3.287 \(\int \frac{x^{10} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=384 \[ \frac{x^2 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e -
3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2)
+ (a*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) + (a^(2/3)*(20*b^3*c - 44*a*b^2*
d + 77*a^2*b*e - 119*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(20/3)) + (a^(2/3)
*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) - (a^(2/3)*(20*b^3*c
- 44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(20/3))

________________________________________________________________________________________

Rubi [A]  time = 1.05046, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1851, 1836, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac{a^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e -
3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2)
+ (a*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) + (a^(2/3)*(20*b^3*c - 44*a*b^2*
d + 77*a^2*b*e - 119*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(20/3)) + (a^(2/3)
*(20*b^3*c - 44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) - (a^(2/3)*(20*b^3*c
- 44*a*b^2*d + 77*a^2*b*e - 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(20/3))

Rule 1828

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 1851

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /;
FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m
]]

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

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]

Rubi steps

\begin{align*} \int \frac{x^{10} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}-\frac{\int \frac{-2 a^3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x+6 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-6 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7-6 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^{10}-6 a b^5 (b e-a f) x^{13}-6 a b^6 f x^{16}}{\left (a+b x^3\right )^2} \, dx}{6 a b^7}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}-\frac{\int \frac{x \left (-2 a^3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+6 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-6 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^9-6 a b^5 (b e-a f) x^{12}-6 a b^6 f x^{15}\right )}{\left (a+b x^3\right )^2} \, dx}{6 a b^7}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\int \frac{-2 a^3 b^7 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right ) x+18 a^2 b^8 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4+18 a^2 b^9 \left (b^2 d-2 a b e+3 a^2 f\right ) x^7+18 a^2 b^{10} (b e-2 a f) x^{10}+18 a^2 b^{11} f x^{13}}{a+b x^3} \, dx}{18 a^2 b^{13}}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\int \frac{x \left (-2 a^3 b^7 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+18 a^2 b^8 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+18 a^2 b^9 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+18 a^2 b^{10} (b e-2 a f) x^9+18 a^2 b^{11} f x^{12}\right )}{a+b x^3} \, dx}{18 a^2 b^{13}}\\ &=\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\int \frac{x \left (-22 a^3 b^8 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+198 a^2 b^9 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+198 a^2 b^{10} \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+198 a^2 b^{11} (b e-3 a f) x^9\right )}{a+b x^3} \, dx}{198 a^2 b^{14}}\\ &=\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\int \frac{x \left (-176 a^3 b^9 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+1584 a^2 b^{10} \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+1584 a^2 b^{11} \left (b^2 d-3 a b e+6 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{1584 a^2 b^{15}}\\ &=\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\int \left (1584 a^2 b^9 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x+1584 a^2 b^{10} \left (b^2 d-3 a b e+6 a^2 f\right ) x^4+\frac{176 \left (-20 a^3 b^{12} c+44 a^4 b^{11} d-77 a^5 b^{10} e+119 a^6 b^9 f\right ) x}{a+b x^3}\right ) \, dx}{1584 a^2 b^{15}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}-\frac{\left (a \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{9 b^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^{19/3}}-\frac{\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 b^{19/3}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac{\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\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 b^{20/3}}-\frac{\left (a \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{19/3}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}}-\frac{\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 b^{20/3}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac{(b e-3 a f) x^8}{8 b^4}+\frac{f x^{11}}{11 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{20/3}}+\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac{a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}}\\ \end{align*}

Mathematica [A]  time = 0.402336, size = 380, normalized size = 0.99 \[ \frac{x^2 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac{a x^2 \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}+\frac{a^2 x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-77 a^2 b e+119 a^3 f+44 a b^2 d-20 b^3 c\right )}{54 b^{20/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-77 a^2 b e+119 a^3 f+44 a b^2 d-20 b^3 c\right )}{27 b^{20/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-77 a^2 b e+119 a^3 f+44 a b^2 d-20 b^3 c\right )}{9 \sqrt{3} b^{20/3}}+\frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e -
3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*b^6*(a + b*x^3)
^2) + (a*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) - (a^(2/3)*(-20*b^3*c + 44*a*
b^2*d - 77*a^2*b*e + 119*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(20/3)) - (a^(2/3)*(
-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) + (a^(2/3)*(-20*b^3*c
+ 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(20/3))

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Maple [B]  time = 0.013, size = 668, 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(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-17/18*a^3/b^4/(b*x^3+a)^2*x^2*d+11/18*a^2/b^3/(b*x^3+a)^2*x^2*c+7/9*a/b^2/(b*x^3+a)^2*x^5*c-29/18*a^5/b^6/(b*
x^3+a)^2*x^2*f+23/18*a^4/b^5/(b*x^3+a)^2*x^2*e-10/27*a/b^4*c/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3
))-119/27*a^4/b^7*f/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+119/54*a^4/b^7*f/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1
/b*a)^(2/3))+77/27*a^3/b^6*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-77/54*a^3/b^6*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1
/3)*x+(1/b*a)^(2/3))-44/27*a^2/b^5*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-16/9*a^4/b^5/(b*x^3+a)^2*x^5*f+13/9*a^3
/b^4/(b*x^3+a)^2*x^5*e-10/9*a^2/b^3/(b*x^3+a)^2*x^5*d-77/27*a^3/b^6*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)
*(2/(1/b*a)^(1/3)*x-1))+1/11*f*x^11/b^3+3/b^5*x^2*a^2*e-3/2/b^4*x^2*a*d-3/8/b^4*x^8*a*f+6/5/b^5*x^5*a^2*f-3/5/
b^4*x^5*a*e-5/b^6*x^2*a^3*f+22/27*a^2/b^5*d/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+20/27*a/b^4*c/
(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/8/b^3*x^8*e+1/5/b^3*x^5*d+1/2/b^3*x^2*c+44/27*a^2/b^5*d*3^(1/2)/(1/b*a)^(1
/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-20/27*a/b^4*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a
)^(1/3)*x-1))+119/27*a^4/b^7*f*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))

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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(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35236, size = 1496, normalized size = 3.9 \begin{align*} \frac{1080 \, b^{5} f x^{17} + 135 \,{\left (11 \, b^{5} e - 17 \, a b^{4} f\right )} x^{14} + 54 \,{\left (44 \, b^{5} d - 77 \, a b^{4} e + 119 \, a^{2} b^{3} f\right )} x^{11} + 297 \,{\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{8} + 1056 \,{\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{5} + 660 \,{\left (20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f\right )} x^{2} - 440 \, \sqrt{3}{\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \,{\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + \sqrt{3} a}{3 \, a}\right ) + 220 \,{\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \,{\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 440 \,{\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \,{\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{11880 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/11880*(1080*b^5*f*x^17 + 135*(11*b^5*e - 17*a*b^4*f)*x^14 + 54*(44*b^5*d - 77*a*b^4*e + 119*a^2*b^3*f)*x^11
+ 297*(20*b^5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3*b^2*f)*x^8 + 1056*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^
2*e - 119*a^4*b*f)*x^5 + 660*(20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f)*x^2 - 440*sqrt(3)*((20*b^5
*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2
*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a^
2/b^2)^(1/3) + sqrt(3)*a)/a) + 220*((20*b^5*c - 44*a*b^4*d + 77*a^2*b^3*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c
- 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2*(20*a*b^4*c - 44*a^2*b^3*d + 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^
2/b^2)^(1/3)*log(a*x^2 - b*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*((20*b^5*c - 44*a*b^4*d + 77*a^2*b^3
*e - 119*a^3*b^2*f)*x^6 + 20*a^2*b^3*c - 44*a^3*b^2*d + 77*a^4*b*e - 119*a^5*f + 2*(20*a*b^4*c - 44*a^2*b^3*d
+ 77*a^3*b^2*e - 119*a^4*b*f)*x^3)*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)))/(b^8*x^6 + 2*a*b^7*x^3 + a^
2*b^6)

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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(x**10*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.09179, size = 663, normalized size = 1.73 \begin{align*} \frac{{\left (20 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 44 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 119 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 77 \, a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{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 b^{6}} + \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} 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 \, b^{8}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} 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 \, b^{8}} + \frac{14 \, a b^{4} c x^{5} - 20 \, a^{2} b^{3} d x^{5} - 32 \, a^{4} b f x^{5} + 26 \, a^{3} b^{2} x^{5} e + 11 \, a^{2} b^{3} c x^{2} - 17 \, a^{3} b^{2} d x^{2} - 29 \, a^{5} f x^{2} + 23 \, a^{4} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{40 \, b^{30} f x^{11} - 165 \, a b^{29} f x^{8} + 55 \, b^{30} x^{8} e + 88 \, b^{30} d x^{5} + 528 \, a^{2} b^{28} f x^{5} - 264 \, a b^{29} x^{5} e + 220 \, b^{30} c x^{2} - 660 \, a b^{29} d x^{2} - 2200 \, a^{3} b^{27} f x^{2} + 1320 \, a^{2} b^{28} x^{2} e}{440 \, b^{33}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*(20*a*b^3*c*(-a/b)^(1/3) - 44*a^2*b^2*d*(-a/b)^(1/3) - 119*a^4*f*(-a/b)^(1/3) + 77*a^3*b*(-a/b)^(1/3)*e)*
(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^6) + 1/27*sqrt(3)*(20*(-a*b^2)^(2/3)*b^3*c - 44*(-a*b^2)^(2/3)*a*
b^2*d - 119*(-a*b^2)^(2/3)*a^3*f + 77*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(
1/3))/b^8 - 1/54*(20*(-a*b^2)^(2/3)*b^3*c - 44*(-a*b^2)^(2/3)*a*b^2*d - 119*(-a*b^2)^(2/3)*a^3*f + 77*(-a*b^2)
^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^8 + 1/18*(14*a*b^4*c*x^5 - 20*a^2*b^3*d*x^5 - 32*a^
4*b*f*x^5 + 26*a^3*b^2*x^5*e + 11*a^2*b^3*c*x^2 - 17*a^3*b^2*d*x^2 - 29*a^5*f*x^2 + 23*a^4*b*x^2*e)/((b*x^3 +
a)^2*b^6) + 1/440*(40*b^30*f*x^11 - 165*a*b^29*f*x^8 + 55*b^30*x^8*e + 88*b^30*d*x^5 + 528*a^2*b^28*f*x^5 - 26
4*a*b^29*x^5*e + 220*b^30*c*x^2 - 660*a*b^29*d*x^2 - 2200*a^3*b^27*f*x^2 + 1320*a^2*b^28*x^2*e)/b^33