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

Optimal. Leaf size=297 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.430108, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1823, 1858, 1860, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2} \]

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)^3,x]

[Out]

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

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 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 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]

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

Mathematica [A]  time = 0.234061, size = 287, normalized size = 0.97 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e-2 a \sqrt [3]{b} g-b^{4/3} d\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^{4/3} h-\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{5/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{5/3}}-\frac{9 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} (b x (d+2 e x)-a (6 f+x (7 g+8 h x)))}{a \left (a+b x^3\right )}}{54 b^{8/3}} \]

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)^3,x]

[Out]

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

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Maple [A]  time = 0.011, size = 490, 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)^3,x)

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 19.2287, size = 15354, normalized size = 51.7 \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)^3,x, algorithm="fricas")

[Out]

-1/108*(36*a*b*f*x^3 - 12*(b^2*e - 4*a*b*h)*x^5 - 6*(b^2*d - 7*a*b*g)*x^4 + 18*a*b*c + 18*a^2*f + 6*(a*b*e + 5
*a^2*h)*x^2 + 2*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a
*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^
4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b
^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 +
a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/
(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g
)*a*b^3)/(a^5*b^8))^(1/3)))*log(2*a*b^3*d*e^2 + 4*a^2*b^2*e^2*g + 1/4*(a^4*b^6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(
I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*
a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*
h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a
*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*
b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4
*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2 + 50*(a^3*b*d + 2*a^4*g)*h^2 - 1/2*(a^
2*b^5*d^2 + 4*a^3*b^4*d*g + 4*a^4*b^3*g^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g
+ 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125
*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)
- 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 +
6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) +
(b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a
^5*b^8))^(1/3))) + 20*(a^2*b^2*d*e + 2*a^3*b*e*g)*h + (b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2
+ 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)*x) + 12*(a*b*d + 2*a^2*g)*x - ((a*b^4*x^6 + 2
*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2
+ 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 -
75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^
2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12
*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4
*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))) +
3*sqrt(1/3)*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 +
6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) +
(b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^
5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3
+ a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^
3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^
2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2*a^3*b^5 + 16*b^2*d*e + 32*a*b*e*g + 80*(a*b*d + 2*a^2*g)*h)/(a^3*b^5)))*log(-
2*a*b^3*d*e^2 - 4*a^2*b^2*e^2*g - 1/4*(a^4*b^6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3
*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*
b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b
^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((
b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125
*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3
- 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2 - 50*(a^3*b*d + 2*a^4*g)*h^2 + 1/2*(a^2*b^5*d^2 + 4*a^3*b^4*d*g + 4*a^
4*b^3*g^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3
+ 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^
3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^
2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^
2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3
- 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))) - 20*(a^2*b^2*d
*e + 2*a^3*b*e*g)*h + 2*(b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2
*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)*x + 3/4*sqrt(1/3)*(2*a^2*b^5*d^2 + 8*a^3*b^4*d*g + 8*a^4*b^3*g^2 + (a^4*b^6
*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^
3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*
h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e
+ 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b
^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 +
(8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))))*sqrt(-(
((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b
^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*
d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*
e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b
*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2
)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2*a^3*b^5 + 16*b^2*d*e + 3
2*a*b*e*g + 80*(a*b*d + 2*a^2*g)*h)/(a^3*b^5))) - ((a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(
3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e
*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*
b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I
*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2
*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2
- 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))) - 3*sqrt(1/3)*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*
b^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^
3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a
^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a
^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g
^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^
3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2*a^3*b^5 + 16
*b^2*d*e + 32*a*b*e*g + 80*(a*b*d + 2*a^2*g)*h)/(a^3*b^5)))*log(-2*a*b^3*d*e^2 - 4*a^2*b^2*e^2*g - 1/4*(a^4*b^
6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a
^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e
*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e
+ 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*
b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3
+ (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2 - 50*
(a^3*b*d + 2*a^4*g)*h^2 + 1/2*(a^2*b^5*d^2 + 4*a^3*b^4*d*g + 4*a^4*b^3*g^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4
*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^
4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 -
6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)
/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b
*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^
2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))) - 20*(a^2*b^2*d*e + 2*a^3*b*e*g)*h + 2*(b^4*d^3 + a*b^3*e^3
+ 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)*x - 3/4*sq
rt(1/3)*(2*a^2*b^5*d^2 + 8*a^3*b^4*d*g + 8*a^4*b^3*g^2 + (a^4*b^6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(I*sqrt(3) + 1
)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 +
125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 -
(e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(
3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 7
5*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^
2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))))*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^
3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5
*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*
b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*(
(b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 12
5*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^
3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2*a^3*b^5 + 16*b^2*d*e + 32*a*b*e*g + 80*(a*b*d + 2*a^2*g)*h)/(a^3*b^5)
)))/(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.12213, size = 454, normalized size = 1.53 \begin{align*} -\frac{{\left (5 \, a h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + b d + 2 \, a g\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^{2} b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 5 \, \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 )}{27 \, a^{2} b^{4}} - \frac{8 \, a b h x^{5} - 2 \, b^{2} x^{5} e - b^{2} d x^{4} + 7 \, a b g x^{4} + 6 \, a b f x^{3} + 5 \, a^{2} h x^{2} + a b x^{2} e + 2 \, a b d x + 4 \, a^{2} g x + 3 \, a b c + 3 \, a^{2} f}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 5 \, \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 )}{54 \, a^{2} b^{4}} \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)^3,x, algorithm="giac")

[Out]

-1/27*(5*a*h*(-a/b)^(1/3) + b*(-a/b)^(1/3)*e + b*d + 2*a*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2)
+ 1/27*sqrt(3)*((-a*b^2)^(1/3)*b^2*d + 2*(-a*b^2)^(1/3)*a*b*g - 5*(-a*b^2)^(2/3)*a*h - (-a*b^2)^(2/3)*b*e)*arc
tan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) - 1/18*(8*a*b*h*x^5 - 2*b^2*x^5*e - b^2*d*x^4 + 7
*a*b*g*x^4 + 6*a*b*f*x^3 + 5*a^2*h*x^2 + a*b*x^2*e + 2*a*b*d*x + 4*a^2*g*x + 3*a*b*c + 3*a^2*f)/((b*x^3 + a)^2
*a*b^2) + 1/54*((-a*b^2)^(1/3)*b^2*d + 2*(-a*b^2)^(1/3)*a*b*g + 5*(-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^4)