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

Optimal. Leaf size=289 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{18 a^{5/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.559415, antiderivative size = 287, normalized size of antiderivative = 0.99, number of steps used = 11, 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} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (2 a h+b e)}{\sqrt [3]{b}}+a g+2 b d\right )}{18 a^{5/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.197893, size = 269, normalized size = 0.93 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e-a \sqrt [3]{b} g-2 b^{4/3} d\right )}{b^{5/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{b^{5/3}}-\frac{6 a (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )}-6 c \log \left (a+b x^3\right )+18 c \log (x)}{18 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.012, size = 507, 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((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^2,x)

[Out]

-1/3/(b*x^3+a)/b*x^2*h+1/3/a*x^2/(b*x^3+a)*e-1/3/(b*x^3+a)/b*x*g+1/3/a*x/(b*x^3+a)*d-1/3/b/(b*x^3+a)*f+1/3/a/(
b*x^3+a)*c+1/9/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*g+2/9/b/a*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-1/18/b^2/(1
/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*g-1/9/b/a*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/
3))+1/9/b^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*g+2/9/b/a*d/(1/b*a)^(2/3)*3^(1/2)*
arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-2/9/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*h-1/9/a/b/(1/b*a)^(1/3)*ln
(x+(1/b*a)^(1/3))*e+1/9/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*h+1/18/a/b/(1/b*a)^(1/3)*ln(x^
2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e+2/9/b^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*h+1
/9/a/b*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e-1/3*c*ln(b*x^3+a)/a^2+c*ln(x)/a^2

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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/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 156.896, size = 28260, normalized size = 97.79 \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/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/324*(108*a*b*c - 108*a^2*f + 108*(a*b*e - a^2*h)*x^2 - 2*(a^2*b^2*x^3 + a^3*b)*((-I*sqrt(3) + 1)*(9*c^2/a^4
- (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2
*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6
*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*
a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*
a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 +
2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g +
6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 +
8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c
)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)*log(12*b^4*c*d^2 + 9*b^4*c^2*e + 4*a*b^3*d
*e^2 + 3*a^2*b^2*c*g^2 + 1/324*(a^4*b^4*e + 2*a^5*b^3*h)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d
*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h +
(e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g
^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^
2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c
*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h
+ (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b
*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*
h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9
*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)^2 + 8*(2*a^3*b*d + a^4*g)*h^2 - 1/18*(4*a^2*b^4*d^2 + 6*a^2*b^4*c*
e + 4*a^3*b^3*d*g + a^4*b^2*g^2 + 12*a^3*b^3*c*h)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*
a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g +
4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*
a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*
b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a
*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g
+ 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 +
6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^
4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)
*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2) + 2*(6*a*b^3*c*d + a^2*b^2*e^2)*g + 2*(9*a*b^3*c^2 + 8*a^2*b^2*d*e + 4*a^
3*b*e*g)*h + (8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*
b*e*h^2 + 8*a^4*h^3)*x) + 108*(a*b*d - a^2*g)*x - (162*b^2*c*x^3 + 162*a*b*c - (a^2*b^2*x^3 + a^3*b)*((-I*sqrt
(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 +
1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3
+ 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1
458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g +
9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6
+ 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^
3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1
/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g
+ 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2) - 3*sqrt(1/3)*(a^2*b^2
*x^3 + a^3*b)*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b
)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) +
1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^
2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*
h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(
I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3)
+ 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*
h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*
g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54
*c/a^2)^2*a^4*b^3 - 108*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^
2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3)
+ 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e
*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c
*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 8
1*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^
3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b
*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3
*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) +
54*c/a^2)*a^2*b^3*c + 2916*b^3*c^2 + 2592*a*b^2*d*e + 1296*a^2*b*e*g + 2592*(2*a^2*b*d + a^3*g)*h)/(a^4*b^3))
)*log(-12*b^4*c*d^2 - 9*b^4*c^2*e - 4*a*b^3*d*e^2 - 3*a^2*b^2*c*g^2 - 1/324*(a^4*b^4*e + 2*a^5*b^3*h)*((-I*sqr
t(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 +
1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3
+ 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/
1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g
+ 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6
+ 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e
^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) -
1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*
g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)^2 - 8*(2*a^3*b*d + a^
4*g)*h^2 + 1/18*(4*a^2*b^4*d^2 + 6*a^2*b^4*c*e + 4*a^3*b^3*d*g + a^4*b^2*g^2 + 12*a^3*b^3*c*h)*((-I*sqrt(3) +
1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*
(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a
*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(2
7*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*
g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/16
2*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12
*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*
(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(
e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2) - 2*(6*a*b^3*c*d + a^2*b^2*e
^2)*g - 2*(9*a*b^3*c^2 + 8*a^2*b^2*d*e + 4*a^3*b*e*g)*h + 2*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^
2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)*x + 1/108*sqrt(1/3)*(72*a^2*b^4*d^2 - 54*a
^2*b^4*c*e + 72*a^3*b^3*d*g + 18*a^4*b^2*g^2 - 108*a^3*b^3*c*h + (a^4*b^4*e + 2*a^5*b^3*h)*((-I*sqrt(3) + 1)*(
9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b
^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3
*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^
5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g +
4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9
*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b
^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*
b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g
+ 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2))*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2
/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^
2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*
g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3
+ 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h
)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*
c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^
2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c
^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d
*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)^2*a^4*b^3 - 108*((-I*sqrt(3) + 1)*(9*
c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3
*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d
^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*
c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*
d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b
^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3
*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^
5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g +
4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)*a^2*b^3*c + 2916*b^3*c^2 + 2592*a*
b^2*d*e + 1296*a^2*b*e*g + 2592*(2*a^2*b*d + a^3*g)*h)/(a^4*b^3))) - (162*b^2*c*x^3 + 162*a*b*c - (a^2*b^2*x^3
+ a^3*b)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3)
)/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b
^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h
^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2
+ (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) +
1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8
*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4
*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^
2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2) + 3
*sqrt(1/3)*(a^2*b^2*x^3 + a^3*b)*sqrt(-(((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h +
(e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^
2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^
2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g
^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6
*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*
a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*
e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d
*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a
^6*b^5))^(1/3) + 54*c/a^2)^2*a^4*b^3 - 108*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h
+ (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)
*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2
*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(
d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(
a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*
h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b
^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6
*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)
/(a^6*b^5))^(1/3) + 54*c/a^2)*a^2*b^3*c + 2916*b^3*c^2 + 2592*a*b^2*d*e + 1296*a^2*b*e*g + 2592*(2*a^2*b*d + a
^3*g)*h)/(a^4*b^3)))*log(-12*b^4*c*d^2 - 9*b^4*c^2*e - 4*a*b^3*d*e^2 - 3*a^2*b^2*c*g^2 - 1/324*(a^4*b^4*e + 2*
a^5*b^3*h)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3
))/(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*
b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*
h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2
+ (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3)
+ 1)*(-1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(
8*b^4*d^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^
4*h^3)/(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b
^2 + (e^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)^2
- 8*(2*a^3*b*d + a^4*g)*h^2 + 1/18*(4*a^2*b^4*d^2 + 6*a^2*b^4*c*e + 4*a^3*b^3*d*g + a^4*b^2*g^2 + 12*a^3*b^3*c
*h)*((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/
27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3
+ a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a
^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3
- 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-
1/27*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d
^3 + a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/
(a^5*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e
^3 - 12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2) - 2*(6*a*
b^3*c*d + a^2*b^2*e^2)*g - 2*(9*a*b^3*c^2 + 8*a^2*b^2*d*e + 4*a^3*b*e*g)*h + 2*(8*b^4*d^3 + a*b^3*e^3 + 12*a*b
^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)*x - 1/108*sqrt(1/3)*(72
*a^2*b^4*d^2 - 54*a^2*b^4*c*e + 72*a^3*b^3*d*g + 18*a^4*b^2*g^2 - 108*a^3*b^3*c*h + (a^4*b^4*e + 2*a^5*b^3*h)*
((-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c
^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a
*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b
^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 1
2*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27
*c^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 +
a*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5
*b^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 -
12*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2))*sqrt(-(((-I*
sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3/a^
6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^3*
e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5) -
1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d^2
*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c^3/
a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b^
3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5)
- 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*d
^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)^2*a^4*b^3 - 108*((
-I*sqrt(3) + 1)*(9*c^2/a^4 - (9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)/(a^4*b^3))/(-1/27*c^3
/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a*b
^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b^5
) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 12*
d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*c
^3/a^6 + 1/162*(9*b^3*c^2 + 2*a*b^2*d*e + 2*a^3*g*h + (e*g + 4*d*h)*a^2*b)*c/(a^6*b^3) + 1/1458*(8*b^4*d^3 + a
*b^3*e^3 + 12*a*b^3*d^2*g + 6*a^2*b^2*d*g^2 + a^3*b*g^3 + 6*a^2*b^2*e^2*h + 12*a^3*b*e*h^2 + 8*a^4*h^3)/(a^5*b
^5) - 1/1458*(27*b^5*c^3 + 8*a^5*h^3 - (g^3 - 12*e*h^2)*a^4*b - 6*(d*g^2 - e^2*h - 3*c*g*h)*a^3*b^2 + (e^3 - 1
2*d^2*g + 9*(e*g + 4*d*h)*c)*a^2*b^3 - 2*(4*d^3 - 9*c*d*e)*a*b^4)/(a^6*b^5))^(1/3) + 54*c/a^2)*a^2*b^3*c + 291
6*b^3*c^2 + 2592*a*b^2*d*e + 1296*a^2*b*e*g + 2592*(2*a^2*b*d + a^3*g)*h)/(a^4*b^3))) + 324*(b^2*c*x^3 + a*b*c
)*log(x))/(a^2*b^2*x^3 + a^3*b)

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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/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.08568, size = 452, normalized size = 1.56 \begin{align*} -\frac{c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{c \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\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 )}{9 \, a^{2} b^{3}} + \frac{a b c - a^{2} f -{\left (a^{2} h - a b e\right )} x^{2} +{\left (a b d - a^{2} g\right )} x}{3 \,{\left (b x^{3} + a\right )} a^{2} b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} - \frac{{\left (2 \, a^{4} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 2 \, a^{3} b^{3} d + a^{4} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b^{3}} \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/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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