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

Optimal. Leaf size=290 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} h+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.497935, antiderivative size = 288, 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 = {1823, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}-4 a g+b d\right )}{18 a^{2/3} b^{7/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} h+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1823

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Pq*(a + b*x^n)^(p + 1))/(b*n*(p + 1)),
x] - Dist[1/(b*n*(p + 1)), Int[D[Pq, x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x]
&& EqQ[m - n + 1, 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^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\int \frac{d+2 e x+3 f x^2+4 g x^3+5 h x^4}{a+b x^3} \, dx}{3 b}\\ &=-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\int \left (\frac{4 g}{b}+\frac{5 h x}{b}+\frac{b d-4 a g+(2 b e-5 a h) x+3 b f x^2}{b \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\int \frac{b d-4 a g+(2 b e-5 a h) x+3 b f x^2}{a+b x^3} \, dx}{3 b^2}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\int \frac{b d-4 a g+(2 b e-5 a h) x}{a+b x^3} \, dx}{3 b^2}+\frac{f \int \frac{x^2}{a+b x^3} \, dx}{b}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{f \log \left (a+b x^3\right )}{3 b^2}+\frac{\int \frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} (b d-4 a g)+\sqrt [3]{a} (2 b e-5 a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{b} (b d-4 a g)+\sqrt [3]{a} (2 b e-5 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{2/3} b^{7/3}}+\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^2}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 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 \sqrt [3]{a} b^{7/3}}-\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 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^{2/3} b^{7/3}}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 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^{2/3} b^{7/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 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^{2/3} b^{8/3}}\\ &=\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}-\frac{\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 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^{2/3} b^{8/3}}+\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{\left (b d-4 a g-\frac{\sqrt [3]{a} (2 b e-5 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^{2/3} b^{7/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.198048, size = 280, normalized size = 0.97 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e+4 a \sqrt [3]{b} g-b^{4/3} d\right )}{a^{2/3}}-\frac{6 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{a+b x^3}+6 b^{2/3} f \log \left (a+b x^3\right )+18 b^{2/3} g x+9 b^{2/3} h x^2}{18 b^{8/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 506, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*h*x^2/b^2+g*x/b^2+1/3/b^2/(b*x^3+a)*x^2*a*h-1/3/b*x^2/(b*x^3+a)*e+1/3/b^2/(b*x^3+a)*a*g*x-1/3/b*x/(b*x^3+a
)*d+1/3*a/b^2/(b*x^3+a)*f-1/3/b/(b*x^3+a)*c-4/9/b^3*a*g/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+2/9/b^3*a*g/(1/b*a)^
(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-4/9/b^3*a*g/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1
/3)*x-1))+1/9/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d-1/18/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2
/3))*d+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))*d+5/9/b^3*a*h/(1/b*a)^(1/3)*ln(
x+(1/b*a)^(1/3))-5/18/b^3*a*h/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-5/9/b^3*a*h*3^(1/2)/(1/b*a)^
(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-2/9/b^2*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/9/b^2*e/(1/b*a)^
(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/b^2*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3
)*x-1))+1/3*f*ln(b*x^3+a)/b^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(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 11.4732, size = 27702, normalized size = 95.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(18*b*h*x^5 + 36*b*g*x^4 - 6*(2*b*e - 5*a*h)*x^2 - 2*(b^3*x^3 + a*b^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9
*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2
*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 6
4*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3
- 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d
*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h
+ (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3
*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90
*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f
+ 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)*log(-8*a*b^3*d*e^2 + 3*a*b^3*d^2*f - 18*a^2*b^2*e*f^2 + 48*a^3*b
*f*g^2 - 1/4*(2*a^2*b^6*e - 5*a^3*b^5*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h
+ (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)
*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 15
0*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(
9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(
1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a
*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3
*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3
- 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3)
- 6*f/b^2)^2 - 50*(a^3*b*d - 4*a^4*g)*h^2 + 1/2*(a*b^5*d^2 - 12*a^2*b^4*e*f - 8*a^2*b^4*d*g + 16*a^3*b^3*g^2 +
30*a^3*b^3*f*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h
)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 +
8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*
h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^
2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sq
rt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b
^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/
(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h +
(16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2) + 8*(4*a^2*b^
2*e^2 - 3*a^2*b^2*d*f)*g + 5*(8*a^2*b^2*d*e + 9*a^3*b*f^2 - 32*a^3*b*e*g)*h - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^
3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)*x) - 12*b*c + 12
*a*f - 12*(b*d - 4*a*g)*x + (18*b*f*x^3 + (b^3*x^3 + a*b^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^
2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 -
8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60
*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75
*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*
a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g
- 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*
b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^
2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3
)/(a^2*b^8))^(1/3) - 6*f/b^2) + 18*a*f - 3*sqrt(1/3)*(b^3*x^3 + a*b^2)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*
(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a
^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 -
64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g
^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9
*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*
h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a
^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 -
90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*
f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)^2*a*b^5 + 12*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b
^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 -
8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 6
0*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 7
5*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)
*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*
g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2
*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h
^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^
3)/(a^2*b^8))^(1/3) - 6*f/b^2)*a*b^3*f + 32*b^2*d*e + 36*a*b*f^2 - 128*a*b*e*g - 80*(a*b*d - 4*a^2*g)*h)/(a*b^
5)))*log(8*a*b^3*d*e^2 - 3*a*b^3*d^2*f + 18*a^2*b^2*e*f^2 - 48*a^3*b*f*g^2 + 1/4*(2*a^2*b^6*e - 5*a^3*b^5*h)*(
2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(5
4*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a
*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (
b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15
*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3
/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*
d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d
^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)
*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)^2 + 50*(a^3*b*d - 4*a^4*g)*h^2
- 1/2*(a*b^5*d^2 - 12*a^2*b^4*e*f - 8*a^2*b^4*d*g + 16*a^3*b^3*g^2 + 30*a^3*b^3*f*h)*(2*(1/2)^(2/3)*(-I*sqrt(3
) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e
+ 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*
d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 -
2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*
e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20
*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2
- 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32
*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 -
9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2) - 8*(4*a^2*b^2*e^2 - 3*a^2*b^2*d*f)*g - 5*(8*a^2*b^2*d*
e + 9*a^3*b*f^2 - 32*a^3*b*e*g)*h - 2*(b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^
3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)*x + 3/4*sqrt(1/3)*(2*a*b^5*d^2 + 12*a^2*b^4*e*f - 16*a^2
*b^4*d*g + 32*a^3*b^3*g^2 - 30*a^3*b^3*f*h + (2*a^2*b^6*e - 5*a^3*b^5*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^
2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*
h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a
^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 -
90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*
f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (
9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*
g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*
g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6
*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2))*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 2
0*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5
*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*
e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a
^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a
^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)
*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h
+ 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b
+ 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^
8))^(1/3) - 6*f/b^2)^2*a*b^5 + 12*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^
2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b
^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b
*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 -
24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) +
(1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) -
(b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^
2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e
*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b
^2)*a*b^3*f + 32*b^2*d*e + 36*a*b*f^2 - 128*a*b*e*g - 80*(a*b*d - 4*a^2*g)*h)/(a*b^5))) + (18*b*f*x^3 + (b^3*x
^3 + a*b^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b
)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b
^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/
(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h +
(16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3)
+ 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^
3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*
b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*
g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2) + 18*a*f + 3*sqrt(
1/3)*(b^3*x^3 + a*b^2)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 -
8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7)
- (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h
^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*
e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2
)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^
4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 -
125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g
+ 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)^
2*a*b^5 + 12*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*
b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*
b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)
/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h
+ (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3
) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e
^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2
*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16
*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)*a*b^3*f + 32*b^2*
d*e + 36*a*b*f^2 - 128*a*b*e*g - 80*(a*b*d - 4*a^2*g)*h)/(a*b^5)))*log(8*a*b^3*d*e^2 - 3*a*b^3*d^2*f + 18*a^2*
b^2*e*f^2 - 48*a^3*b*f*g^2 + 1/4*(2*a^2*b^6*e - 5*a^3*b^5*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b
^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 -
8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 6
0*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 7
5*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)
*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*
g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2
*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h
^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^
3)/(a^2*b^8))^(1/3) - 6*f/b^2)^2 + 50*(a^3*b*d - 4*a^4*g)*h^2 - 1/2*(a*b^5*d^2 - 12*a^2*b^4*e*f - 8*a^2*b^4*d*
g + 16*a^3*b^3*g^2 + 30*a^3*b^3*f*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9
*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(
a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^
3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^
3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3)
+ (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7
) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e
*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 2
4*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*
f/b^2) - 8*(4*a^2*b^2*e^2 - 3*a^2*b^2*d*f)*g - 5*(8*a^2*b^2*d*e + 9*a^3*b*f^2 - 32*a^3*b*e*g)*h - 2*(b^4*d^3 +
8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4
*h^3)*x - 3/4*sqrt(1/3)*(2*a*b^5*d^2 + 12*a^2*b^4*e*f - 16*a^2*b^4*d*g + 32*a^3*b^3*g^2 - 30*a^3*b^3*f*h + (2*
a^2*b^6*e - 5*a^3*b^5*h)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g
- 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^
4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 -
125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g
+ 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/
3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3
+ 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a
^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20
*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2))*sqrt
(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5)
)/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 -
12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8)
+ (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2
- 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54
*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*
b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b
^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*
f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)^2*a*b^5 + 12*(2*(1/2)^(2/3)
*(-I*sqrt(3) + 1)*(9*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9
*(2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g +
48*a^2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125
*a^4*h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*
b^2 - 2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b
^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^
2*b^2*d*g^2 - 64*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*
h^3 - 2*(32*g^3 - 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 -
2*(4*e^3 - 9*d*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) - 6*f/b^2)*a*b^3*f + 32*b^2*d*e + 36*a*b*f^2 - 128*a*b*
e*g - 80*(a*b*d - 4*a^2*g)*h)/(a*b^5))))/(b^3*x^3 + a*b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.09203, size = 444, normalized size = 1.53 \begin{align*} \frac{f \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{{\left (a h - b e\right )} x^{2} - b c + a f -{\left (b d - a g\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - 2 \, \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 b^{4}} + \frac{b^{2} h x^{2} + 2 \, b^{2} g x}{2 \, b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + 2 \, \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 b^{4}} + \frac{{\left (5 \, a b^{3} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - b^{4} d + 4 \, a b^{3} 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 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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