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

Optimal. Leaf size=347 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{54 a^{8/3} b^{5/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {c \log (x)}{a^3}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.72, antiderivative size = 345, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{54 a^{8/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {c \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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]

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 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 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 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 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 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]

Rubi steps

\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^2 c-b (5 b d+a g) x-2 b (2 b e+a h) x^2+3 b^2 \left (\frac {b c}{a}-f\right ) x^3}{x \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+2 b^2 (5 b d+a g) x+2 b^2 (2 b e+a h) x^2}{x \left (a+b x^3\right )} \, dx}{18 a^2 b^3}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x}+\frac {2 b^2 \left (a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {a (5 b d+a g)+a (2 b e+a h) x-9 b^2 c x^2}{a+b x^3} \, dx}{9 a^3 b}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\int \frac {a (5 b d+a g)+a (2 b e+a h) x}{a+b x^3} \, dx}{9 a^3 b}-\frac {(b c) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (5 b d+a g)+a^{4/3} (2 b e+a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} b^{4/3}}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3} b}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+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^{7/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+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}{54 a^{8/3} b^{4/3}}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}+\frac {\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+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^{8/3} b^{5/3}}\\ &=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+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^{8/3} b^{5/3}}+\frac {c \log (x)}{a^3}+\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac {\left (5 b d+a g-\frac {\sqrt [3]{a} (2 b e+a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 311, normalized size = 0.90 \[ \frac {\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{5/3}}+\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac {2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac {9 a^2 (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )^2}+\frac {3 a (a x (g+2 h x)+6 b c+b x (5 d+4 e x))}{b \left (a+b x^3\right )}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [C]  time = 55.84, size = 12815, normalized size = 36.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2916*(972*a*b^2*c*x^3 + 324*(2*a*b^2*e + a^2*b*h)*x^5 + 162*(5*a*b^2*d + a^2*b*g)*x^4 + 1458*a^2*b*c - 486*a
^3*f + 162*(7*a^2*b*e - a^3*h)*x^2 - 2*(a^3*b^3*x^6 + 2*a^4*b^2*x^3 + a^5*b)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (
81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 +
10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^
2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*
c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*
g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1
/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b
^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5)
- 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 -
75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)*log(225
*b^4*c*d^2 + 162*b^4*c^2*e + 40*a*b^3*d*e^2 + 9*a^2*b^2*c*g^2 + 1/2916*(2*a^6*b^4*e + a^7*b^3*h)*((-I*sqrt(3)
+ 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1
/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b
^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5)
- 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 -
75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1
)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*
(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2
+ a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h
)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3)
+ 486*c/a^3)^2 + 2*(5*a^3*b*d + a^4*g)*h^2 - 1/54*(25*a^3*b^4*d^2 + 36*a^3*b^4*c*e + 10*a^4*b^3*d*g + a^5*b^2*
g^2 + 18*a^4*b^3*c*h)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a
^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*
b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h +
6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*
e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a
^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5
*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3
+ 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4
*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 -
54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3) + 2*(45*a*b^3*c*d + 4*a^2*b^2*e^2)*g + (81*a*b^3*c^2 + 40*a^2*
b^2*d*e + 8*a^3*b*e*g)*h + (125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2
*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)*x) + 324*(4*a^2*b*d - a^3*g)*x - (1458*b^3*c*x^6 + 2916*a*b^2*c*x^3 + 14
58*a^2*b*c - (a^3*b^3*x^6 + 2*a^4*b^2*x^3 + a^5*b)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e
+ a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (
2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a
^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e
*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*
(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b
^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g +
15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 +
a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*
d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3) - 3*sqrt(1/3)*(a^3*b^3*x^6 + 2*a^
4*b^2*x^3 + a^5*b)*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*
h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(
a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2
*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2
- 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4
)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g
+ 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*
g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)
*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d
^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)^2*a^6*b^3 - 972*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^
2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*
d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*
a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5
*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h
)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81
*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 +
75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/3936
6*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g
+ 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)*a^3*b^3*c + 2361
96*b^3*c^2 + 116640*a*b^2*d*e + 23328*a^2*b*e*g + 11664*(5*a^2*b*d + a^3*g)*h)/(a^6*b^3)))*log(-225*b^4*c*d^2
- 162*b^4*c^2*e - 40*a*b^3*d*e^2 - 9*a^2*b^2*c*g^2 - 1/2916*(2*a^6*b^4*e + a^7*b^3*h)*((-I*sqrt(3) + 1)*(81*c^
2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b
^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75
*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*
(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g +
27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^
3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^
3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/
(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 +
(8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3
)^2 - 2*(5*a^3*b*d + a^4*g)*h^2 + 1/54*(25*a^3*b^4*d^2 + 36*a^3*b^4*c*e + 10*a^4*b^3*d*g + a^5*b^2*g^2 + 18*a^
4*b^3*c*h)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*
b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39
366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*
h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c
*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1
/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b
)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^
2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d
*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*
a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3) - 2*(45*a*b^3*c*d + 4*a^2*b^2*e^2)*g - (81*a*b^3*c^2 + 40*a^2*b^2*d*e + 8
*a^3*b*e*g)*h + 2*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*
h + 6*a^3*b*e*h^2 + a^4*h^3)*x + 1/972*sqrt(1/3)*(1350*a^3*b^4*d^2 - 972*a^3*b^4*c*e + 540*a^4*b^3*d*g + 54*a^
5*b^2*g^2 - 486*a^4*b^3*c*h + (2*a^6*b^4*e + a^7*b^3*h)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2
*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*
h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^
2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3
- 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3
- 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 1
0*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2
*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c
^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g
+ 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3))*sqrt(-(((-I*sqrt(3) + 1)*(8
1*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(
81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3
+ 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39
366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2
*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/2
7*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^
4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h
^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b
^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c
/a^3)^2*a^6*b^3 - 972*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a
^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*
b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h +
6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*
e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a
^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5
*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3
+ 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4
*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 -
54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)*a^3*b^3*c + 236196*b^3*c^2 + 116640*a*b^2*d*e + 23328*a^2*b*e*
g + 11664*(5*a^2*b*d + a^3*g)*h)/(a^6*b^3))) - (1458*b^3*c*x^6 + 2916*a*b^2*c*x^3 + 1458*a^2*b*c - (a^3*b^3*x^
6 + 2*a^4*b^2*x^3 + a^5*b)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d
*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/
(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^
2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2
- 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^
4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*
g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b
*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2
)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*
d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3) + 3*sqrt(1/3)*(a^3*b^3*x^6 + 2*a^4*b^2*x^3 + a^5*b)*sqrt(
-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1
/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*
b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4
*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3
*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729
*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*
b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h +
6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*
e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a
^9*b^5))^(1/3) + 486*c/a^3)^2*a^6*b^3 - 972*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g
*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g +
5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^
3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a
^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3
- 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e
+ a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*
b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3
- (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)
*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)*a^3*b^3*c + 236196*b^3*c^2 + 116640*a*b^
2*d*e + 23328*a^2*b*e*g + 11664*(5*a^2*b*d + a^3*g)*h)/(a^6*b^3)))*log(-225*b^4*c*d^2 - 162*b^4*c^2*e - 40*a*b
^3*d*e^2 - 9*a^2*b^2*c*g^2 - 1/2916*(2*a^6*b^4*e + a^7*b^3*h)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10
*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e +
a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^
2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 -
(g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a
^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c
^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b
^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729
*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*
(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)^2 - 2*(5*a^3*b*d + a^4
*g)*h^2 + 1/54*(25*a^3*b^4*d^2 + 36*a^3*b^4*c*e + 10*a^4*b^3*d*g + a^5*b^2*g^2 + 18*a^4*b^3*c*h)*((-I*sqrt(3)
+ 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1
/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b
^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5)
- 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 -
75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1
)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*
(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2
+ a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h
)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3)
+ 486*c/a^3) - 2*(45*a*b^3*c*d + 4*a^2*b^2*e^2)*g - (81*a*b^3*c^2 + 40*a^2*b^2*d*e + 8*a^3*b*e*g)*h + 2*(125*b
^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*
h^3)*x - 1/972*sqrt(1/3)*(1350*a^3*b^4*d^2 - 972*a^3*b^4*c*e + 540*a^4*b^3*d*g + 54*a^5*b^2*g^2 - 486*a^4*b^3*
c*h + (2*a^6*b^4*e + a^7*b^3*h)*((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g
+ 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*
b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b
^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*
d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)
*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h +
(2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 +
a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*
e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5
*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3))*sqrt(-(((-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2
+ 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*
e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^
2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h
^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*
c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b
^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75
*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*
(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g +
27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 486*c/a^3)^2*a^6*b^3 - 972*((
-I*sqrt(3) + 1)*(81*c^2/a^6 - (81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)/(a^6*b^3))/(-1/27*
c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3) + 1/39366*(125*b^4*
d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a^3*b*e*h^2 + a^4*h^3
)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*h - 9*c*g*h)*a^3*b^2
+ (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b^5))^(1/3) + 729*(I*
sqrt(3) + 1)*(-1/27*c^3/a^9 + 1/1458*(81*b^3*c^2 + 10*a*b^2*d*e + a^3*g*h + (2*e*g + 5*d*h)*a^2*b)*c/(a^9*b^3)
+ 1/39366*(125*b^4*d^3 + 8*a*b^3*e^3 + 75*a*b^3*d^2*g + 15*a^2*b^2*d*g^2 + a^3*b*g^3 + 12*a^2*b^2*e^2*h + 6*a
^3*b*e*h^2 + a^4*h^3)/(a^8*b^5) - 1/39366*(729*b^5*c^3 + a^5*h^3 - (g^3 - 6*e*h^2)*a^4*b - 3*(5*d*g^2 - 4*e^2*
h - 9*c*g*h)*a^3*b^2 + (8*e^3 - 75*d^2*g + 27*(2*e*g + 5*d*h)*c)*a^2*b^3 - 5*(25*d^3 - 54*c*d*e)*a*b^4)/(a^9*b
^5))^(1/3) + 486*c/a^3)*a^3*b^3*c + 236196*b^3*c^2 + 116640*a*b^2*d*e + 23328*a^2*b*e*g + 11664*(5*a^2*b*d + a
^3*g)*h)/(a^6*b^3))) + 2916*(b^3*c*x^6 + 2*a*b^2*c*x^3 + a^2*b*c)*log(x))/(a^3*b^3*x^6 + 2*a^4*b^2*x^3 + a^5*b
)

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giac [A]  time = 0.27, size = 376, normalized size = 1.08 \[ -\frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (5 \, b^{2} d + a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (5 \, b^{2} d + a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} + \frac {6 \, a b^{2} c x^{3} + 2 \, {\left (a^{2} b h + 2 \, a b^{2} e\right )} x^{5} + {\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f - {\left (a^{3} h - 7 \, a^{2} b e\right )} x^{2} + 2 \, {\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac {{\left (a^{5} b^{2} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{4} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 618, normalized size = 1.78 \[ \frac {h \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a}+\frac {2 b e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {g \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a}+\frac {5 b d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {b c \,x^{3}}{3 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {7 e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a}-\frac {h \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b}+\frac {4 d x}{9 \left (b \,x^{3}+a \right )^{2} a}-\frac {g x}{9 \left (b \,x^{3}+a \right )^{2} b}+\frac {c}{2 \left (b \,x^{3}+a \right )^{2} a}-\frac {f}{6 \left (b \,x^{3}+a \right )^{2} b}+\frac {\sqrt {3}\, g \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {g \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}-\frac {g \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {\sqrt {3}\, h \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}-\frac {h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {h \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {c \ln \relax (x )}{a^{3}}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.11, size = 368, normalized size = 1.06 \[ \frac {6 \, b^{2} c x^{3} + 2 \, {\left (2 \, b^{2} e + a b h\right )} x^{5} + {\left (5 \, b^{2} d + a b g\right )} x^{4} + 9 \, a b c - 3 \, a^{2} f + {\left (7 \, a b e - a^{2} h\right )} x^{2} + 2 \, {\left (4 \, a b d - a^{2} g\right )} x}{18 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {c \log \relax (x)}{a^{3}} + \frac {\sqrt {3} {\left (2 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} - \frac {{\left (18 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b d + a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b d - a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.70, size = 1716, normalized size = 4.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*b*c - a*f)/(6*a*b) + (x^4*(5*b*d + a*g))/(18*a^2) + (x^5*(2*b*e + a*h))/(9*a^2) + (x*(4*b*d - a*g))/(9*a*b
) + (x^2*(7*b*e - a*h))/(18*a*b) + (b*c*x^3)/(3*a^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + symsum(log((c*(25*b^2*d^2
+ a^2*g^2 - 18*b^2*c*e - 9*a*b*c*h + 10*a*b*d*g))/(81*a^6) - (root(19683*a^9*b^5*z^3 + 19683*a^6*b^5*c*z^2 + 8
1*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c
*d*e + 27*a^3*b^2*c*g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3
*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)*(a^3*g^2
+ 25*a*b^2*d^2 + 324*b^3*c^2*x + 2916*root(19683*a^9*b^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^
5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h
+ 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^
2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)^2*a^6*b^3*x - 27*root(19683*a^9*b
^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z +
6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2
+ 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^
5*c^3 + a^5*h^3, z, k)*a^5*b*h + 36*a*b^2*c*e + 18*a^2*b*c*h + 10*a^2*b*d*g + 10*a^3*g*h*x - 54*root(19683*a^9
*b^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z
+ 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h
^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*
b^5*c^3 + a^5*h^3, z, k)*a^4*b^2*e + 1944*root(19683*a^9*b^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 40
5*a^5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*
g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*
d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)*a^3*b^3*c*x + 100*a*b^2*d*e*x
+ 50*a^2*b*d*h*x + 20*a^2*b*e*g*x))/(81*a^4) - (x*(a^4*h^3 - 125*b^4*d^3 + 8*a*b^3*e^3 - a^3*b*g^3 - 15*a^2*b
^2*d*g^2 + 12*a^2*b^2*e^2*h + 180*b^4*c*d*e - 75*a*b^3*d^2*g + 6*a^3*b*e*h^2 + 18*a^2*b^2*c*g*h + 90*a*b^3*c*d
*h + 36*a*b^3*c*e*g))/(729*a^6*b^2))*root(19683*a^9*b^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^5
*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h +
135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^2
+ 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k), k, 1, 3) + (c*log(x))/a^3

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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