3.5.26 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x (a+b x^3)^3} \, dx\) [426]

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

[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 = 0.48, 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 = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\text {ArcTan}\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 {\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 {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} \end {gather*}

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

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 631

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 642

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 648

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 1843

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)/a)*Coeff[R, x, i]*x^(i - m), {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[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1848

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 1874

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 1885

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 ) \text {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 = 0.18, size = 311, normalized size = 0.90 \begin {gather*} \frac {\frac {3 a (6 b c+b x (5 d+4 e x)+a x (g+2 h x))}{b \left (a+b x^3\right )}-\frac {9 a^2 (-b (c+x (d+e x))+a (f+x (g+h x)))}{b \left (a+b x^3\right )^2}-\frac {2 \sqrt {3} \sqrt [3]{a} \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 {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+54 c \log (x)+\frac {2 \sqrt [3]{a} \left (5 b^{4/3} d-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g-a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}+\frac {\sqrt [3]{a} \left (-5 b^{4/3} d+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-18 c \log \left (a+b x^3\right )}{54 a^3} \end {gather*}

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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Maple [A]
time = 0.42, size = 339, normalized size = 0.98

method	result	size
default	\(\frac {\frac {\left (\frac {1}{9} a^{2} h +\frac {2}{9} a b e \right ) x^{5}+\left (\frac {1}{18} a^{2} g +\frac {5}{18} a b d \right ) x^{4}+\frac {a b c \,x^{3}}{3}-\frac {a^{2} \left (a h -7 b e \right ) x^{2}}{18 b}-\frac {a^{2} \left (a g -4 b d \right ) x}{9 b}-\frac {a^{2} \left (a f -3 b c \right )}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a^{2} g +5 a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h +2 a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 b c \ln \left (b \,x^{3}+a \right )}{9 b}}{a^{3}}+\frac {c \ln \left (x \right )}{a^{3}}\)	\(339\)
risch	\(\frac {\frac {\left (a h +2 b e \right ) x^{5}}{9 a^{2}}+\frac {\left (a g +5 b d \right ) x^{4}}{18 a^{2}}+\frac {b c \,x^{3}}{3 a^{2}}-\frac {\left (a h -7 b e \right ) x^{2}}{18 a b}-\frac {\left (a g -4 b d \right ) x}{9 a b}-\frac {a f -3 b c}{6 a b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {c \ln \left (-x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{9} b^{5} \textit {\_Z}^{3}+27 a^{6} b^{5} c \,\textit {\_Z}^{2}+\left (3 a^{6} b^{2} g h +15 a^{5} b^{3} d h +6 a^{5} b^{3} e g +30 a^{4} b^{4} d e +243 a^{3} b^{5} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}+6 a^{4} b e \,h^{2}-a^{4} b \,g^{3}+27 a^{3} b^{2} c g h -15 a^{3} b^{2} d \,g^{2}+12 a^{3} b^{2} e^{2} h +135 a^{2} b^{3} c d h +54 a^{2} b^{3} c e g -75 a^{2} b^{3} d^{2} g +8 a^{2} b^{3} e^{3}+270 a \,b^{4} c d e -125 a \,b^{4} d^{3}+729 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b^{5}-72 \textit {\_R}^{2} a^{5} b^{5} c +\left (-10 a^{5} b^{2} g h -50 a^{4} b^{3} d h -20 a^{4} b^{3} e g -100 a^{3} b^{4} d e -324 a^{2} b^{5} c^{2}\right ) \textit {\_R} -3 a^{4} h^{3}-18 a^{3} b e \,h^{2}+3 a^{3} b \,g^{3}-54 a^{2} b^{2} c g h +45 a^{2} b^{2} d \,g^{2}-36 a^{2} b^{2} e^{2} h -270 a \,b^{3} c d h -108 a \,b^{3} c e g +225 a \,b^{3} d^{2} g -24 a \,b^{3} e^{3}-540 b^{4} c d e +375 b^{4} d^{3}\right ) x +\left (a^{7} b^{3} h +2 b^{4} e \,a^{6}\right ) \textit {\_R}^{2}+\left (-a^{5} b^{2} g^{2}-18 a^{4} b^{3} c h -10 a^{4} b^{3} d g -36 a^{3} b^{4} c e -25 a^{3} b^{4} d^{2}\right ) \textit {\_R} +27 a^{2} b^{2} c \,g^{2}-243 a \,b^{3} c^{2} h +270 a \,b^{3} c d g -486 b^{4} c^{2} e +675 b^{4} c \,d^{2}\right )\right )}{27}\)	\(656\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 373, normalized size = 1.07 \begin {gather*} \frac {6 \, b^{2} c x^{3} + 2 \, {\left (a b h + 2 \, b^{2} e\right )} x^{5} + {\left (5 \, b^{2} d + a b g\right )} x^{4} + 9 \, a b c - 3 \, a^{2} f - {\left (a^{2} h - 7 \, a b e\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 \left (x\right )}{a^{3}} + \frac {\sqrt {3} {\left (a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e + 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}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + 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}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 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}}} \end {gather*}

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*(a*b*h + 2*b^2*e)*x^5 + (5*b^2*d + a*b*g)*x^4 + 9*a*b*c - 3*a^2*f - (a^2*h - 7*a*b*e)*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)*(a^2*h*(a/b)^(2
/3) + 2*a*b*(a/b)^(2/3)*e + 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) - a^2*h*(a/b)^(1/3) - 2*a*b*(a/b)^(1/3)*e + 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) + a^2*h*(a/b)^(1/3) +
2*a*b*(a/b)^(1/3)*e - 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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Fricas [C] Result contains complex when optimal does not.
time = 21.81, size = 12815, normalized size = 36.93 \begin {gather*} \text {too large to display} \end {gather*}

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 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [A]
time = 0.49, size = 376, normalized size = 1.08 \begin {gather*} -\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}} \end {gather*}

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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Mupad [B]
time = 5.70, size = 1716, normalized size = 4.95 \begin {gather*} \frac {\frac {3\,b\,c-a\,f}{6\,a\,b}+\frac {x^4\,\left (5\,b\,d+a\,g\right )}{18\,a^2}+\frac {x^5\,\left (2\,b\,e+a\,h\right )}{9\,a^2}+\frac {x\,\left (4\,b\,d-a\,g\right )}{9\,a\,b}+\frac {x^2\,\left (7\,b\,e-a\,h\right )}{18\,a\,b}+\frac {b\,c\,x^3}{3\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\frac {c\,\left (a^2\,g^2+10\,a\,b\,d\,g-9\,c\,h\,a\,b+25\,b^2\,d^2-18\,c\,e\,b^2\right )}{81\,a^6}-\frac {\mathrm {root}\left (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\right )\,\left (a^3\,g^2+25\,a\,b^2\,d^2+324\,b^3\,c^2\,x+{\mathrm {root}\left (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\right )}^2\,a^6\,b^3\,x\,2916-\mathrm {root}\left (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\right )\,a^5\,b\,h\,27+36\,a\,b^2\,c\,e+18\,a^2\,b\,c\,h+10\,a^2\,b\,d\,g+10\,a^3\,g\,h\,x-\mathrm {root}\left (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\right )\,a^4\,b^2\,e\,54+\mathrm {root}\left (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\right )\,a^3\,b^3\,c\,x\,1944+100\,a\,b^2\,d\,e\,x+50\,a^2\,b\,d\,h\,x+20\,a^2\,b\,e\,g\,x\right )}{a^4\,81}-\frac {x\,\left (a^4\,h^3+6\,a^3\,b\,e\,h^2-a^3\,b\,g^3-15\,a^2\,b^2\,d\,g^2+12\,a^2\,b^2\,e^2\,h+18\,c\,a^2\,b^2\,g\,h-75\,a\,b^3\,d^2\,g+90\,c\,a\,b^3\,d\,h+8\,a\,b^3\,e^3+36\,c\,a\,b^3\,e\,g-125\,b^4\,d^3+180\,c\,b^4\,d\,e\right )}{729\,a^6\,b^2}\right )\,\mathrm {root}\left (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\right )\right )+\frac {c\,\ln \left (x\right )}{a^3} \end {gather*}

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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