2} c^{3} b \right ) \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )+\frac {\left (4 a^{3} c \,e^{2}+6 a^{2} b^{2} e^{2}-12 a b \,c^{2} d e -4 b^{3} c d e +c^{4} d^{2}\right ) \ln \left (e \,x^{3}+d \right )}{3 e}}{e^{2}}\)	\(489\)

[In]

int((c*x^2+b*x+a)^4/(e*x^3+d),x,method=_RETURNVERBOSE)

[Out]

1/6*c^4*x^6/e+4/5*b*c^3*x^5/e+1/e*a*c^3*x^4+3/2/e*b^2*c^2*x^4+4/e*a*b*c^2*x^3+4/3/e*b^3*c*x^3-1/3/e^2*c^4*d*x^
3+3/e*a^2*c^2*x^2+6/e*a*b^2*c*x^2+1/2/e*b^4*x^2-2/e^2*b*c^3*d*x^2+12/e*a^2*b*c*x+4/e*a*b^3*x-4/e^2*a*c^3*d*x-6
/e^2*x*b^2*c^2*d+1/3/e^3*sum(((4*a^3*c*e^2+6*a^2*b^2*e^2-12*a*b*c^2*d*e-4*b^3*c*d*e+c^4*d^2)*_R^2+_R*(4*a^3*b*
e^2-6*a^2*c^2*d*e-12*a*b^2*c*d*e-b^4*d*e+4*b*c^3*d^2)+a^4*e^2-12*a^2*b*c*d*e-4*a*b^3*d*e+4*a*c^3*d^2+6*b^2*c^2
*d^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3*e+d))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 91.41 (sec) , antiderivative size = 47284, normalized size of antiderivative = 73.31 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x^3+d),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**4/(e*x**3+d),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x^3+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=-\frac {\sqrt {3} {\left (6 \, b^{2} c^{2} d^{2} e + 4 \, a c^{3} d^{2} e - 4 \, a b^{3} d e^{2} - 12 \, a^{2} b c d e^{2} + a^{4} e^{3} - 4 \, \left (-d e^{2}\right )^{\frac {1}{3}} b c^{3} d^{2} + \left (-d e^{2}\right )^{\frac {1}{3}} b^{4} d e + 12 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b^{2} c d e + 6 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} c^{2} d e - 4 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{3} b e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}} e^{2}} - \frac {{\left (6 \, b^{2} c^{2} d^{2} e + 4 \, a c^{3} d^{2} e - 4 \, a b^{3} d e^{2} - 12 \, a^{2} b c d e^{2} + a^{4} e^{3} + 4 \, \left (-d e^{2}\right )^{\frac {1}{3}} b c^{3} d^{2} - \left (-d e^{2}\right )^{\frac {1}{3}} b^{4} d e - 12 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b^{2} c d e - 6 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} c^{2} d e + 4 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{3} b e^{2}\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}} e^{2}} + \frac {{\left (c^{4} d^{2} - 4 \, b^{3} c d e - 12 \, a b c^{2} d e + 6 \, a^{2} b^{2} e^{2} + 4 \, a^{3} c e^{2}\right )} \log \left ({\left | e x^{3} + d \right |}\right )}{3 \, e^{3}} + \frac {5 \, c^{4} e^{5} x^{6} + 24 \, b c^{3} e^{5} x^{5} + 45 \, b^{2} c^{2} e^{5} x^{4} + 30 \, a c^{3} e^{5} x^{4} - 10 \, c^{4} d e^{4} x^{3} + 40 \, b^{3} c e^{5} x^{3} + 120 \, a b c^{2} e^{5} x^{3} - 60 \, b c^{3} d e^{4} x^{2} + 15 \, b^{4} e^{5} x^{2} + 180 \, a b^{2} c e^{5} x^{2} + 90 \, a^{2} c^{2} e^{5} x^{2} - 180 \, b^{2} c^{2} d e^{4} x - 120 \, a c^{3} d e^{4} x + 120 \, a b^{3} e^{5} x + 360 \, a^{2} b c e^{5} x}{30 \, e^{6}} - \frac {{\left (4 \, b c^{3} d^{2} e^{11} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - b^{4} d e^{12} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - 12 \, a b^{2} c d e^{12} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - 6 \, a^{2} c^{2} d e^{12} \left (-\frac {d}{e}\right )^{\frac {1}{3}} + 4 \, a^{3} b e^{13} \left (-\frac {d}{e}\right )^{\frac {1}{3}} + 6 \, b^{2} c^{2} d^{2} e^{11} + 4 \, a c^{3} d^{2} e^{11} - 4 \, a b^{3} d e^{12} - 12 \, a^{2} b c d e^{12} + a^{4} e^{13}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d e^{13}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x^3+d),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*(6*b^2*c^2*d^2*e + 4*a*c^3*d^2*e - 4*a*b^3*d*e^2 - 12*a^2*b*c*d*e^2 + a^4*e^3 - 4*(-d*e^2)^(1/3)*
b*c^3*d^2 + (-d*e^2)^(1/3)*b^4*d*e + 12*(-d*e^2)^(1/3)*a*b^2*c*d*e + 6*(-d*e^2)^(1/3)*a^2*c^2*d*e - 4*(-d*e^2)
^(1/3)*a^3*b*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d/e)^(1/3))/(-d/e)^(1/3))/((-d*e^2)^(2/3)*e^2) - 1/6*(6*b^2*c^2*
d^2*e + 4*a*c^3*d^2*e - 4*a*b^3*d*e^2 - 12*a^2*b*c*d*e^2 + a^4*e^3 + 4*(-d*e^2)^(1/3)*b*c^3*d^2 - (-d*e^2)^(1/
3)*b^4*d*e - 12*(-d*e^2)^(1/3)*a*b^2*c*d*e - 6*(-d*e^2)^(1/3)*a^2*c^2*d*e + 4*(-d*e^2)^(1/3)*a^3*b*e^2)*log(x^
2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/((-d*e^2)^(2/3)*e^2) + 1/3*(c^4*d^2 - 4*b^3*c*d*e - 12*a*b*c^2*d*e + 6*a^2*
b^2*e^2 + 4*a^3*c*e^2)*log(abs(e*x^3 + d))/e^3 + 1/30*(5*c^4*e^5*x^6 + 24*b*c^3*e^5*x^5 + 45*b^2*c^2*e^5*x^4 +
30*a*c^3*e^5*x^4 - 10*c^4*d*e^4*x^3 + 40*b^3*c*e^5*x^3 + 120*a*b*c^2*e^5*x^3 - 60*b*c^3*d*e^4*x^2 + 15*b^4*e^
5*x^2 + 180*a*b^2*c*e^5*x^2 + 90*a^2*c^2*e^5*x^2 - 180*b^2*c^2*d*e^4*x - 120*a*c^3*d*e^4*x + 120*a*b^3*e^5*x +
360*a^2*b*c*e^5*x)/e^6 - 1/3*(4*b*c^3*d^2*e^11*(-d/e)^(1/3) - b^4*d*e^12*(-d/e)^(1/3) - 12*a*b^2*c*d*e^12*(-d
/e)^(1/3) - 6*a^2*c^2*d*e^12*(-d/e)^(1/3) + 4*a^3*b*e^13*(-d/e)^(1/3) + 6*b^2*c^2*d^2*e^11 + 4*a*c^3*d^2*e^11
- 4*a*b^3*d*e^12 - 12*a^2*b*c*d*e^12 + a^4*e^13)*(-d/e)^(1/3)*log(abs(x - (-d/e)^(1/3)))/(d*e^13)

Mupad [B] (verification not implemented)

Time = 9.24 (sec) , antiderivative size = 2971, normalized size of antiderivative = 4.61 \[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx=\text {Too large to display} \]

[In]

int((a + b*x + c*x^2)^4/(d + e*x^3),x)

[Out]

x^2*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(2*e) - (2*b*c^3*d)/e^2) - x^3*((c^4*d)/(3*e^2) - (4*b*c*(3*a*c + b^2))/(3
*e)) + symsum(log(root(27*d^2*e^9*z^3 + 324*a*b*c^2*d^3*e^7*z^2 + 108*b^3*c*d^3*e^7*z^2 - 108*a^3*c*d^2*e^8*z^
2 - 162*a^2*b^2*d^2*e^8*z^2 - 27*c^4*d^4*e^6*z^2 - 72*a*b*c^6*d^5*e^4*z + 216*a^2*b^2*c^4*d^4*e^5*z + 144*a^3*
b^3*c^2*d^3*e^6*z - 108*a^5*b^2*c*d^2*e^7*z + 108*a^2*b^5*c*d^3*e^6*z - 36*a^4*b*c^3*d^3*e^6*z + 36*a*b^4*c^3*
d^4*e^5*z + 144*b^3*c^5*d^5*e^4*z + 90*b^6*c^2*d^4*e^5*z - 144*a^3*c^5*d^4*e^5*z + 90*a^6*c^2*d^2*e^7*z + 171*
a^4*b^4*d^2*e^7*z + 36*a*b^7*d^3*e^6*z + 36*a^7*b*d*e^8*z + 9*c^8*d^6*e^3*z + 36*a^7*b^4*c*d^2*e^6 - 36*a^7*b*
c^4*d^3*e^5 - 36*a^4*b^7*c*d^3*e^5 - 36*a^4*b*c^7*d^5*e^3 - 36*a*b^7*c^4*d^5*e^3 + 36*a*b^4*c^7*d^6*e^2 + 12*a
*b^10*c*d^4*e^4 + 108*a^5*b^5*c^2*d^3*e^5 - 108*a^5*b^2*c^5*d^4*e^4 + 108*a^2*b^5*c^5*d^5*e^3 - 96*a^6*b^3*c^3
*d^3*e^5 + 96*a^3*b^6*c^3*d^4*e^4 - 96*a^3*b^3*c^6*d^5*e^3 - 54*a^8*b^2*c^2*d^2*e^6 - 54*a^2*b^8*c^2*d^4*e^4 -
54*a^2*b^2*c^8*d^6*e^2 - 9*a^4*b^4*c^4*d^4*e^4 - 12*a^10*b*c*d*e^7 - 12*a*b*c^10*d^7*e - 6*b^6*c^6*d^6*e^2 +
4*b^9*c^3*d^5*e^3 - 6*a^6*c^6*d^4*e^4 - 4*a^9*c^3*d^2*e^6 - 4*a^3*c^9*d^6*e^2 - 6*a^6*b^6*d^2*e^6 + 4*a^3*b^9*
d^3*e^5 + 4*b^3*c^9*d^7*e + 4*a^9*b^3*d*e^7 - b^12*d^4*e^4 - c^12*d^8 - a^12*e^8, z, k)*((x*(3*a^4*e^5 + 12*a*
c^3*d^2*e^3 + 18*b^2*c^2*d^2*e^3 - 12*a*b^3*d*e^4 - 36*a^2*b*c*d*e^4))/e^3 - (6*c^4*d^3*e^3 + 36*a^2*b^2*d*e^5
- 24*b^3*c*d^2*e^4 + 24*a^3*c*d*e^5 - 72*a*b*c^2*d^2*e^4)/e^4 + 9*root(27*d^2*e^9*z^3 + 324*a*b*c^2*d^3*e^7*z
^2 + 108*b^3*c*d^3*e^7*z^2 - 108*a^3*c*d^2*e^8*z^2 - 162*a^2*b^2*d^2*e^8*z^2 - 27*c^4*d^4*e^6*z^2 - 72*a*b*c^6
*d^5*e^4*z + 216*a^2*b^2*c^4*d^4*e^5*z + 144*a^3*b^3*c^2*d^3*e^6*z - 108*a^5*b^2*c*d^2*e^7*z + 108*a^2*b^5*c*d
^3*e^6*z - 36*a^4*b*c^3*d^3*e^6*z + 36*a*b^4*c^3*d^4*e^5*z + 144*b^3*c^5*d^5*e^4*z + 90*b^6*c^2*d^4*e^5*z - 14
4*a^3*c^5*d^4*e^5*z + 90*a^6*c^2*d^2*e^7*z + 171*a^4*b^4*d^2*e^7*z + 36*a*b^7*d^3*e^6*z + 36*a^7*b*d*e^8*z + 9
*c^8*d^6*e^3*z + 36*a^7*b^4*c*d^2*e^6 - 36*a^7*b*c^4*d^3*e^5 - 36*a^4*b^7*c*d^3*e^5 - 36*a^4*b*c^7*d^5*e^3 - 3
6*a*b^7*c^4*d^5*e^3 + 36*a*b^4*c^7*d^6*e^2 + 12*a*b^10*c*d^4*e^4 + 108*a^5*b^5*c^2*d^3*e^5 - 108*a^5*b^2*c^5*d
^4*e^4 + 108*a^2*b^5*c^5*d^5*e^3 - 96*a^6*b^3*c^3*d^3*e^5 + 96*a^3*b^6*c^3*d^4*e^4 - 96*a^3*b^3*c^6*d^5*e^3 -
54*a^8*b^2*c^2*d^2*e^6 - 54*a^2*b^8*c^2*d^4*e^4 - 54*a^2*b^2*c^8*d^6*e^2 - 9*a^4*b^4*c^4*d^4*e^4 - 12*a^10*b*c
*d*e^7 - 12*a*b*c^10*d^7*e - 6*b^6*c^6*d^6*e^2 + 4*b^9*c^3*d^5*e^3 - 6*a^6*c^6*d^4*e^4 - 4*a^9*c^3*d^2*e^6 - 4
*a^3*c^9*d^6*e^2 - 6*a^6*b^6*d^2*e^6 + 4*a^3*b^9*d^3*e^5 + 4*b^3*c^9*d^7*e + 4*a^9*b^3*d*e^7 - b^12*d^4*e^4 -
c^12*d^8 - a^12*e^8, z, k)*d*e^2) + (c^8*d^5 + 4*a^7*b*e^5 + 4*a*b^7*d^2*e^3 + 19*a^4*b^4*d*e^4 + 10*a^6*c^2*d
*e^4 + 16*b^3*c^5*d^4*e - 16*a^3*c^5*d^3*e^2 + 10*b^6*c^2*d^3*e^2 - 8*a*b*c^6*d^4*e + 24*a^2*b^2*c^4*d^3*e^2 +
16*a^3*b^3*c^2*d^2*e^3 - 12*a^5*b^2*c*d*e^4 + 4*a*b^4*c^3*d^3*e^2 + 12*a^2*b^5*c*d^2*e^3 - 4*a^4*b*c^3*d^2*e^
3)/e^4 + (x*(10*a^6*b^2*e^4 - 4*a^7*c*e^4 - 4*a*c^7*d^4 + 10*b^2*c^6*d^4 + b^8*d^2*e^2 + 16*a^3*b^5*d*e^3 + 16
*b^5*c^3*d^3*e + 19*a^4*c^4*d^2*e^2 + 24*a^2*b^4*c^2*d^2*e^2 - 16*a^3*b^2*c^3*d^2*e^2 - 4*a*b^3*c^4*d^3*e + 8*
a*b^6*c*d^2*e^2 + 12*a^2*b*c^5*d^3*e - 4*a^4*b^3*c*d*e^3 + 12*a^5*b*c^2*d*e^3))/e^3)*root(27*d^2*e^9*z^3 + 324
*a*b*c^2*d^3*e^7*z^2 + 108*b^3*c*d^3*e^7*z^2 - 108*a^3*c*d^2*e^8*z^2 - 162*a^2*b^2*d^2*e^8*z^2 - 27*c^4*d^4*e^
6*z^2 - 72*a*b*c^6*d^5*e^4*z + 216*a^2*b^2*c^4*d^4*e^5*z + 144*a^3*b^3*c^2*d^3*e^6*z - 108*a^5*b^2*c*d^2*e^7*z
+ 108*a^2*b^5*c*d^3*e^6*z - 36*a^4*b*c^3*d^3*e^6*z + 36*a*b^4*c^3*d^4*e^5*z + 144*b^3*c^5*d^5*e^4*z + 90*b^6*
c^2*d^4*e^5*z - 144*a^3*c^5*d^4*e^5*z + 90*a^6*c^2*d^2*e^7*z + 171*a^4*b^4*d^2*e^7*z + 36*a*b^7*d^3*e^6*z + 36
*a^7*b*d*e^8*z + 9*c^8*d^6*e^3*z + 36*a^7*b^4*c*d^2*e^6 - 36*a^7*b*c^4*d^3*e^5 - 36*a^4*b^7*c*d^3*e^5 - 36*a^4
*b*c^7*d^5*e^3 - 36*a*b^7*c^4*d^5*e^3 + 36*a*b^4*c^7*d^6*e^2 + 12*a*b^10*c*d^4*e^4 + 108*a^5*b^5*c^2*d^3*e^5 -
108*a^5*b^2*c^5*d^4*e^4 + 108*a^2*b^5*c^5*d^5*e^3 - 96*a^6*b^3*c^3*d^3*e^5 + 96*a^3*b^6*c^3*d^4*e^4 - 96*a^3*
b^3*c^6*d^5*e^3 - 54*a^8*b^2*c^2*d^2*e^6 - 54*a^2*b^8*c^2*d^4*e^4 - 54*a^2*b^2*c^8*d^6*e^2 - 9*a^4*b^4*c^4*d^4
*e^4 - 12*a^10*b*c*d*e^7 - 12*a*b*c^10*d^7*e - 6*b^6*c^6*d^6*e^2 + 4*b^9*c^3*d^5*e^3 - 6*a^6*c^6*d^4*e^4 - 4*a
^9*c^3*d^2*e^6 - 4*a^3*c^9*d^6*e^2 - 6*a^6*b^6*d^2*e^6 + 4*a^3*b^9*d^3*e^5 + 4*b^3*c^9*d^7*e + 4*a^9*b^3*d*e^7
- b^12*d^4*e^4 - c^12*d^8 - a^12*e^8, z, k), k, 1, 3) - x*((d*(4*a*c^3 + 6*b^2*c^2))/e^2 - (4*a*b*(3*a*c + b^
2))/e) + (c^4*x^6)/(6*e) + (x^4*(4*a*c^3 + 6*b^2*c^2))/(4*e) + (4*b*c^3*x^5)/(5*e)