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

Optimal. Leaf size=325 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac{h \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

-(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*b^2*(a + b*x^3)^2) + (x*(b*c - 7*a*f + 2*(b*d - 4*a*g)*x
+ 3*(b*e - 3*a*h)*x^2))/(18*a*b^2*(a + b*x^3)) - ((b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*Arc
Tan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(8/3)) + ((b^(1/3)*(b*c + 2*a*f) - a^(1/3
)*(b*d + 5*a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(8/3)) - ((b^(1/3)*(b*c + 2*a*f) - a^(1/3)*(b*d + 5*a
*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(8/3)) + (h*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 0.642445, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1828, 1858, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 b^2 \left (a+b x^3\right )^2}+\frac{h \log \left (a+b x^3\right )}{3 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.253258, size = 315, normalized size = 0.97 \[ \frac{-\frac{9 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{\left (a+b x^3\right )^2}+\frac{36 a^2 h-3 a b (6 e+x (7 f+8 g x))+3 b^2 x (c+2 d x)}{a \left (a+b x^3\right )}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d-2 a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{5/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^{4/3} g-\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{5/3}}+18 h \log \left (a+b x^3\right )}{54 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 + (36*a^2*h + 3*b^2*x*(c + 2*d*x) - 3*a*
b*(6*e + x*(7*f + 8*g*x)))/(a*(a + b*x^3)) - (2*Sqrt[3]*b^(1/3)*(b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a
^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d + 2*a*b^(
1/3)*f - 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-(b^(4/3)*c) + a^(1/3)*b*d - 2*a*b^(1/3)*f
+ 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 18*h*Log[a + b*x^3])/(54*b^3)

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Maple [A]  time = 0.012, size = 515, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/9*(4*a*g-b*d)/a/b*x^5-1/18*(7*a*f-b*c)/a/b*x^4+1/3*(2*a*h-b*e)/b^2*x^3-1/18*(5*a*g+b*d)/b^2*x^2-1/9*(2*a*f
+b*c)/b^2*x+1/6*a*(3*a*h-b*e)/b^3)/(b*x^3+a)^2+2/27/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f+1/27/b^2/a/(1/b*a)
^(2/3)*ln(x+(1/b*a)^(1/3))*c-1/27/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f-1/54/b^2/a/(1/b*a)
^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+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))*f+1/27/b^2/a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-5/27/b^3/(1/b*a)^(1/3
)*ln(x+(1/b*a)^(1/3))*g-1/27/b^2/a/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*d+5/54/b^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(
1/3)*x+(1/b*a)^(2/3))*g+1/54/b^2/a/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*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))*g+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))*d+1/3*h*ln(b*x^3+a)/b^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(x^3*(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 = 28.7256, size = 29083, normalized size = 89.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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*(12*(b^3*d - 4*a*b^2*g)*x^5 + 6*(b^3*c - 7*a*b^2*f)*x^4 - 18*a^2*b*e + 54*a^3*h - 36*(a*b^2*e - 2*a^2*b*
h)*x^3 - 6*(a*b^2*d + 5*a^2*b*g)*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*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^
3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f
+ 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729
*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4
*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9
- 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b
^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*
c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d
^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)*log(2*a*b^4*c*d^2 + 4*
a^2*b^3*d^2*f + 1/4*(a^4*b^7*d + 5*a^5*b^6*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*
b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2
+ (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3
+ 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*
a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 -
6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 8
1*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a
^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 5
4*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3
- (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2 + 50*(a^3*b^2*c + 2*a^4*b*f)*g^2 + 81*(a^4*b*d + 5*a^
5*g)*h^2 - 1/2*(a^2*b^6*c^2 + 4*a^3*b^5*c*f + 4*a^4*b^4*f^2 - 18*(a^4*b^4*d + 5*a^5*b^3*g)*h)*(2*(1/2)^(2/3)*(
-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h
^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 +
6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) +
(b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 -
3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1
)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*
b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a
^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a
^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3) + 20*(a
^2*b^3*c*d + 2*a^3*b^2*d*f)*g - 9*(a^2*b^3*c^2 + 4*a^3*b^2*c*f + 4*a^4*b*f^2)*h + (b^5*c^3 + a*b^4*d^3 + 6*a*b
^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 + 15*a^2*b^3*d^2*g + 75*a^3*b^2*d*g^2 + 125*a^4*b*g^3)*x) - 12*(a*
b^2*c + 2*a^2*b*f)*x + (54*a*b^2*h*x^6 + 108*a^2*b*h*x^3 + 54*a^3*h + (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*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3
*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3
+ a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^
3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)
*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*
(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) +
(b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 1
25*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2
- 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*
h/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*h^2/b^6
- (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^
2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*
c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*
(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)
*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*
d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12
*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5
*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2
+ 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2*a^3*b^6 + 36*(2*(1/2)^(2/3)*(-I*s
qrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b
^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a
*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^
5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5
*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1
458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*
d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b
^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b
^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)*a^3*b^3*h +
16*b^3*c*d + 32*a*b^2*d*f + 324*a^3*h^2 + 80*(a*b^2*c + 2*a^2*b*f)*g)/(a^3*b^6)))*log(-2*a*b^4*c*d^2 - 4*a^2*
b^3*d^2*f - 1/4*(a^4*b^7*d + 5*a^5*b^6*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*
g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (
2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15
*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*
b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c
^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^
3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b
*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*
g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (
d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2 - 50*(a^3*b^2*c + 2*a^4*b*f)*g^2 - 81*(a^4*b*d + 5*a^5*g)
*h^2 + 1/2*(a^2*b^6*c^2 + 4*a^3*b^5*c*f + 4*a^4*b^4*f^2 - 18*(a^4*b^4*d + 5*a^5*b^3*g)*h)*(2*(1/2)^(2/3)*(-I*s
qrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b
^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a
*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^
5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5
*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1
458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*
d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b
^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b
^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3) - 20*(a^2*b
^3*c*d + 2*a^3*b^2*d*f)*g + 9*(a^2*b^3*c^2 + 4*a^3*b^2*c*f + 4*a^4*b*f^2)*h + 2*(b^5*c^3 + a*b^4*d^3 + 6*a*b^4
*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 + 15*a^2*b^3*d^2*g + 75*a^3*b^2*d*g^2 + 125*a^4*b*g^3)*x + 3/4*sqrt(
1/3)*(2*a^2*b^6*c^2 + 8*a^3*b^5*c*f + 8*a^4*b^4*f^2 + (a^4*b^7*d + 5*a^5*b^6*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1
)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b
^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f
+ 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 72
9*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (
4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^
9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*
b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5
*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*
d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3) + 18*(a^4*b^4*d + 5*a
^5*b^3*g)*h)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*
f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*
h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^
3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h
- 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9)
)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g
)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*
g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 +
135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/
(a^5*b^9))^(1/3) - 18*h/b^3)^2*a^3*b^6 + 36*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*
f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 +
(2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 +
15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^
4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6
*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*
a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3
*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*
f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 -
(d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)*a^3*b^3*h + 16*b^3*c*d + 32*a*b^2*d*f + 324*a^3*h^2 + 80*
(a*b^2*c + 2*a^2*b*f)*g)/(a^3*b^6))) + (54*a*b^2*h*x^6 + 108*a^2*b*h*x^3 + 54*a^3*h + (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*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*
c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3
*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*
g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(
25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3
) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^
2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75
*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*
g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b
^9))^(1/3) - 18*h/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*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*
(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2
*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 +
729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g -
(4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/
b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*
a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b
^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(
5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2*a^3*b^6 + 36*(2*(
1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b
^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 +
a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)
/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d
)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I
*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (
b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125
*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 -
18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/
b^3)*a^3*b^3*h + 16*b^3*c*d + 32*a*b^2*d*f + 324*a^3*h^2 + 80*(a*b^2*c + 2*a^2*b*f)*g)/(a^3*b^6)))*log(-2*a*b^
4*c*d^2 - 4*a^2*b^3*d^2*f - 1/4*(a^4*b^7*d + 5*a^5*b^6*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c
*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g
+ 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 +
8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3
- 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*
b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a
^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2
*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5
*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h
)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2 - 50*(a^3*b^2*c + 2*a^4*b*f)*g^2 - 81*(a^
4*b*d + 5*a^5*g)*h^2 + 1/2*(a^2*b^6*c^2 + 4*a^3*b^5*c*f + 4*a^4*b^4*f^2 - 18*(a^4*b^4*d + 5*a^5*b^3*g)*h)*(2*(
1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b
^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 +
a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)
/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d
)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I
*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (
b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125
*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 -
18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/
b^3) - 20*(a^2*b^3*c*d + 2*a^3*b^2*d*f)*g + 9*(a^2*b^3*c^2 + 4*a^3*b^2*c*f + 4*a^4*b*f^2)*h + 2*(b^5*c^3 + a*b
^4*d^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 + 15*a^2*b^3*d^2*g + 75*a^3*b^2*d*g^2 + 125*a^4*b*g^
3)*x - 3/4*sqrt(1/3)*(2*a^2*b^6*c^2 + 8*a^3*b^5*c*f + 8*a^4*b^4*f^2 + (a^4*b^7*d + 5*a^5*b^6*g)*(2*(1/2)^(2/3)
*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458
*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3
+ 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8)
+ (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2
- 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) +
1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 +
a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/
(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)
*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3) + 18*
(a^4*b^4*d + 5*a^5*b^3*g)*h)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3*c*d + 10*a^2*b*f*g + 81
*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f
+ 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b
^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a^4*b + (8
*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*
a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*g + 81*a^3*h^2
+ (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2 + 8*a^3*b*f^3 +
15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^3 - 54*f*g*h)*a
^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^2*b^3 - (d^3 -
6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)^2*a^3*b^6 + 36*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(81*h^2/b^6 - (b^3
*c*d + 10*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)/(a^3*b^6))/(1458*h^3/b^9 - 27*(b^3*c*d + 10*a^2*b*f*
g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b^2*c*f^2
+ 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 - 5*(25*g^
3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d*h)*c)*a^
2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(1458*h^3/b^9 - 27*(b^3*c*d + 10
*a^2*b*f*g + 81*a^3*h^2 + (2*d*f + 5*c*g)*a*b^2)*h/(a^3*b^9) + (b^4*c^3 + a*b^3*d^3 + 6*a*b^3*c^2*f + 12*a^2*b
^2*c*f^2 + 8*a^3*b*f^3 + 15*a^2*b^2*d^2*g + 75*a^3*b*d*g^2 + 125*a^4*g^3)/(a^5*b^8) + (b^5*c^3 + 729*a^5*h^3 -
5*(25*g^3 - 54*f*g*h)*a^4*b + (8*f^3 + 135*c*g*h - 3*(25*g^2 - 18*f*h)*d)*a^3*b^2 - 3*(5*d^2*g - (4*f^2 + 9*d
*h)*c)*a^2*b^3 - (d^3 - 6*c^2*f)*a*b^4)/(a^5*b^9))^(1/3) - 18*h/b^3)*a^3*b^3*h + 16*b^3*c*d + 32*a*b^2*d*f + 3
24*a^3*h^2 + 80*(a*b^2*c + 2*a^2*b*f)*g)/(a^3*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**3*(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 = 1.08958, size = 512, normalized size = 1.58 \begin{align*} \frac{h \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d - 5 \, \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^{2} b^{4}} + \frac{2 \,{\left (b^{2} d - 4 \, a b g\right )} x^{5} +{\left (b^{2} c - 7 \, a b f\right )} x^{4} + 6 \,{\left (2 \, a^{2} h - a b e\right )} x^{3} -{\left (a b d + 5 \, a^{2} g\right )} x^{2} - 2 \,{\left (a b c + 2 \, a^{2} f\right )} x + \frac{3 \,{\left (3 \, a^{3} h - a^{2} b e\right )}}{b}}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d + 5 \, \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^{2} b^{4}} - \frac{{\left (a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{3} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{4} c + 2 \, a^{2} b^{3} 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^{3} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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