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

Optimal. Leaf size=395 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x} \]

[Out]

-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d - a*g + (b*e - a*h)*x - b*((b*c)/a - f)*x^2))/(6*a^2*(a +
b*x^3)^2) - (x*(11*b*d - 5*a*g + 2*(5*b*e - 2*a*h)*x - 3*b*((5*b*c)/a - 3*f)*x^2))/(18*a^3*(a + b*x^3)) + ((2
0*b^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*a^(11/3)*b^(2/3)) - ((3*b*c - a*f)*Log[x])/a^4 - ((5*b^(1/3)*(4*b*d - a*g) - 2*a^(1/3)*(7*b*e - a*
h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(2/3)) + ((5*b^(1/3)*(4*b*d - a*g) - 2*a^(1/3)*(7*b*e - a*h))*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(2/3)) + ((3*b*c - a*f)*Log[a + b*x^3])/(3*a^4)

________________________________________________________________________________________

Rubi [A]  time = 1.00616, antiderivative size = 392, 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} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}-5 a g+20 b d\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d - a*g + (b*e - a*h)*x - b*((b*c)/a - f)*x^2))/(6*a^2*(a +
b*x^3)^2) - (x*(11*b*d - 5*a*g + 2*(5*b*e - 2*a*h)*x - 3*b*((5*b*c)/a - 3*f)*x^2))/(18*a^3*(a + b*x^3)) + ((2
0*b^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*a^(11/3)*b^(2/3)) - ((3*b*c - a*f)*Log[x])/a^4 - ((5*b^(1/3)*(4*b*d - a*g) - 2*a^(1/3)*(7*b*e - a*
h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(2/3)) + ((20*b*d - 5*a*g - (2*a^(1/3)*(7*b*e - a*h))/b^(1/3))*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(1/3)) + ((3*b*c - a*f)*Log[a + b*x^3])/(3*a^4)

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 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

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

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left (a+b x^3\right )^3} \, dx &=-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^2 c-6 b^2 d x-6 b^2 e x^2+6 b^2 \left (\frac{b c}{a}-f\right ) x^3+5 b^2 \left (\frac{b d}{a}-g\right ) x^4+4 b^2 \left (\frac{b e}{a}-h\right ) x^5-\frac{3 b^3 (b c-a f) x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \frac{18 b^4 c+18 b^4 d x+18 b^4 e x^2-18 b^4 \left (\frac{2 b c}{a}-f\right ) x^3-2 b^4 \left (\frac{11 b d}{a}-5 g\right ) x^4-2 b^4 \left (\frac{5 b e}{a}-2 h\right ) x^5}{x^4 \left (a+b x^3\right )} \, dx}{18 a^2 b^4}\\ &=-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^4 c}{a x^4}+\frac{18 b^4 d}{a x^3}+\frac{18 b^4 e}{a x^2}+\frac{18 b^4 (-3 b c+a f)}{a^2 x}+\frac{2 b^4 \left (-5 a (4 b d-a g)-2 a (7 b e-a h) x+9 b (3 b c-a f) x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{(3 b c-a f) \log (x)}{a^4}+\frac{\int \frac{-5 a (4 b d-a g)-2 a (7 b e-a h) x+9 b (3 b c-a f) x^2}{a+b x^3} \, dx}{9 a^4}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{(3 b c-a f) \log (x)}{a^4}+\frac{\int \frac{-5 a (4 b d-a g)-2 a (7 b e-a h) x}{a+b x^3} \, dx}{9 a^4}+\frac{(b (3 b c-a f)) \int \frac{x^2}{a+b x^3} \, dx}{a^4}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{(3 b c-a f) \log (x)}{a^4}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}+\frac{\int \frac{\sqrt [3]{a} \left (-10 a \sqrt [3]{b} (4 b d-a g)-2 a^{4/3} (7 b e-a h)\right )+\sqrt [3]{b} \left (5 a \sqrt [3]{b} (4 b d-a g)-2 a^{4/3} (7 b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{14/3} \sqrt [3]{b}}-\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{(3 b c-a f) \log (x)}{a^4}-\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 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^{10/3} \sqrt [3]{b}}+\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 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^{11/3} \sqrt [3]{b}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{(3 b c-a f) \log (x)}{a^4}-\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 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^{11/3} \sqrt [3]{b}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 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^{11/3} b^{2/3}}\\ &=-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac{x \left (11 b d-5 a g+2 (5 b e-2 a h) x-3 b \left (\frac{5 b c}{a}-3 f\right ) x^2\right )}{18 a^3 \left (a+b x^3\right )}+\frac{\left (20 b^{4/3} d+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g-2 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^{11/3} b^{2/3}}-\frac{(3 b c-a f) \log (x)}{a^4}-\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{\left (20 b d-5 a g-\frac{2 \sqrt [3]{a} (7 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^{11/3} \sqrt [3]{b}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}\\ \end{align*}

Mathematica [A]  time = 0.590882, size = 352, normalized size = 0.89 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{a^2 (9 a (f+x (g+h x))-9 b (c+x (d+e x)))}{\left (a+b x^3\right )^2}+\frac{3 a (6 a f+a x (5 g+4 h x)-12 b c-b x (11 d+10 e x))}{a+b x^3}+18 (3 b c-a f) \log \left (a+b x^3\right )+54 \log (x) (a f-3 b c)-\frac{18 a c}{x^3}-\frac{27 a d}{x^2}-\frac{54 a e}{x}}{54 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-18*a*c)/x^3 - (27*a*d)/x^2 - (54*a*e)/x + (3*a*(-12*b*c + 6*a*f - b*x*(11*d + 10*e*x) + a*x*(5*g + 4*h*x)))
/(a + b*x^3) + (a^2*(-9*b*(c + x*(d + e*x)) + 9*a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (2*Sqrt[3]*a^(1/3)*(20*b
^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3)
+ 54*(-3*b*c + a*f)*Log[x] - (2*a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(1
/3) + b^(1/3)*x])/b^(2/3) + (a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 18*(3*b*c - a*f)*Log[a + b*x^3])/(54*a^4)

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Maple [B]  time = 0.017, size = 680, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a/(b*x^3+a)^2*f-1/3/a^3*ln(b*x^3+a)*f+1/a^3*ln(x)*f+2/27/a^2*h*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*
(2/(1/b*a)^(1/3)*x-1))+5/27/a^2*g/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/3*c/a^3/
x^3+5/27/a^2*g/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-5/54/a^2*g/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(
2/3))-5/9/a^3/(b*x^3+a)^2*x^5*e*b^2-20/27/a^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*
d-11/18/a^3/(b*x^3+a)^2*x^4*b^2*d-1/2*d/a^3/x^2-e/a^3/x-5/6/a^2*b/(b*x^3+a)^2*c-7/9/a^2/(b*x^3+a)^2*b*x*d-13/1
8/a^2/(b*x^3+a)^2*x^2*b*e-14/27/a^3*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+14/27/a^
3*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-7/27/a^3*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-20/27/a^3
/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d+10/27/a^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+7/18/a/(b
*x^3+a)^2*x^2*h+4/9/a/(b*x^3+a)^2*g*x+5/18/a^2/(b*x^3+a)^2*x^4*b*g+2/9/a^2/(b*x^3+a)^2*x^5*b*h-2/27/a^2*h/b/(1
/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/3/a^2/(b*x^3+a)^2*x^3*b*f+1/27/a^2*h/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+
(1/b*a)^(2/3))-3*b*c*ln(x)/a^4+b*c*ln(b*x^3+a)/a^4-2/3/a^3*b^2/(b*x^3+a)^2*c*x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.09566, size = 612, normalized size = 1.55 \begin{align*} \frac{{\left (3 \, b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} - \frac{{\left (3 \, b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} - \frac{{\left (2 \, a^{6} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 20 \, a^{5} b^{2} d + 5 \, a^{6} b 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^{9} b} + \frac{4 \,{\left (a^{2} b h - 7 \, a b^{2} e\right )} x^{8} - 5 \,{\left (4 \, a b^{2} d - a^{2} b g\right )} x^{7} - 6 \,{\left (3 \, a b^{2} c - a^{2} b f\right )} x^{6} + 7 \,{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{5} - 18 \, a^{3} x^{2} e - 9 \, a^{3} d x - 8 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x^{4} - 6 \, a^{3} c - 9 \,{\left (3 \, a^{2} b c - a^{3} f\right )} x^{3}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \end{align*}

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^4/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*(3*b*c - a*f)*log(abs(b*x^3 + a))/a^4 - (3*b*c - a*f)*log(abs(x))/a^4 - 1/27*sqrt(3)*(20*(-a*b^2)^(1/3)*b^
2*d - 5*(-a*b^2)^(1/3)*a*b*g + 2*(-a*b^2)^(2/3)*a*h - 14*(-a*b^2)^(2/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^
(1/3))/(-a/b)^(1/3))/(a^4*b^2) - 1/54*(20*(-a*b^2)^(1/3)*b^2*d - 5*(-a*b^2)^(1/3)*a*b*g - 2*(-a*b^2)^(2/3)*a*h
+ 14*(-a*b^2)^(2/3)*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^2) - 1/27*(2*a^6*b*h*(-a/b)^(1/3) -
14*a^5*b^2*(-a/b)^(1/3)*e - 20*a^5*b^2*d + 5*a^6*b*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^9*b) + 1/18*(
4*(a^2*b*h - 7*a*b^2*e)*x^8 - 5*(4*a*b^2*d - a^2*b*g)*x^7 - 6*(3*a*b^2*c - a^2*b*f)*x^6 + 7*(a^3*h - 7*a^2*b*e
)*x^5 - 18*a^3*x^2*e - 9*a^3*d*x - 8*(4*a^2*b*d - a^3*g)*x^4 - 6*a^3*c - 9*(3*a^2*b*c - a^3*f)*x^3)/((b*x^3 +
a)^2*a^4*x^3)