\(\int \frac {x^4 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^3} \, dx\) [421]

Optimal result

Integrand size = 38, antiderivative size = 345 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac {\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 a^{5/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{10/3}}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {1842, 1872, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{9 \sqrt {3} a^{4/3} b^{10/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{54 a^{4/3} b^{10/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{27 a^{4/3} b^{10/3}}-\frac {x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}+\frac {g \log \left (a+b x^3\right )}{3 b^3}+\frac {h x}{b^3} \]

[In]

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

[Out]

(h*x)/b^3 + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a + b*x^3)^2) - (x*(a*(7*b*e - 1
3*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)*x^2))/(18*a*b^3*(a + b*x^3)) - ((b^(5/3)*c + 2*a^(2/3)*b*e +
5*a*b^(2/3)*f - 14*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(10/3))
- ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(10/3)) + ((b^(2/
3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(10/
3)) + (g*Log[a + b*x^3])/(3*b^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 1842

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(
q - 1)/n] + 1))), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m,
0]

Rule 1872

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

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]

Rule 1901

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {\int \frac {a^2 (b e-a h)-2 a b (b c-a f) x-3 a b (b d-a g) x^2-6 a b (b e-a h) x^3-6 a b^2 f x^4-6 a b^2 g x^5-6 a b^2 h x^6}{\left (a+b x^3\right )^2} \, dx}{6 a b^3} \\ & = \frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {\int \frac {2 a^2 b^2 (2 b e-5 a h)+2 a b^3 (b c+5 a f) x+18 a^2 b^3 g x^2+18 a^2 b^3 h x^3}{a+b x^3} \, dx}{18 a^2 b^5} \\ & = \frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {\int \left (18 a^2 b^2 h+\frac {2 \left (2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x+9 a^2 b^3 g x^2\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^5} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {\int \frac {2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x+9 a^2 b^3 g x^2}{a+b x^3} \, dx}{9 a^2 b^5} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {\int \frac {2 a^2 b^2 (b e-7 a h)+a b^3 (b c+5 a f) x}{a+b x^3} \, dx}{9 a^2 b^5}+\frac {g \int \frac {x^2}{a+b x^3} \, dx}{b^2} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}+\frac {g \log \left (a+b x^3\right )}{3 b^3}+\frac {\int \frac {\sqrt [3]{a} \left (a^{4/3} b^3 (b c+5 a f)+4 a^2 b^{7/3} (b e-7 a h)\right )+\sqrt [3]{b} \left (a^{4/3} b^3 (b c+5 a f)-2 a^2 b^{7/3} (b e-7 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{8/3} b^{16/3}}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^3} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 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 b^3}+\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 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^{4/3} b^{10/3}} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 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^{4/3} b^{10/3}} \\ & = \frac {h x}{b^3}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 b^3 \left (a+b x^3\right )^2}-\frac {x \left (a (7 b e-13 a h)-2 b (b c-4 a f) x-3 b (b d-3 a g) x^2\right )}{18 a b^3 \left (a+b x^3\right )}-\frac {\left (b^{5/3} c+2 a^{2/3} b e+5 a b^{2/3} f-14 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^{4/3} b^{10/3}}-\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{10/3}}+\frac {\left (b^{2/3} (b c+5 a f)-2 a^{2/3} (b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{10/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {54 b^{2/3} h x-\frac {9 b^{2/3} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{\left (a+b x^3\right )^2}+\frac {3 b^{2/3} \left (2 b^2 c x^2+a^2 (12 g+13 h x)-a b (6 d+x (7 e+8 f x))\right )}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^2 c+2 a^{2/3} b^{4/3} e+5 a b f-14 a^{5/3} \sqrt [3]{b} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {2 \left (b^2 c-2 a^{2/3} b^{4/3} e+5 a b f+14 a^{5/3} \sqrt [3]{b} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {\left (b^2 c-2 a^{2/3} b^{4/3} e+5 a b f+14 a^{5/3} \sqrt [3]{b} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}+18 b^{2/3} g \log \left (a+b x^3\right )}{54 b^{11/3}} \]

[In]

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

[Out]

(54*b^(2/3)*h*x - (9*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(
2*b^2*c*x^2 + a^2*(12*g + 13*h*x) - a*b*(6*d + x*(7*e + 8*f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + 2*a^(2
/3)*b^(4/3)*e + 5*a*b*f - 14*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*(b^2
*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2*c - 2*a^(
2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) + 18*
b^(2/3)*g*Log[a + b*x^3])/(54*b^(11/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.49

method	result	size
risch	\(\frac {h x}{b^{3}}+\frac {-\frac {b^{2} \left (4 a f -b c \right ) x^{5}}{9 a}+\left (\frac {13}{18} a b h -\frac {7}{18} b^{2} e \right ) x^{4}+\left (\frac {2}{3} a b g -\frac {1}{3} b^{2} d \right ) x^{3}-\frac {b \left (5 a f +b c \right ) x^{2}}{18}+\frac {a \left (5 a h -2 b e \right ) x}{9}+\frac {a^{2} g}{2}-\frac {a b d}{6}}{b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (9 g b \,\textit {\_R}^{2}+\frac {b \left (5 a f +b c \right ) \textit {\_R}}{a}-14 a h +2 b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b^{4}}\)	\(168\)
default	\(\frac {h x}{b^{3}}-\frac {\frac {\frac {b^{2} \left (4 a f -b c \right ) x^{5}}{9 a}+\left (-\frac {13}{18} a b h +\frac {7}{18} b^{2} e \right ) x^{4}+\left (-\frac {2}{3} a b g +\frac {1}{3} b^{2} d \right ) x^{3}+\frac {b \left (5 a f +b c \right ) x^{2}}{18}-\frac {a \left (5 a h -2 b e \right ) x}{9}-\frac {a^{2} g}{2}+\frac {a b d}{6}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (14 a^{2} h -2 a e b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-5 a f b -b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 a g \ln \left (b \,x^{3}+a \right )}{9 a}}{b^{3}}\)	\(339\)

[In]

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

[Out]

h*x/b^3+(-1/9*b^2*(4*a*f-b*c)/a*x^5+(13/18*a*b*h-7/18*b^2*e)*x^4+(2/3*a*b*g-1/3*b^2*d)*x^3-1/18*b*(5*a*f+b*c)*
x^2+1/9*a*(5*a*h-2*b*e)*x+1/2*a^2*g-1/6*a*b*d)/b^3/(b*x^3+a)^2+1/27/b^4*sum((9*g*b*_R^2+b*(5*a*f+b*c)/a*_R-14*
a*h+2*b*e)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.15 (sec) , antiderivative size = 12967, normalized size of antiderivative = 37.59 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.13 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} - {\left (7 \, a b^{2} e - 13 \, a^{2} b h\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \, {\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} - {\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} - 2 \, {\left (2 \, a^{2} b e - 5 \, a^{3} h\right )} x}{18 \, {\left (a b^{5} x^{6} + 2 \, a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}} + \frac {h x}{b^{3}} + \frac {\sqrt {3} {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac {{\left (18 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b e + 14 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (9 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b e - 14 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.10 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {h x}{b^{3}} + \frac {g \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (2 \, a b e - 14 \, a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (2 \, a b e - 14 \, a^{2} h + \left (-a b^{2}\right )^{\frac {1}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} + \frac {2 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} - {\left (7 \, a b^{2} e - 13 \, a^{2} b h\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \, {\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} - {\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} - 2 \, {\left (2 \, a^{2} b e - 5 \, a^{3} h\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{3}} - \frac {{\left (a b^{6} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b^{5} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b^{5} e - 14 \, a^{3} b^{4} h\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{7}} \]

[In]

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

[Out]

h*x/b^3 + 1/3*g*log(abs(b*x^3 + a))/b^3 - 1/27*sqrt(3)*(2*a*b*e - 14*a^2*h - (-a*b^2)^(1/3)*b*c - 5*(-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*b^2) - 1/54*(2*a*b*e - 14*a^
2*h + (-a*b^2)^(1/3)*b*c + 5*(-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*b^
2) + 1/18*(2*(b^3*c - 4*a*b^2*f)*x^5 - (7*a*b^2*e - 13*a^2*b*h)*x^4 - 3*a^2*b*d + 9*a^3*g - 6*(a*b^2*d - 2*a^2
*b*g)*x^3 - (a*b^2*c + 5*a^2*b*f)*x^2 - 2*(2*a^2*b*e - 5*a^3*h)*x)/((b*x^3 + a)^2*a*b^3) - 1/27*(a*b^6*c*(-a/b
)^(1/3) + 5*a^2*b^5*f*(-a/b)^(1/3) + 2*a^2*b^5*e - 14*a^3*b^4*h)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*
b^7)

Mupad [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 916, normalized size of antiderivative = 2.66 \[ \int \frac {x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^4\,b^{10}\,z^3-19683\,a^4\,b^7\,g\,z^2-5670\,a^4\,b^4\,f\,h\,z-1134\,a^3\,b^5\,c\,h\,z+810\,a^3\,b^5\,e\,f\,z+162\,a^2\,b^6\,c\,e\,z+6561\,a^4\,b^4\,g^2\,z+1890\,a^4\,b\,f\,g\,h+378\,a^3\,b^2\,c\,g\,h-270\,a^3\,b^2\,e\,f\,g-54\,a^2\,b^3\,c\,e\,g-1176\,a^4\,b\,e\,h^2+15\,a\,b^4\,c^2\,f+168\,a^3\,b^2\,e^2\,h+75\,a^2\,b^3\,c\,f^2+125\,a^3\,b^2\,f^3-8\,a^2\,b^3\,e^3-729\,a^4\,b\,g^3+2744\,a^5\,h^3+b^5\,c^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^4\,b^{10}\,z^3-19683\,a^4\,b^7\,g\,z^2-5670\,a^4\,b^4\,f\,h\,z-1134\,a^3\,b^5\,c\,h\,z+810\,a^3\,b^5\,e\,f\,z+162\,a^2\,b^6\,c\,e\,z+6561\,a^4\,b^4\,g^2\,z+1890\,a^4\,b\,f\,g\,h+378\,a^3\,b^2\,c\,g\,h-270\,a^3\,b^2\,e\,f\,g-54\,a^2\,b^3\,c\,e\,g-1176\,a^4\,b\,e\,h^2+15\,a\,b^4\,c^2\,f+168\,a^3\,b^2\,e^2\,h+75\,a^2\,b^3\,c\,f^2+125\,a^3\,b^2\,f^3-8\,a^2\,b^3\,e^3-729\,a^4\,b\,g^3+2744\,a^5\,h^3+b^5\,c^3,z,k\right )\,a\,b^2\,9-\frac {6\,a\,g}{b}+\frac {x\,\left (54\,a^2\,b^4\,e-378\,a^3\,b^3\,h\right )}{81\,a^2\,b^4}\right )+\frac {81\,a^2\,g^2+2\,b^2\,c\,e-70\,a^2\,f\,h-14\,a\,b\,c\,h+10\,a\,b\,e\,f}{81\,a\,b^4}+\frac {x\,\left (126\,g\,h\,a^3+25\,a^2\,b\,f^2-18\,e\,g\,a^2\,b+10\,a\,b^2\,c\,f+b^3\,c^2\right )}{81\,a^2\,b^4}\right )\,\mathrm {root}\left (19683\,a^4\,b^{10}\,z^3-19683\,a^4\,b^7\,g\,z^2-5670\,a^4\,b^4\,f\,h\,z-1134\,a^3\,b^5\,c\,h\,z+810\,a^3\,b^5\,e\,f\,z+162\,a^2\,b^6\,c\,e\,z+6561\,a^4\,b^4\,g^2\,z+1890\,a^4\,b\,f\,g\,h+378\,a^3\,b^2\,c\,g\,h-270\,a^3\,b^2\,e\,f\,g-54\,a^2\,b^3\,c\,e\,g-1176\,a^4\,b\,e\,h^2+15\,a\,b^4\,c^2\,f+168\,a^3\,b^2\,e^2\,h+75\,a^2\,b^3\,c\,f^2+125\,a^3\,b^2\,f^3-8\,a^2\,b^3\,e^3-729\,a^4\,b\,g^3+2744\,a^5\,h^3+b^5\,c^3,z,k\right )\right )-\frac {x^2\,\left (\frac {c\,b^2}{18}+\frac {5\,a\,f\,b}{18}\right )-\frac {a^2\,g}{2}-x\,\left (\frac {5\,a^2\,h}{9}-\frac {2\,a\,b\,e}{9}\right )+x^3\,\left (\frac {b^2\,d}{3}-\frac {2\,a\,b\,g}{3}\right )+\frac {b\,x^4\,\left (7\,b\,e-13\,a\,h\right )}{18}+\frac {a\,b\,d}{6}-\frac {b\,x^5\,\left (b^2\,c-4\,a\,b\,f\right )}{9\,a}}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {h\,x}{b^3} \]

[In]

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

[Out]

symsum(log(root(19683*a^4*b^10*z^3 - 19683*a^4*b^7*g*z^2 - 5670*a^4*b^4*f*h*z - 1134*a^3*b^5*c*h*z + 810*a^3*b
^5*e*f*z + 162*a^2*b^6*c*e*z + 6561*a^4*b^4*g^2*z + 1890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270*a^3*b^2*e*f*g -
54*a^2*b^3*c*e*g - 1176*a^4*b*e*h^2 + 15*a*b^4*c^2*f + 168*a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3
- 8*a^2*b^3*e^3 - 729*a^4*b*g^3 + 2744*a^5*h^3 + b^5*c^3, z, k)*(9*root(19683*a^4*b^10*z^3 - 19683*a^4*b^7*g*
z^2 - 5670*a^4*b^4*f*h*z - 1134*a^3*b^5*c*h*z + 810*a^3*b^5*e*f*z + 162*a^2*b^6*c*e*z + 6561*a^4*b^4*g^2*z + 1
890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270*a^3*b^2*e*f*g - 54*a^2*b^3*c*e*g - 1176*a^4*b*e*h^2 + 15*a*b^4*c^2*f
+ 168*a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 + 125*a^3*b^2*f^3 - 8*a^2*b^3*e^3 - 729*a^4*b*g^3 + 2744*a^5*h^3 + b^5
*c^3, z, k)*a*b^2 - (6*a*g)/b + (x*(54*a^2*b^4*e - 378*a^3*b^3*h))/(81*a^2*b^4)) + (81*a^2*g^2 + 2*b^2*c*e - 7
0*a^2*f*h - 14*a*b*c*h + 10*a*b*e*f)/(81*a*b^4) + (x*(b^3*c^2 + 25*a^2*b*f^2 + 126*a^3*g*h + 10*a*b^2*c*f - 18
*a^2*b*e*g))/(81*a^2*b^4))*root(19683*a^4*b^10*z^3 - 19683*a^4*b^7*g*z^2 - 5670*a^4*b^4*f*h*z - 1134*a^3*b^5*c
*h*z + 810*a^3*b^5*e*f*z + 162*a^2*b^6*c*e*z + 6561*a^4*b^4*g^2*z + 1890*a^4*b*f*g*h + 378*a^3*b^2*c*g*h - 270
*a^3*b^2*e*f*g - 54*a^2*b^3*c*e*g - 1176*a^4*b*e*h^2 + 15*a*b^4*c^2*f + 168*a^3*b^2*e^2*h + 75*a^2*b^3*c*f^2 +
125*a^3*b^2*f^3 - 8*a^2*b^3*e^3 - 729*a^4*b*g^3 + 2744*a^5*h^3 + b^5*c^3, z, k), k, 1, 3) - (x^2*((b^2*c)/18
+ (5*a*b*f)/18) - (a^2*g)/2 - x*((5*a^2*h)/9 - (2*a*b*e)/9) + x^3*((b^2*d)/3 - (2*a*b*g)/3) + (b*x^4*(7*b*e -
13*a*h))/18 + (a*b*d)/6 - (b*x^5*(b^2*c - 4*a*b*f))/(9*a))/(a^2*b^3 + b^5*x^6 + 2*a*b^4*x^3) + (h*x)/b^3