3.5.27 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 (a+b x^3)^3} \, dx\) [427]

Optimal. Leaf size=362 \[ -\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{4/3}}+\frac {d \log (x)}{a^3}+\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} b^{4/3}}-\frac {d \log \left (a+b x^3\right )}{3 a^3} \]

[Out]

-c/a^3/x+1/6*x*(a*(-a*h+b*e)-b*(-a*f+b*c)*x-b*(-a*g+b*d)*x^2)/a^2/b/(b*x^3+a)^2+1/18*x*(a*(a*h+5*b*e)-2*b*(-2*
a*f+5*b*c)*x-3*b*(-a*g+3*b*d)*x^2)/a^3/b/(b*x^3+a)+d*ln(x)/a^3+1/27*(2*b^(2/3)*(-a*f+7*b*c)+a^(2/3)*(a*h+5*b*e
))*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(4/3)-1/54*(2*b^(2/3)*(-a*f+7*b*c)+a^(2/3)*(a*h+5*b*e))*ln(a^(2/3)-a^(1/3)
*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/b^(4/3)-1/3*d*ln(b*x^3+a)/a^3+1/27*(14*b^(5/3)*c-5*a^(2/3)*b*e-2*a*b^(2/3)*f-
a^(5/3)*h)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)/b^(4/3)*3^(1/2)

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Rubi [A]
time = 0.89, antiderivative size = 362, normalized size of antiderivative = 1.00, 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 (-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right )}{9 \sqrt {3} a^{10/3} b^{4/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{54 a^{10/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{27 a^{10/3} b^{4/3}}+\frac {x \left (-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {c}{a^3 x}+\frac {d \log (x)}{a^3}+\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (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^2*(a + b*x^3)^3),x]

[Out]

-(c/(a^3*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a^2*b*(a + b*x^3)^2) + (x*(a*(5*b*
e + a*h) - 2*b*(5*b*c - 2*a*f)*x - 3*b*(3*b*d - a*g)*x^2))/(18*a^3*b*(a + b*x^3)) + ((14*b^(5/3)*c - 5*a^(2/3)
*b*e - 2*a*b^(2/3)*f - a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(4/
3)) + (d*Log[x])/a^3 + ((2*b^(2/3)*(7*b*c - a*f) + a^(2/3)*(5*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/
3)*b^(4/3)) - ((2*b^(2/3)*(7*b*c - a*f) + a^(2/3)*(5*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
])/(54*a^(10/3)*b^(4/3)) - (d*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^2 \left (a+b x^3\right )^3} \, dx &=\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^2 c-6 b^2 d x-b (5 b e+a h) x^2+4 b^2 \left (\frac {b c}{a}-f\right ) x^3+3 b^2 \left (\frac {b d}{a}-g\right ) x^4}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \frac {18 b^4 c+18 b^4 d x+2 b^3 (5 b e+a h) x^2-2 b^4 \left (\frac {5 b c}{a}-2 f\right ) x^3}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^4}\\ &=\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^4 c}{a x^2}+\frac {18 b^4 d}{a x}+\frac {2 b^3 \left (a (5 b e+a h)-2 b (7 b c-a f) x-9 b^2 d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {a (5 b e+a h)-2 b (7 b c-a f) x-9 b^2 d x^2}{a+b x^3} \, dx}{9 a^3 b}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\int \frac {a (5 b e+a h)-2 b (7 b c-a f) x}{a+b x^3} \, dx}{9 a^3 b}-\frac {(b d) \int \frac {x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}-\frac {d \log \left (a+b x^3\right )}{3 a^3}+\frac {\int \frac {\sqrt [3]{a} \left (-2 \sqrt [3]{a} b (7 b c-a f)+2 a \sqrt [3]{b} (5 b e+a h)\right )+\sqrt [3]{b} \left (-2 \sqrt [3]{a} b (7 b c-a f)-a \sqrt [3]{b} (5 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 (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3} b}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3 b}-\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+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^{10/3} b^{4/3}}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {d \log (x)}{a^3}+\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 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^{10/3} b^{4/3}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}-\frac {\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} 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^{10/3} b^{4/3}}\\ &=-\frac {c}{a^3 x}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac {\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{4/3}}+\frac {d \log (x)}{a^3}+\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac {\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 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^{10/3} b^{4/3}}-\frac {d \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 336, normalized size = 0.93 \begin {gather*} -\frac {\frac {54 a c}{x}+\frac {9 a^2 \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{b \left (a+b x^3\right )^2}-\frac {3 a \left (a^2 h x-10 b^2 c x^2+a b (6 d+x (5 e+4 f x))\right )}{b \left (a+b x^3\right )}+\frac {2 \sqrt {3} a^{2/3} \left (-14 b^{5/3} c+5 a^{2/3} b e+2 a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{4/3}}-54 a d \log (x)-\frac {2 a^{2/3} \left (14 b^{5/3} c+5 a^{2/3} b e-2 a b^{2/3} f+a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}+\frac {a^{2/3} \left (14 b^{5/3} c+5 a^{2/3} b e-2 a b^{2/3} f+a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}+18 a d \log \left (a+b x^3\right )}{54 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*((54*a*c)/x + (9*a^2*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(b*(a + b*x^3)^2) - (3*a*(a^2*
h*x - 10*b^2*c*x^2 + a*b*(6*d + x*(5*e + 4*f*x))))/(b*(a + b*x^3)) + (2*Sqrt[3]*a^(2/3)*(-14*b^(5/3)*c + 5*a^(
2/3)*b*e + 2*a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) - 54*a*d*Log[x] - (
2*a^(2/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) + (a^(2
/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/b^(4/3) + 18*a*d*Log[a + b*x^3])/a^4

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Maple [A]
time = 0.42, size = 345, normalized size = 0.95

method	result	size
default	\(\frac {\frac {\left (\frac {2}{9} a b f -\frac {5}{9} b^{2} c \right ) x^{5}+\left (\frac {1}{18} a^{2} h +\frac {5}{18} a b e \right ) x^{4}+\frac {a b d \,x^{3}}{3}+\frac {a \left (7 a f -13 b c \right ) x^{2}}{18}-\frac {a^{2} \left (a h -4 b e \right ) x}{9 b}-\frac {a^{2} \left (a g -3 b d \right )}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a^{2} h +5 a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (2 a b f -14 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 b d \ln \left (b \,x^{3}+a \right )}{9 b}}{a^{3}}-\frac {c}{a^{3} x}+\frac {d \ln \left (x \right )}{a^{3}}\)	\(345\)
risch	\(\frac {\frac {2 b \left (a f -7 b c \right ) x^{6}}{9 a^{3}}+\frac {\left (a h +5 b e \right ) x^{5}}{18 a^{2}}+\frac {b d \,x^{4}}{3 a^{2}}+\frac {7 \left (a f -7 b c \right ) x^{3}}{18 a^{2}}-\frac {\left (a h -4 b e \right ) x^{2}}{9 a b}-\frac {\left (a g -3 b d \right ) x}{6 a b}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )^{2}}+\frac {d \ln \left (x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{10} b^{4} \textit {\_Z}^{3}+27 a^{7} b^{4} d \,\textit {\_Z}^{2}+\left (6 a^{6} b^{2} f h -42 a^{5} b^{3} c h +30 a^{5} b^{3} e f -210 a^{4} b^{4} c e +243 a^{4} b^{4} d^{2}\right ) \textit {\_Z} -a^{5} h^{3}-15 a^{4} b e \,h^{2}+54 a^{3} b^{2} d f h -75 a^{3} b^{2} e^{2} h +8 a^{3} b^{2} f^{3}-378 a^{2} b^{3} c d h -168 a^{2} b^{3} c \,f^{2}+270 a^{2} b^{3} d e f -125 a^{2} b^{3} e^{3}+1176 a \,b^{4} c^{2} f -1890 a \,b^{4} c d e +729 a \,b^{4} d^{3}-2744 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b^{4}-72 \textit {\_R}^{2} a^{7} b^{4} d +\left (-20 a^{6} b^{2} f h +140 a^{5} b^{3} c h -100 a^{5} b^{3} e f +700 a^{4} b^{4} c e -324 a^{4} b^{4} d^{2}\right ) \textit {\_R} +3 a^{5} h^{3}+45 a^{4} b e \,h^{2}-108 a^{3} b^{2} d f h +225 a^{3} b^{2} e^{2} h -24 a^{3} b^{2} f^{3}+756 a^{2} b^{3} c d h +504 a^{2} b^{3} c \,f^{2}-540 a^{2} b^{3} d e f +375 a^{2} b^{3} e^{3}-3528 a \,b^{4} c^{2} f +3780 a \,b^{4} c d e +8232 b^{5} c^{3}\right ) x +\left (2 b^{3} f \,a^{8}-14 b^{4} c \,a^{7}\right ) \textit {\_R}^{2}+\left (-h^{2} a^{7} b -10 b^{2} e h \,a^{6}-36 b^{3} d f \,a^{5}-25 b^{3} e^{2} a^{5}+252 b^{4} c d \,a^{4}\right ) \textit {\_R} +27 a^{4} b d \,h^{2}+270 a^{3} b^{2} d e h -486 a^{2} b^{3} d^{2} f +675 a^{2} b^{3} d \,e^{2}+3402 a \,b^{4} c \,d^{2}\right )\right )}{27}\)	\(679\)

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^2/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(((2/9*a*b*f-5/9*b^2*c)*x^5+(1/18*a^2*h+5/18*a*b*e)*x^4+1/3*a*b*d*x^3+1/18*a*(7*a*f-13*b*c)*x^2-1/9*a^2*
(a*h-4*b*e)/b*x-1/6*a^2*(a*g-3*b*d)/b)/(b*x^3+a)^2+1/9/b*((a^2*h+5*a*b*e)*(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)))+(2*a*b*f-14*b^2*c)*(-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*d*ln(b*x^3+a)))-c/a^3/x+d*ln(x
)/a^3

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Maxima [A]
time = 0.51, size = 404, normalized size = 1.12 \begin {gather*} \frac {6 \, a b^{2} d x^{4} - 4 \, {\left (7 \, b^{3} c - a b^{2} f\right )} x^{6} + {\left (a^{2} b h + 5 \, a b^{2} e\right )} x^{5} - 18 \, a^{2} b c - 7 \, {\left (7 \, a b^{2} c - a^{2} b f\right )} x^{3} - 2 \, {\left (a^{3} h - 4 \, a^{2} b e\right )} x^{2} + 3 \, {\left (3 \, a^{2} b d - a^{3} g\right )} x}{18 \, {\left (a^{3} b^{3} x^{7} + 2 \, a^{4} b^{2} x^{4} + a^{5} b x\right )}} + \frac {d \log \left (x\right )}{a^{3}} - \frac {\sqrt {3} {\left (14 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} 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^{4} b} - \frac {{\left (18 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h + 5 \, a 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^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 14 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h - 5 \, a b e\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^2/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/18*(6*a*b^2*d*x^4 - 4*(7*b^3*c - a*b^2*f)*x^6 + (a^2*b*h + 5*a*b^2*e)*x^5 - 18*a^2*b*c - 7*(7*a*b^2*c - a^2*
b*f)*x^3 - 2*(a^3*h - 4*a^2*b*e)*x^2 + 3*(3*a^2*b*d - a^3*g)*x)/(a^3*b^3*x^7 + 2*a^4*b^2*x^4 + a^5*b*x) + d*lo
g(x)/a^3 - 1/27*sqrt(3)*(14*b^2*c*(a/b)^(2/3) - 2*a*b*f*(a/b)^(2/3) - a^2*h*(a/b)^(1/3) - 5*a*b*(a/b)^(1/3)*e)
*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b) - 1/54*(18*b^2*d*(a/b)^(2/3) + 14*b^2*c*(a/b)^(1/
3) - 2*a*b*f*(a/b)^(1/3) + a^2*h + 5*a*b*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^2*(a/b)^(2/3)) - 1/2
7*(9*b^2*d*(a/b)^(2/3) - 14*b^2*c*(a/b)^(1/3) + 2*a*b*f*(a/b)^(1/3) - a^2*h - 5*a*b*e)*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 = 22.79, size = 12951, normalized size = 35.78 \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^2/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/2916*(972*a*b^2*d*x^4 - 648*(7*b^3*c - a*b^2*f)*x^6 + 162*(5*a*b^2*e + a^2*b*h)*x^5 - 2916*a^2*b*c - 1134*(7
*a*b^2*c - a^2*b*f)*x^3 + 324*(4*a^2*b*e - a^3*h)*x^2 - 2*(a^3*b^3*x^7 + 2*a^4*b^2*x^4 + a^5*b*x)*((-I*sqrt(3)
+ 1)*(81*d^2/a^6 - (2*a^2*f*h + 2*(5*e*f - 7*c*h)*a*b + (81*d^2 - 70*c*e)*b^2)/(a^6*b^2))/(-1/27*d^3/a^9 + 1/
1458*(2*a^2*f*h + 2*(5*e*f - 7*c*h)*a*b + (81*d^2 - 70*c*e)*b^2)*d/(a^9*b^2) - 1/39366*(2744*b^5*c^3 - 125*a^2
*b^3*e^3 - 1176*a*b^4*c^2*f + 168*a^2*b^3*c*f^2 - 8*a^3*b^2*f^3 - 75*a^3*b^2*e^2*h - 15*a^4*b*e*h^2 - a^5*h^3)
/(a^10*b^4) + 1/39366*(2744*b^5*c^3 + 15*a^4*b*e*h^2 + a^5*h^3 - (8*f^3 - 75*e^2*h + 54*d*f*h)*a^3*b^2 + (125*
e^3 - 270*d*e*f + 42*(4*f^2 + 9*d*h)*c)*a^2*b^3 - 3*(243*d^3 - 630*c*d*e + 392*c^2*f)*a*b^4)/(a^10*b^4))^(1/3)
+ 729*(I*sqrt(3) + 1)*(-1/27*d^3/a^9 + 1/1458*(2*a^2*f*h + 2*(5*e*f - 7*c*h)*a*b + (81*d^2 - 70*c*e)*b^2)*d/(
a^9*b^2) - 1/39366*(2744*b^5*c^3 - 125*a^2*b^3*e^3 - 1176*a*b^4*c^2*f + 168*a^2*b^3*c*f^2 - 8*a^3*b^2*f^3 - 75
*a^3*b^2*e ...

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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**2/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.47, size = 390, normalized size = 1.08 \begin {gather*} -\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (a^{2} h + 5 \, a b e + 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\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^{3}} - \frac {{\left (a^{2} h + 5 \, a b e - 14 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\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^{3}} + \frac {6 \, a b^{2} d x^{4} - 4 \, {\left (7 \, b^{3} c - a b^{2} f\right )} x^{6} + {\left (a^{2} b h + 5 \, a b^{2} e\right )} x^{5} - 18 \, a^{2} b c - 7 \, {\left (7 \, a b^{2} c - a^{2} b f\right )} x^{3} - 2 \, {\left (a^{3} h - 4 \, a^{2} b e\right )} x^{2} + 3 \, {\left (3 \, a^{2} b d - a^{3} g\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} b x} + \frac {{\left (14 \, a^{3} b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{4} b^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{5} b^{2} h - 5 \, a^{4} b^{3} e\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^2/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/3*d*log(abs(b*x^3 + a))/a^3 + d*log(abs(x))/a^3 - 1/27*sqrt(3)*(a^2*h + 5*a*b*e + 14*(-a*b^2)^(1/3)*b*c - 2
*(-a*b^2)^(1/3)*a*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a^3) - 1/54*(a^2*h
+ 5*a*b*e - 14*(-a*b^2)^(1/3)*b*c + 2*(-a*b^2)^(1/3)*a*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(
2/3)*a^3) + 1/18*(6*a*b^2*d*x^4 - 4*(7*b^3*c - a*b^2*f)*x^6 + (a^2*b*h + 5*a*b^2*e)*x^5 - 18*a^2*b*c - 7*(7*a*
b^2*c - a^2*b*f)*x^3 - 2*(a^3*h - 4*a^2*b*e)*x^2 + 3*(3*a^2*b*d - a^3*g)*x)/((b*x^3 + a)^2*a^3*b*x) + 1/27*(14
*a^3*b^4*c*(-a/b)^(1/3) - 2*a^4*b^3*f*(-a/b)^(1/3) - a^5*b^2*h - 5*a^4*b^3*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^
(1/3)))/(a^7*b^3)

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Mupad [B]
time = 5.75, size = 1747, normalized size = 4.83 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {d\,\left (a^3\,h^2+10\,a^2\,b\,e\,h+25\,a\,b^2\,e^2-18\,d\,f\,a\,b^2+126\,c\,d\,b^3\right )}{81\,a^7}-\frac {\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )\,\left (a^3\,h^2+25\,a\,b^2\,e^2+324\,b^3\,d^2\,x-252\,b^3\,c\,d+{\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )}^2\,a^6\,b^3\,x\,2916+36\,a\,b^2\,d\,f+10\,a^2\,b\,e\,h-700\,b^3\,c\,e\,x+\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )\,a^3\,b^3\,c\,378-\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )\,a^4\,b^2\,f\,54+\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )\,a^3\,b^3\,d\,x\,1944-140\,a\,b^2\,c\,h\,x+100\,a\,b^2\,e\,f\,x+20\,a^2\,b\,f\,h\,x\right )}{a^4\,81}+\frac {x\,\left (a^5\,h^3+15\,a^4\,b\,e\,h^2+75\,a^3\,b^2\,e^2\,h-8\,a^3\,b^2\,f^3-36\,d\,a^3\,b^2\,f\,h+168\,a^2\,b^3\,c\,f^2+252\,d\,a^2\,b^3\,c\,h+125\,a^2\,b^3\,e^3-180\,d\,a^2\,b^3\,e\,f-1176\,a\,b^4\,c^2\,f+1260\,d\,a\,b^4\,c\,e+2744\,b^5\,c^3\right )}{729\,a^8\,b}\right )\,\mathrm {root}\left (19683\,a^{10}\,b^4\,z^3+19683\,a^7\,b^4\,d\,z^2+162\,a^6\,b^2\,f\,h\,z-1134\,a^5\,b^3\,c\,h\,z+810\,a^5\,b^3\,e\,f\,z-5670\,a^4\,b^4\,c\,e\,z+6561\,a^4\,b^4\,d^2\,z-1890\,a\,b^4\,c\,d\,e+54\,a^3\,b^2\,d\,f\,h-378\,a^2\,b^3\,c\,d\,h+270\,a^2\,b^3\,d\,e\,f-15\,a^4\,b\,e\,h^2+1176\,a\,b^4\,c^2\,f-75\,a^3\,b^2\,e^2\,h-168\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-125\,a^2\,b^3\,e^3+729\,a\,b^4\,d^3-a^5\,h^3-2744\,b^5\,c^3,z,k\right )\right )+\frac {\frac {x^5\,\left (5\,b\,e+a\,h\right )}{18\,a^2}-\frac {7\,x^3\,\left (7\,b\,c-a\,f\right )}{18\,a^2}-\frac {c}{a}-\frac {2\,b\,x^6\,\left (7\,b\,c-a\,f\right )}{9\,a^3}+\frac {x\,\left (3\,b\,d-a\,g\right )}{6\,a\,b}+\frac {x^2\,\left (4\,b\,e-a\,h\right )}{9\,a\,b}+\frac {b\,d\,x^4}{3\,a^2}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\frac {d\,\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^2*(a + b*x^3)^3),x)

[Out]

symsum(log((d*(a^3*h^2 + 25*a*b^2*e^2 + 126*b^3*c*d - 18*a*b^2*d*f + 10*a^2*b*e*h))/(81*a^7) - (root(19683*a^1
0*b^4*z^3 + 19683*a^7*b^4*d*z^2 + 162*a^6*b^2*f*h*z - 1134*a^5*b^3*c*h*z + 810*a^5*b^3*e*f*z - 5670*a^4*b^4*c*
e*z + 6561*a^4*b^4*d^2*z - 1890*a*b^4*c*d*e + 54*a^3*b^2*d*f*h - 378*a^2*b^3*c*d*h + 270*a^2*b^3*d*e*f - 15*a^
4*b*e*h^2 + 1176*a*b^4*c^2*f - 75*a^3*b^2*e^2*h - 168*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 125*a^2*b^3*e^3 + 729*a*
b^4*d^3 - a^5*h^3 - 2744*b^5*c^3, z, k)*(a^3*h^2 + 25*a*b^2*e^2 + 324*b^3*d^2*x - 252*b^3*c*d + 2916*root(1968
3*a^10*b^4*z^3 + 19683*a^7*b^4*d*z^2 + 162*a^6*b^2*f*h*z - 1134*a^5*b^3*c*h*z + 810*a^5*b^3*e*f*z - 5670*a^4*b
^4*c*e*z + 6561*a^4*b^4*d^2*z - 1890*a*b^4*c*d*e + 54*a^3*b^2*d*f*h - 378*a^2*b^3*c*d*h + 270*a^2*b^3*d*e*f -
15*a^4*b*e*h^2 + 1176*a*b^4*c^2*f - 75*a^3*b^2*e^2*h - 168*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 125*a^2*b^3*e^3 + 7
29*a*b^4*d^3 - a^5*h^3 - 2744*b^5*c^3, z, k)^2*a^6*b^3*x + 36*a*b^2*d*f + 10*a^2*b*e*h - 700*b^3*c*e*x + 378*r
oot(19683*a^10*b^4*z^3 + 19683*a^7*b^4*d*z^2 + 162*a^6*b^2*f*h*z - 1134*a^5*b^3*c*h*z + 810*a^5*b^3*e*f*z - 56
70*a^4*b^4*c*e*z + 6561*a^4*b^4*d^2*z - 1890*a*b^4*c*d*e + 54*a^3*b^2*d*f*h - 378*a^2*b^3*c*d*h + 270*a^2*b^3*
d*e*f - 15*a^4*b*e*h^2 + 1176*a*b^4*c^2*f - 75*a^3*b^2*e^2*h - 168*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 125*a^2*b^3
*e^3 + 729*a*b^4*d^3 - a^5*h^3 - 2744*b^5*c^3, z, k)*a^3*b^3*c - 54*root(19683*a^10*b^4*z^3 + 19683*a^7*b^4*d*
z^2 + 162*a^6*b^2*f*h*z - 1134*a^5*b^3*c*h*z + 810*a^5*b^3*e*f*z - 5670*a^4*b^4*c*e*z + 6561*a^4*b^4*d^2*z - 1
890*a*b^4*c*d*e + 54*a^3*b^2*d*f*h - 378*a^2*b^3*c*d*h + 270*a^2*b^3*d*e*f - 15*a^4*b*e*h^2 + 1176*a*b^4*c^2*f
- 75*a^3*b^2*e^2*h - 168*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 125*a^2*b^3*e^3 + 729*a*b^4*d^3 - a^5*h^3 - 2744*b^5
*c^3, z, k)*a^4*b^2*f + 1944*root(19683*a^10*b^4*z^3 + 19683*a^7*b^4*d*z^2 + 162*a^6*b^2*f*h*z - 1134*a^5*b^3*
c*h*z + 810*a^5*b^3*e*f*z - 5670*a^4*b^4*c*e*z + 6561*a^4*b^4*d^2*z - 1890*a*b^4*c*d*e + 54*a^3*b^2*d*f*h - 37
8*a^2*b^3*c*d*h + 270*a^2*b^3*d*e*f - 15*a^4*b*e*h^2 + 1176*a*b^4*c^2*f - 75*a^3*b^2*e^2*h - 168*a^2*b^3*c*f^2
+ 8*a^3*b^2*f^3 - 125*a^2*b^3*e^3 + 729*a*b^4*d^3 - a^5*h^3 - 2744*b^5*c^3, z, k)*a^3*b^3*d*x - 140*a*b^2*c*h
*x + 100*a*b^2*e*f*x + 20*a^2*b*f*h*x))/(81*a^4) + (x*(2744*b^5*c^3 + a^5*h^3 + 125*a^2*b^3*e^3 - 8*a^3*b^2*f^
3 + 168*a^2*b^3*c*f^2 + 75*a^3*b^2*e^2*h - 1176*a*b^4*c^2*f + 15*a^4*b*e*h^2 + 252*a^2*b^3*c*d*h - 180*a^2*b^3
*d*e*f - 36*a^3*b^2*d*f*h + 1260*a*b^4*c*d*e))/(729*a^8*b))*root(19683*a^10*b^4*z^3 + 19683*a^7*b^4*d*z^2 + 16
2*a^6*b^2*f*h*z - 1134*a^5*b^3*c*h*z + 810*a^5*b^3*e*f*z - 5670*a^4*b^4*c*e*z + 6561*a^4*b^4*d^2*z - 1890*a*b^
4*c*d*e + 54*a^3*b^2*d*f*h - 378*a^2*b^3*c*d*h + 270*a^2*b^3*d*e*f - 15*a^4*b*e*h^2 + 1176*a*b^4*c^2*f - 75*a^
3*b^2*e^2*h - 168*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 125*a^2*b^3*e^3 + 729*a*b^4*d^3 - a^5*h^3 - 2744*b^5*c^3, z,
k), k, 1, 3) + ((x^5*(5*b*e + a*h))/(18*a^2) - (7*x^3*(7*b*c - a*f))/(18*a^2) - c/a - (2*b*x^6*(7*b*c - a*f))
/(9*a^3) + (x*(3*b*d - a*g))/(6*a*b) + (x^2*(4*b*e - a*h))/(9*a*b) + (b*d*x^4)/(3*a^2))/(a^2*x + b^2*x^7 + 2*a
*b*x^4) + (d*log(x))/a^3

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