3.286 \(\int \frac{x^{12} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=416 \[ \frac{x^4 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{4 b^6}-\frac{a^2 x \left (31 a^2 b e-37 a^3 f-25 a b^2 d+19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{54 b^{22/3}}-\frac{a x \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{b^7}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{27 b^{22/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

[Out]

-((a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x)/b^7) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/
(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3
*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) - (a^2*(19*b^3*c - 25*a*b^2*d + 31*a^2*b*e - 37*
a^3*f)*x)/(18*b^7*(a + b*x^3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(22/3)) + (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^
3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(22/3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

________________________________________________________________________________________

Rubi [A]  time = 0.742162, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1828, 1858, 1887, 200, 31, 634, 617, 204, 628} \[ \frac{x^4 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{4 b^6}-\frac{a^2 x \left (31 a^2 b e-37 a^3 f-25 a b^2 d+19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{54 b^{22/3}}-\frac{a x \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{b^7}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{27 b^{22/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (104 a^2 b e-152 a^3 f-65 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

-((a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x)/b^7) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/
(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3
*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) - (a^2*(19*b^3*c - 25*a*b^2*d + 31*a^2*b*e - 37*
a^3*f)*x)/(18*b^7*(a + b*x^3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(22/3)) + (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^
3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(22/3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

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 1858

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]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1887

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]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{\int \frac{a^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a^3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3+6 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-6 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9-6 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^{12}-6 a b^5 (b e-a f) x^{15}-6 a b^6 f x^{18}}{\left (a+b x^3\right )^2} \, dx}{6 a b^7}\\ &=\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{\int \frac{2 a^4 b^6 \left (8 b^3 c-11 a b^2 d+14 a^2 b e-17 a^3 f\right )-18 a^3 b^7 \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x^3+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^6+18 a^2 b^9 \left (b^2 d-2 a b e+3 a^2 f\right ) x^9+18 a^2 b^{10} (b e-2 a f) x^{12}+18 a^2 b^{11} f x^{15}}{a+b x^3} \, dx}{18 a^2 b^{13}}\\ &=\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{\int \left (-18 a^3 b^6 \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right )+18 a^2 b^7 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^3+18 a^2 b^8 \left (b^2 d-3 a b e+6 a^2 f\right ) x^6+18 a^2 b^9 (b e-3 a f) x^9+18 a^2 b^{10} f x^{12}-\frac{2 \left (-35 a^4 b^9 c+65 a^5 b^8 d-104 a^6 b^7 e+152 a^7 b^6 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^{13}}\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac{(b e-3 a f) x^{10}}{10 b^4}+\frac{f x^{13}}{13 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{\left (a^2 \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{9 b^7}\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac{(b e-3 a f) x^{10}}{10 b^4}+\frac{f x^{13}}{13 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{\left (a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^7}+\frac{\left (a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right )\right ) \int \frac{2 \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^7}\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac{(b e-3 a f) x^{10}}{10 b^4}+\frac{f x^{13}}{13 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{22/3}}-\frac{\left (a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 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^{22/3}}+\frac{\left (a^{5/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 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^7}\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac{(b e-3 a f) x^{10}}{10 b^4}+\frac{f x^{13}}{13 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}+\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{22/3}}-\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{22/3}}+\frac{\left (a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 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^{22/3}}\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^4}{4 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^7}{7 b^5}+\frac{(b e-3 a f) x^{10}}{10 b^4}+\frac{f x^{13}}{13 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (19 b^3 c-25 a b^2 d+31 a^2 b e-37 a^3 f\right ) x}{18 b^7 \left (a+b x^3\right )}-\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 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^{22/3}}+\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{22/3}}-\frac{a^{4/3} \left (35 b^3 c-65 a b^2 d+104 a^2 b e-152 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{22/3}}\\ \end{align*}

Mathematica [A]  time = 0.482441, size = 411, normalized size = 0.99 \[ \frac{x^4 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{4 b^6}+\frac{a^2 x \left (-31 a^2 b e+37 a^3 f+25 a b^2 d-19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-104 a^2 b e+152 a^3 f+65 a b^2 d-35 b^3 c\right )}{54 b^{22/3}}+\frac{a x \left (-10 a^2 b e+15 a^3 f+6 a b^2 d-3 b^3 c\right )}{b^7}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-104 a^2 b e+152 a^3 f+65 a b^2 d-35 b^3 c\right )}{27 b^{22/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-104 a^2 b e+152 a^3 f+65 a b^2 d-35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/(4
*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(
b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) + (a^2*(-19*b^3*c + 25*a*b^2*d - 31*a^2*b*e + 37*a
^3*f)*x)/(18*b^7*(a + b*x^3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*ArcTan[(1 - (2*b^(
1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(22/3)) - (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(27*b^(22/3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(2
/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

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Maple [A]  time = 0.013, size = 706, 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^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-19/18*a^2/b^3/(b*x^3+a)^2*x^4*c+17/9*a^6/b^7/(b*x^3+a)^2*f*x-14/9*a^5/b^6/(b*x^3+a)^2*e*x+11/9*a^4/b^5/(b*x^3
+a)^2*d*x-8/9*a^3/b^4/(b*x^3+a)^2*c*x+1/13*f*x^13/b^3+37/18*a^5/b^6/(b*x^3+a)^2*x^4*f-31/18*a^4/b^5/(b*x^3+a)^
2*x^4*e+25/18*a^3/b^4/(b*x^3+a)^2*x^4*d+104/27*a^4/b^7*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-35/54*a^2/b^5*c/(1/
b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-65/27*a^3/b^6*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+65/54*a^3/b
^6*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-152/27*a^5/b^8*f/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+76
/27*a^5/b^8*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+35/27*a^2/b^5*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(
1/3))-52/27*a^4/b^7*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-3/10/b^4*x^10*a*f+6/7/b^5*x^7*a^2*f-
3/7/b^4*x^7*a*e-5/2/b^6*x^4*a^3*f+3/2/b^5*x^4*a^2*e-3/4/b^4*x^4*a*d+15/b^7*a^4*f*x-10/b^6*a^3*e*x+6/b^5*a^2*d*
x-3/b^4*a*c*x+1/10/b^3*x^10*e+1/7/b^3*x^7*d+1/4/b^3*x^4*c-152/27*a^5/b^8*f/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^
(1/2)*(2/(1/b*a)^(1/3)*x-1))+104/27*a^4/b^7*e/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-
65/27*a^3/b^6*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+35/27*a^2/b^5*c/(1/b*a)^(2/3)*
3^(1/2)*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^12*(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 = 1.32608, size = 1589, normalized size = 3.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/49140*(3780*b^6*f*x^19 + 378*(13*b^6*e - 19*a*b^5*f)*x^16 + 108*(65*b^6*d - 104*a*b^5*e + 152*a^2*b^4*f)*x^1
3 + 351*(35*b^6*c - 65*a*b^5*d + 104*a^2*b^4*e - 152*a^3*b^3*f)*x^10 - 3510*(35*a*b^5*c - 65*a^2*b^4*d + 104*a
^3*b^3*e - 152*a^4*b^2*f)*x^7 - 9555*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^4 - 1820*sq
rt(3)*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 15
2*a^4*b^2*f)*x^6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^3)*(-a/b)^(1/3)*arctan(1/3*
(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 910*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*b*e - 152*a^6*f + (35
*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e
- 152*a^5*b*f)*x^3)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*(35*a^3*b^3*c - 65*a^4*b^2*d
+ 104*a^5*b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^6 + 2*(35*a^2*b^4*c
- 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 5460*(35*a^3*b^3*c - 6
5*a^4*b^2*d + 104*a^5*b*e - 152*a^6*f)*x)/(b^9*x^6 + 2*a*b^8*x^3 + a^2*b^7)

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

[Out]

Timed out

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Giac [A]  time = 1.09094, size = 675, normalized size = 1.62 \begin{align*} \frac{\sqrt{3}{\left (35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{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 \, b^{8}} - \frac{{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d - 152 \, a^{5} f + 104 \, a^{4} b 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^{7}} + \frac{{\left (35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{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 \, b^{8}} - \frac{19 \, a^{2} b^{4} c x^{4} - 25 \, a^{3} b^{3} d x^{4} - 37 \, a^{5} b f x^{4} + 31 \, a^{4} b^{2} x^{4} e + 16 \, a^{3} b^{3} c x - 22 \, a^{4} b^{2} d x - 34 \, a^{6} f x + 28 \, a^{5} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{7}} + \frac{140 \, b^{36} f x^{13} - 546 \, a b^{35} f x^{10} + 182 \, b^{36} x^{10} e + 260 \, b^{36} d x^{7} + 1560 \, a^{2} b^{34} f x^{7} - 780 \, a b^{35} x^{7} e + 455 \, b^{36} c x^{4} - 1365 \, a b^{35} d x^{4} - 4550 \, a^{3} b^{33} f x^{4} + 2730 \, a^{2} b^{34} x^{4} e - 5460 \, a b^{35} c x + 10920 \, a^{2} b^{34} d x + 27300 \, a^{4} b^{32} f x - 18200 \, a^{3} b^{33} x e}{1820 \, b^{39}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(35*(-a*b^2)^(1/3)*a*b^3*c - 65*(-a*b^2)^(1/3)*a^2*b^2*d - 152*(-a*b^2)^(1/3)*a^4*f + 104*(-a*b^2
)^(1/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^8 - 1/27*(35*a^2*b^3*c - 65*a^3*b^2*d
- 152*a^5*f + 104*a^4*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/54*(35*(-a*b^2)^(1/3)*a*b^3*c
- 65*(-a*b^2)^(1/3)*a^2*b^2*d - 152*(-a*b^2)^(1/3)*a^4*f + 104*(-a*b^2)^(1/3)*a^3*b*e)*log(x^2 + x*(-a/b)^(1/3
) + (-a/b)^(2/3))/b^8 - 1/18*(19*a^2*b^4*c*x^4 - 25*a^3*b^3*d*x^4 - 37*a^5*b*f*x^4 + 31*a^4*b^2*x^4*e + 16*a^3
*b^3*c*x - 22*a^4*b^2*d*x - 34*a^6*f*x + 28*a^5*b*x*e)/((b*x^3 + a)^2*b^7) + 1/1820*(140*b^36*f*x^13 - 546*a*b
^35*f*x^10 + 182*b^36*x^10*e + 260*b^36*d*x^7 + 1560*a^2*b^34*f*x^7 - 780*a*b^35*x^7*e + 455*b^36*c*x^4 - 1365
*a*b^35*d*x^4 - 4550*a^3*b^33*f*x^4 + 2730*a^2*b^34*x^4*e - 5460*a*b^35*c*x + 10920*a^2*b^34*d*x + 27300*a^4*b
^32*f*x - 18200*a^3*b^33*x*e)/b^39