3.428 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 (a+b x^3)^3} \, dx\)

Optimal. Leaf size=360 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3} \]

[Out]

-c/(2*a^3*x^2) - d/(a^3*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a^2*(a + b*x^3)^2) - (x*(11*
b*c - 5*a*f + 2*(5*b*d - 2*a*g)*x + 3*(3*b*e - a*h)*x^2))/(18*a^3*(a + b*x^3)) + ((20*b^(4/3)*c + 14*a^(1/3)*b
*d - 5*a*b^(1/3)*f - 2*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(2/
3)) + (e*Log[x])/a^3 - ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d - a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1
1/3)*b^(2/3)) + ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d - a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(54*a^(11/3)*b^(2/3)) - (e*Log[a + b*x^3])/(3*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.813874, antiderivative size = 357, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}-5 a f+20 b c\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]

[Out]

-c/(2*a^3*x^2) - d/(a^3*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a^2*(a + b*x^3)^2) - (x*(11*
b*c - 5*a*f + 2*(5*b*d - 2*a*g)*x + 3*(3*b*e - a*h)*x^2))/(18*a^3*(a + b*x^3)) + ((20*b^(4/3)*c + 14*a^(1/3)*b
*d - 5*a*b^(1/3)*f - 2*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(2/
3)) + (e*Log[x])/a^3 - ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d - a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1
1/3)*b^(2/3)) + ((20*b*c - 5*a*f - (2*a^(1/3)*(7*b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(54*a^(11/3)*b^(1/3)) - (e*Log[a + b*x^3])/(3*a^3)

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1871

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 1860

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 31

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

Rule 634

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 617

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 204

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

Rule 628

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 260

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]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^3} \, dx &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^2 c-6 b^2 d x-6 b^2 e x^2+5 b^2 \left (\frac{b c}{a}-f\right ) x^3+4 b^2 \left (\frac{b d}{a}-g\right ) x^4+3 b^2 \left (\frac{b e}{a}-h\right ) x^5}{x^3 \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \frac{18 b^4 c+18 b^4 d x+18 b^4 e x^2-2 b^4 \left (\frac{11 b c}{a}-5 f\right ) x^3-2 b^4 \left (\frac{5 b d}{a}-2 g\right ) x^4}{x^3 \left (a+b x^3\right )} \, dx}{18 a^2 b^4}\\ &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^4 c}{a x^3}+\frac{18 b^4 d}{a x^2}+\frac{18 b^4 e}{a x}+\frac{2 b^4 \left (-5 (4 b c-a f)-2 (7 b d-a g) x-9 b e x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{e \log (x)}{a^3}+\frac{\int \frac{-5 (4 b c-a f)-2 (7 b d-a g) x-9 b e x^2}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{e \log (x)}{a^3}+\frac{\int \frac{-5 (4 b c-a f)-2 (7 b d-a g) x}{a+b x^3} \, dx}{9 a^3}-\frac{(b e) \int \frac{x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{e \log (x)}{a^3}-\frac{e \log \left (a+b x^3\right )}{3 a^3}+\frac{\int \frac{\sqrt [3]{a} \left (-10 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )+\sqrt [3]{b} \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} \sqrt [3]{b}}-\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{e \log (x)}{a^3}-\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3} \sqrt [3]{b}}+\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{11/3} \sqrt [3]{b}}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{e \log (x)}{a^3}-\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{11/3} b^{2/3}}\\ &=-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b c-5 a f+2 (5 b d-2 a g) x+3 (3 b e-a h) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\left (20 b^{4/3} c+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f-2 a^{4/3} g\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{e \log (x)}{a^3}-\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{\left (20 b c-5 a f-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.606096, size = 337, normalized size = 0.94 \[ -\frac{\frac{9 a^2 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{b \left (a+b x^3\right )^2}-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d+5 a \sqrt [3]{b} f-20 b^{4/3} c\right )}{b^{2/3}}-\frac{3 a (6 a e+a x (5 f+4 g x)-b x (11 c+10 d x))}{a+b x^3}+18 a e \log \left (a+b x^3\right )+\frac{27 a c}{x^2}+\frac{54 a d}{x}-54 a e \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]

[Out]

-((27*a*c)/x^2 + (54*a*d)/x - (3*a*(6*a*e - b*x*(11*c + 10*d*x) + a*x*(5*f + 4*g*x)))/(a + b*x^3) + (9*a^2*(a^
2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)^2) + (2*Sqrt[3]*a^(1/3)*(-20*b^(4/3)*c - 14*a^(
1/3)*b*d + 5*a*b^(1/3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - 54*a*e*Log[x] +
(2*a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) -
(a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/
3)*x^2])/b^(2/3) + 18*a*e*Log[a + b*x^3])/(54*a^4)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 626, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^3,x)

[Out]

-20/27/a^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-11/18/a^3/(b*x^3+a)^2*b^2*x^4*c+e
*ln(x)/a^3-1/3*e*ln(b*x^3+a)/a^3+2/27/a^2*g*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-
5/9/a^3/(b*x^3+a)^2*x^5*b^2*d-7/9/a^2/(b*x^3+a)^2*b*x*c-14/27/a^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/
(1/b*a)^(1/3)*x-1))*d-d/a^3/x-1/2*c/a^3/x^2-13/18/a^2/(b*x^3+a)^2*b*x^2*d-2/27/a^2*g/b/(1/b*a)^(1/3)*ln(x+(1/b
*a)^(1/3))+1/27/a^2*g/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-20/27/a^3/(1/b*a)^(2/3)*ln(x+(1/b*
a)^(1/3))*c+10/27/a^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+14/27/a^3/(1/b*a)^(1/3)*ln(x+(1/b*
a)^(1/3))*d-7/27/a^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d-1/6/(b*x^3+a)^2/b*h+2/9/a^2/(b*x^3+
a)^2*x^5*b*g+4/9/a/(b*x^3+a)^2*f*x+7/18/a/(b*x^3+a)^2*x^2*g+1/2/a/(b*x^3+a)^2*e+5/18/a^2/(b*x^3+a)^2*x^4*b*f+5
/27/a^2*f/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-5/54/a^2*f/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))
+5/27/a^2*f/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/3*b/a^2/(b*x^3+a)^2*e*x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 139.296, size = 31293, normalized size = 86.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/2916*(972*a*b^2*e*x^5 - 648*(7*b^3*d - a*b^2*g)*x^7 - 810*(4*b^3*c - a*b^2*f)*x^6 - 2916*a^2*b*d*x - 1134*(7
*a*b^2*d - a^2*b*g)*x^4 - 1458*a^2*b*c - 1296*(4*a*b^2*c - a^2*b*f)*x^3 + 486*(3*a^2*b*e - a^3*h)*x^2 - 2*(a^3
*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70
*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*
b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3
- 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3
- 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(34
3*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2
*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000
*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^
2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 39
2*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486
*e/a^3)*log(-7840*a*b^3*c*d^2 + 3600*a*b^3*c^2*e - 1134*a^2*b^2*d*e^2 + 225*a^3*b*e*f^2 - 1/1458*(7*a^8*b^2*d
- a^9*b*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b)
)/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(80
00*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168
*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a
^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2
*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2
- 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2
*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3
+ 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e
*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)^2 - 40*(4*a^3*b*c -
a^4*f)*g^2 - 1/54*(400*a^4*b^3*c^2 - 252*a^5*b^2*d*e - 200*a^5*b^2*c*f + 25*a^6*b*f^2 + 36*a^6*b*e*g)*((-I*sq
rt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 +
1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*
a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a
^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3
- 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b
^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g
)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b
*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125
*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8
*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3) + 40*(49*a^2*b^2*d^2 - 45*a^2*b^2*c*e
)*f + 2*(1120*a^2*b^2*c*d + 81*a^3*b*e^2 - 280*a^3*b*d*f)*g - (8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*
f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)*x) - (1458*b^3*e*x^
8 + 2916*a*b^2*e*x^5 + 1458*a^2*b*e*x^2 - (a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*((-I*sqrt(3) + 1)*(81*e^2/
a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*
d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*
b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2)
- 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d
^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I
*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) -
1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2
*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g +
168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d
*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3) - 3*sqrt(1/3)*(a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*
sqrt(-(((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-
1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b
^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3
*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b
+ 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*
a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 7
0*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f
^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*
a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*
c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)^2*a^7*b - 972*((-I*sqrt
(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1
/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*
b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4
*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 -
630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2
))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*
a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f
^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f
^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(
343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)*a^4*b*e + 3265920*b^2*c*d + 236196*a*b*
e^2 - 816480*a*b*d*f - 116640*(4*a*b*c - a^2*f)*g)/(a^7*b)))*log(7840*a*b^3*c*d^2 - 3600*a*b^3*c^2*e + 1134*a^
2*b^2*d*e^2 - 225*a^3*b*e*f^2 + 1/1458*(7*a^8*b^2*d - a^9*b*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d +
10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*
e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*
b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c
^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 1
8*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*
e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^
3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*
g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*
(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3
)/(a^11*b^2))^(1/3) + 486*e/a^3)^2 + 40*(4*a^3*b*c - a^4*f)*g^2 + 1/54*(400*a^4*b^3*c^2 - 252*a^5*b^2*d*e - 20
0*a^5*b^2*c*f + 25*a^6*b*f^2 + 36*a^6*b*e*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e
^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*
c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^
3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (
125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2
- 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(
280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3
- 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(
a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e
*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3
) + 486*e/a^3) - 40*(49*a^2*b^2*d^2 - 45*a^2*b^2*c*e)*f - 2*(1120*a^2*b^2*c*d + 81*a^3*b*e^2 - 280*a^3*b*d*f)*
g - 2*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^
2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)*x + 1/486*sqrt(1/3)*(10800*a^4*b^3*c^2 + 3402*a^5*b^2*d*e - 5400*a^5*b^2*c*
f + 675*a^6*b*f^2 - 486*a^6*b*e*g - (7*a^8*b^2*d - a^9*b*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*
a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2
- 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2
*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3
+ 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e
*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3
/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 +
2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2
- 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(24
3*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(
a^11*b^2))^(1/3) + 486*e/a^3))*sqrt(-(((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70
*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*
b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3
- 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3
- 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(34
3*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2
*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000
*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^
2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 39
2*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486
*e/a^3)^2*a^7*b - 972*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a
*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) -
1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*
d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 1
68*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*
e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*
g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1
500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(80
00*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25
*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)*a^4*b*e
+ 3265920*b^2*c*d + 236196*a*b*e^2 - 816480*a*b*d*f - 116640*(4*a*b*c - a^2*f)*g)/(a^7*b))) - (1458*b^3*e*x^8
+ 2916*a*b^2*e*x^5 + 1458*a^2*b*e*x^2 - (a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*((-I*sqrt(3) + 1)*(81*e^2/a^
6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d
+ 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^
3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) -
1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2
*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*s
qrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1
/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d
^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 16
8*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e
+ 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3) + 3*sqrt(1/3)*(a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*sq
rt(-(((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/
27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4
*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b
*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b +
3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*
b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*
d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2
- 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^
4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)
*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)^2*a^7*b - 972*((-I*sqrt(3
) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1
458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^
3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g
^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 63
0*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))
^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*
b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3
- 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3
- 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(34
3*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)*a^4*b*e + 3265920*b^2*c*d + 236196*a*b*e^
2 - 816480*a*b*d*f - 116640*(4*a*b*c - a^2*f)*g)/(a^7*b)))*log(7840*a*b^3*c*d^2 - 3600*a*b^3*c^2*e + 1134*a^2*
b^2*d*e^2 - 225*a^3*b*e*f^2 + 1/1458*(7*a^8*b^2*d - a^9*b*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10
*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^
2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^
2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3
+ 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*
e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^
3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3
+ 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^
2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(2
43*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/
(a^11*b^2))^(1/3) + 486*e/a^3)^2 + 40*(4*a^3*b*c - a^4*f)*g^2 + 1/54*(400*a^4*b^3*c^2 - 252*a^5*b^2*d*e - 200*
a^5*b^2*c*f + 25*a^6*b*f^2 + 36*a^6*b*e*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2
- 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*
g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*
b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (12
5*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 -
8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(28
0*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 -
6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^
11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f
+ 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3)
+ 486*e/a^3) - 40*(49*a^2*b^2*d^2 - 45*a^2*b^2*c*e)*f - 2*(1120*a^2*b^2*c*d + 81*a^3*b*e^2 - 280*a^3*b*d*f)*g
- 2*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*
g + 168*a^3*b*d*g^2 - 8*a^4*g^3)*x - 1/486*sqrt(1/3)*(10800*a^4*b^3*c^2 + 3402*a^5*b^2*d*e - 5400*a^5*b^2*c*f
+ 675*a^6*b*f^2 - 486*a^6*b*e*g - (7*a^8*b^2*d - a^9*b*g)*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^
2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 -
70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c
*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 +
8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g
)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a
^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2
744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 -
8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*
e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^
11*b^2))^(1/3) + 486*e/a^3))*sqrt(-(((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d
*f - 40*c*g)*a*b)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)
*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 -
1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 -
270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*
d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c
*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a
*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2)
- 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*
d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e
/a^3)^2*a^7*b - 972*((-I*sqrt(3) + 1)*(81*e^2/a^6 - (280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b
)/(a^7*b))/(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g + (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/
39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 1500*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^
2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168
*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e
+ 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*e^3/a^9 + 1/1458*(280*b^2*c*d + 10*a^2*f*g
+ (81*e^2 - 70*d*f - 40*c*g)*a*b)*e/(a^10*b) - 1/39366*(8000*b^4*c^3 + 2744*a*b^3*d^3 - 6000*a*b^3*c^2*f + 150
0*a^2*b^2*c*f^2 - 125*a^3*b*f^3 - 1176*a^2*b^2*d^2*g + 168*a^3*b*d*g^2 - 8*a^4*g^3)/(a^11*b^2) - 1/39366*(8000
*b^4*c^3 + 8*a^4*g^3 - (125*f^3 - 270*e*f*g + 168*d*g^2)*a^3*b + 3*(243*e^3 - 630*d*e*f + 392*d^2*g + 20*(25*f
^2 - 18*e*g)*c)*a^2*b^2 - 8*(343*d^3 - 945*c*d*e + 750*c^2*f)*a*b^3)/(a^11*b^2))^(1/3) + 486*e/a^3)*a^4*b*e +
3265920*b^2*c*d + 236196*a*b*e^2 - 816480*a*b*d*f - 116640*(4*a*b*c - a^2*f)*g)/(a^7*b))) + 2916*(b^3*e*x^8 +
2*a*b^2*e*x^5 + a^2*b*e*x^2)*log(x))/(a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.11512, size = 568, normalized size = 1.58 \begin{align*} -\frac{e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{e \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + 2 \, \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 )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - 2 \, \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 )}{54 \, a^{4} b^{2}} - \frac{28 \, b^{3} d x^{7} - 4 \, a b^{2} g x^{7} + 20 \, b^{3} c x^{6} - 5 \, a b^{2} f x^{6} - 6 \, a b^{2} x^{5} e + 49 \, a b^{2} d x^{4} - 7 \, a^{2} b g x^{4} + 32 \, a b^{2} c x^{3} - 8 \, a^{2} b f x^{3} + 3 \, a^{3} h x^{2} - 9 \, a^{2} b x^{2} e + 18 \, a^{2} b d x + 9 \, a^{2} b c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3} b} + \frac{{\left (14 \, a^{3} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{4} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, a^{3} b^{2} c - 5 \, a^{4} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/3*e*log(abs(b*x^3 + a))/a^3 + e*log(abs(x))/a^3 - 1/27*sqrt(3)*(20*(-a*b^2)^(1/3)*b^2*c - 5*(-a*b^2)^(1/3)*
a*b*f - 14*(-a*b^2)^(2/3)*b*d + 2*(-a*b^2)^(2/3)*a*g)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a
^4*b^2) - 1/54*(20*(-a*b^2)^(1/3)*b^2*c - 5*(-a*b^2)^(1/3)*a*b*f + 14*(-a*b^2)^(2/3)*b*d - 2*(-a*b^2)^(2/3)*a*
g)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^2) - 1/18*(28*b^3*d*x^7 - 4*a*b^2*g*x^7 + 20*b^3*c*x^6 - 5*
a*b^2*f*x^6 - 6*a*b^2*x^5*e + 49*a*b^2*d*x^4 - 7*a^2*b*g*x^4 + 32*a*b^2*c*x^3 - 8*a^2*b*f*x^3 + 3*a^3*h*x^2 -
9*a^2*b*x^2*e + 18*a^2*b*d*x + 9*a^2*b*c)/((b*x^4 + a*x)^2*a^3*b) + 1/27*(14*a^3*b^2*d*(-a/b)^(1/3) - 2*a^4*b*
g*(-a/b)^(1/3) + 20*a^3*b^2*c - 5*a^4*b*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b)