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

Optimal. Leaf size=345 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{54 a^{4/3} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{27 a^{4/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{h x}{b^3} \]

[Out]

(h*x)/b^3 + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a + b*x^3)^2) - (x*(a*(7*b*e - 1
3*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)*x^2))/(18*a*b^3*(a + b*x^3)) - ((b^(5/3)*c + 2*a^(2/3)*b*e +
5*a*b^(2/3)*f - 14*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(10/3))
- ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(10/3)) + ((b^(2/
3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(10/
3)) + (g*Log[a + b*x^3])/(3*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.891265, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {1828, 1858, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{54 a^{4/3} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{27 a^{4/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{h x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(h*x)/b^3 + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a + b*x^3)^2) - (x*(a*(7*b*e - 1
3*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)*x^2))/(18*a*b^3*(a + b*x^3)) - ((b^(5/3)*c + 2*a^(2/3)*b*e +
5*a*b^(2/3)*f - 14*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(10/3))
- ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(10/3)) + ((b^(2/
3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(10/
3)) + (g*Log[a + b*x^3])/(3*b^3)

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1887

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]

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{x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{\int \frac{a^2 (b e-a h)-2 a b (b c-a f) x-3 a b (b d-a g) x^2-6 a b (b e-a h) x^3-6 a b^2 f x^4-6 a b^2 g x^5-6 a b^2 h x^6}{\left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{\int \frac{2 a^2 b^2 (2 b e-5 a h)+2 a b^3 (b c+5 a f) x+18 a^2 b^3 g x^2+18 a^2 b^3 h x^3}{a+b x^3} \, dx}{18 a^2 b^5}\\ &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{\int \left (18 a^2 b^2 h+\frac{2 \left (2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x+9 a^2 b^3 g x^2\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^5}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{\int \frac{2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x+9 a^2 b^3 g x^2}{a+b x^3} \, dx}{9 a^2 b^5}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{\int \frac{2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x}{a+b x^3} \, dx}{9 a^2 b^5}+\frac{g \int \frac{x^2}{a+b x^3} \, dx}{b^2}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{\int \frac{\sqrt [3]{a} \left (a^{4/3} b^3 (b c+5 a f)+4 a^2 b^{7/3} (b e-7 a h)\right )+\sqrt [3]{b} \left (a^{4/3} b^3 (b c+5 a f)-2 a^2 b^{7/3} (b e-7 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{8/3} b^{16/3}}-\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^3}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 a^{5/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a b^3}+\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\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^{4/3} b^{10/3}}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 a^{5/3} h\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^{4/3} b^{10/3}}\\ &=\frac{h x}{b^3}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac{x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac{\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 a^{5/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac{\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.305199, size = 342, normalized size = 0.99 \[ \frac{-\frac{9 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} \left (a^2 (12 g+13 h x)-a b (6 d+x (7 e+8 f x))+2 b^2 c x^2\right )}{a \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{2/3} b^{4/3} e-14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}+18 b^{2/3} g \log \left (a+b x^3\right )+54 b^{2/3} h x}{54 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54*b^(2/3)*h*x - (9*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(
2*b^2*c*x^2 + a^2*(12*g + 13*h*x) - a*b*(6*d + x*(7*e + 8*f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + 2*a^(2
/3)*b^(4/3)*e + 5*a*b*f - 14*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*(b^2
*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2*c - 2*a^(
2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) + 18*
b^(2/3)*g*Log[a + b*x^3])/(54*b^(11/3))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 619, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 34.1865, size = 30286, normalized size = 87.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(108*a*b^2*h*x^7 + 12*(b^3*c - 4*a*b^2*f)*x^5 - 42*(a*b^2*e - 7*a^2*b*h)*x^4 - 18*a^2*b*d + 54*a^3*g - 3
6*(a*b^2*d - 2*a^2*b*g)*x^3 - 6*(a*b^2*c + 5*a^2*b*f)*x^2 - 2*(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*(2*(1/2)^(
2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(14
58*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3
*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h
^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f
^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) +
(1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/
(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h
+ 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f
*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)
*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)*log(-4*a*b^4*c^2*e - 40*a^2*b^3*c*e*f - 100*a^3*b^2*e*f^2 + 36*a^3*
b^2*e^2*g + 1764*a^5*g*h^2 - 1/4*(a^3*b^8*c + 5*a^4*b^7*f)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^
2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70
*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2
+ 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*
f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a
^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^
9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15
*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*
b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*
e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3
)^2 - 81*(a^3*b^2*c + 5*a^4*b*f)*g^2 + (2*a^3*b^5*e^2 - 28*a^4*b^4*e*h + 98*a^5*b^3*h^2 - 9*(a^3*b^5*c + 5*a^4
*b^4*f)*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h
)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) -
(b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4
*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*
e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3
)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5
*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^
3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3
- 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3
- 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3) + 28*(a^2*b^3*c^2 + 10*a^3*b^2*c*f + 25*a^4*b*
f^2 - 18*a^4*b*e*g)*h - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a
^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)*x) - 24*(a^2*b*e - 7*a^3*h)*x + (54*a*b^2*g*x^6 + 108*a^2*b*g*
x^3 + 54*a^3*g + (a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*
e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h
)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 12
5*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f +
2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b
^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 -
27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b
^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10
) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*
g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3) -
3*sqrt(1/3)*(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*
c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f
*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 +
125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f
+ 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3
*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9
- 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a
*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^
10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*
f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)^
2*a^2*b^6 + 36*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7
*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^
9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176
*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h +
392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2
*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 +
2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^
2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5
*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*
e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)*a^2*b^3*g + 32*b^2*c*e + 160*a*b*e*f + 324
*a^2*g^2 - 224*(a*b*c + 5*a^2*f)*h)/(a^2*b^6)))*log(4*a*b^4*c^2*e + 40*a^2*b^3*c*e*f + 100*a^3*b^2*e*f^2 - 36*
a^3*b^2*e^2*g - 1764*a^5*g*h^2 + 1/4*(a^3*b^8*c + 5*a^4*b^7*f)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (
2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2
- 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*
f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*
c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*
h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^
3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3
+ 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(
a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 -
270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g
/b^3)^2 + 81*(a^3*b^2*c + 5*a^4*b*f)*g^2 - (2*a^3*b^5*e^2 - 28*a^4*b^4*e*h + 98*a^5*b^3*h^2 - 9*(a^3*b^5*c + 5
*a^4*b^4*f)*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7
*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^
9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176
*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h +
392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2
*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 +
2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^
2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5
*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*
e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3) - 28*(a^2*b^3*c^2 + 10*a^3*b^2*c*f + 25*a^
4*b*f^2 - 18*a^4*b*e*g)*h - 2*(b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 -
168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)*x + 3/4*sqrt(1/3)*(8*a^3*b^5*e^2 - 112*a^4*b^4*e*h + 392
*a^5*b^3*h^2 + (a^3*b^8*c + 5*a^4*b^7*f)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 -
70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e
*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3
- 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 -
3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 -
3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e
+ (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*
a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 +
15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h
+ 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3) + 18*(a^3*b^5*c
+ 5*a^4*b^4*f)*g)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*
(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)
*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^
2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 63
0*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e
*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f
*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 +
125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f
+ 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3
*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)^2*a^2*b^6 + 36*(2*(1/2)^(2/3)*(-
I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/
b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 +
15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^
4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 27
0*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^
(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^
9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176
*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h +
392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2
*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)*a^2*b^3*g + 32*b^2*c*e + 160*a*b*e*f + 324*a^2*g^2 - 224*(a*b*c + 5*a^2*f)
*h)/(a^2*b^6))) + (54*a*b^2*g*x^6 + 108*a^2*b*g*x^3 + 54*a^3*g + (a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*(2*(1/2
)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/
(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*
b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^
5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (12
5*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3
) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)
*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^
2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 63
0*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e
*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3) + 3*sqrt(1/3)*(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)*sqrt(-((2*(1
/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6)
)/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^
2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*
a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (
125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1
/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*
b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*
e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 -
630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18
*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)^2*a^2*b^6 + 36*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (
2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2
- 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*
f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*
c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*
h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^
3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3
+ 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(
a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 -
270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g
/b^3)*a^2*b^3*g + 32*b^2*c*e + 160*a*b*e*f + 324*a^2*g^2 - 224*(a*b*c + 5*a^2*f)*h)/(a^2*b^6)))*log(4*a*b^4*c^
2*e + 40*a^2*b^3*c*e*f + 100*a^3*b^2*e*f^2 - 36*a^3*b^2*e^2*g - 1764*a^5*g*h^2 + 1/4*(a^3*b^8*c + 5*a^4*b^7*f)
*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^
2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3
+ 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 -
2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4
*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^1
0))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c
*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^
3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*
g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^
2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)^2 + 81*(a^3*b^2*c + 5*a^4*b*f)*g^2 - (2*a^3*b^5*e^2 - 28
*a^4*b^4*e*h + 98*a^5*b^3*h^2 - 9*(a^3*b^5*c + 5*a^4*b^4*f)*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (
2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2
- 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*
f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*
c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*
h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^
3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3
+ 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(
a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 -
270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g
/b^3) - 28*(a^2*b^3*c^2 + 10*a^3*b^2*c*f + 25*a^4*b*f^2 - 18*a^4*b*e*g)*h - 2*(b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*
b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)*x - 3/4*
sqrt(1/3)*(8*a^3*b^5*e^2 - 112*a^4*b^4*e*h + 392*a^5*b^3*h^2 + (a^3*b^8*c + 5*a^4*b^7*f)*(2*(1/2)^(2/3)*(-I*sq
rt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9
- 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a
*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^
10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*
f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3
)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) -
(b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4
*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*
e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3
)/(a^4*b^10))^(1/3) - 18*g/b^3) + 18*(a^3*b^5*c + 5*a^4*b^4*f)*g)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g
^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e
+ (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*
a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 +
15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h
+ 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1
)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^
2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*
a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (
125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1
/3) - 18*g/b^3)^2*a^2*b^6 + 36*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*g^2/b^6 - (2*b^2*c*e + (81*g^2 - 70*f*h)*a^
2 + 2*(5*e*f - 7*c*h)*a*b)/(a^2*b^6))/(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2 - 70*f*h)*a^2 + 2*(5*e*f - 7*c*h
)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 168*a^3*
b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*c^2*f + 2744*a^5*h^3 - 3*(243*g^
3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*h)*a^3*b^2 - (8*e^3 - 3*(25*f^2
- 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*g^3/b^9 - 27*(2*b^2*c*e + (81*g^2
- 70*f*h)*a^2 + 2*(5*e*f - 7*c*h)*a*b)*g/(a^2*b^9) - (b^5*c^3 + 8*a^2*b^3*e^3 + 15*a*b^4*c^2*f + 75*a^2*b^3*c*
f^2 + 125*a^3*b^2*f^3 - 168*a^3*b^2*e^2*h + 1176*a^4*b*e*h^2 - 2744*a^5*h^3)/(a^4*b^10) - (b^5*c^3 + 15*a*b^4*
c^2*f + 2744*a^5*h^3 - 3*(243*g^3 - 630*f*g*h + 392*e*h^2)*a^4*b + (125*f^3 - 270*e*f*g + 168*e^2*h + 378*c*g*
h)*a^3*b^2 - (8*e^3 - 3*(25*f^2 - 18*e*g)*c)*a^2*b^3)/(a^4*b^10))^(1/3) - 18*g/b^3)*a^2*b^3*g + 32*b^2*c*e + 1
60*a*b*e*f + 324*a^2*g^2 - 224*(a*b*c + 5*a^2*f)*h)/(a^2*b^6))))/(a*b^5*x^6 + 2*a^2*b^4*x^3 + a^3*b^3)

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

[Out]

Timed out

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Giac [A]  time = 1.08362, size = 541, normalized size = 1.57 \begin{align*} \frac{h x}{b^{3}} + \frac{g \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\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^{2} b^{4}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\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^{2} b^{4}} + \frac{2 \,{\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} +{\left (13 \, a^{2} b h - 7 \, a b^{2} e\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \,{\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} -{\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} + 2 \,{\left (5 \, a^{3} h - 2 \, a^{2} b e\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{3}} - \frac{{\left (a b^{6} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{3} b^{4} h + 2 \, a^{2} b^{5} e\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^{3} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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