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

[Out]

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

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Rubi [A]  time = 0.74, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 31

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

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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.69, size = 411, normalized size = 0.99 \[ \frac {x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}+\frac {a^2 x \left (37 a^3 f-31 a^2 b e+25 a b^2 d-19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}+\frac {a x \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )}{b^7}+\frac {x^4 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{4 b^6}+\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{54 b^{22/3}}-\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{27 b^{22/3}}+\frac {a^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{9 \sqrt {3} b^{22/3}}+\frac {x^{10} (b e-3 a f)}{10 b^4}+\frac {f x^{13}}{13 b^3} \]

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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fricas [A]  time = 0.62, size = 667, normalized size = 1.60 \[ \frac {3780 \, b^{6} f x^{19} + 378 \, {\left (13 \, b^{6} e - 19 \, a b^{5} f\right )} x^{16} + 108 \, {\left (65 \, b^{6} d - 104 \, a b^{5} e + 152 \, a^{2} b^{4} f\right )} x^{13} + 351 \, {\left (35 \, b^{6} c - 65 \, a b^{5} d + 104 \, a^{2} b^{4} e - 152 \, a^{3} b^{3} f\right )} x^{10} - 3510 \, {\left (35 \, a b^{5} c - 65 \, a^{2} b^{4} d + 104 \, a^{3} b^{3} e - 152 \, a^{4} b^{2} f\right )} x^{7} - 9555 \, {\left (35 \, a^{2} b^{4} c - 65 \, a^{3} b^{3} d + 104 \, a^{4} b^{2} e - 152 \, a^{5} b f\right )} x^{4} - 1820 \, \sqrt {3} {\left (35 \, a^{3} b^{3} c - 65 \, a^{4} b^{2} d + 104 \, a^{5} b e - 152 \, a^{6} f + {\left (35 \, a b^{5} c - 65 \, a^{2} b^{4} d + 104 \, a^{3} b^{3} e - 152 \, a^{4} b^{2} f\right )} x^{6} + 2 \, {\left (35 \, a^{2} b^{4} c - 65 \, a^{3} b^{3} d + 104 \, a^{4} b^{2} e - 152 \, a^{5} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 910 \, {\left (35 \, a^{3} b^{3} c - 65 \, a^{4} b^{2} d + 104 \, a^{5} b e - 152 \, a^{6} f + {\left (35 \, a b^{5} c - 65 \, a^{2} b^{4} d + 104 \, a^{3} b^{3} e - 152 \, a^{4} b^{2} f\right )} x^{6} + 2 \, {\left (35 \, a^{2} b^{4} c - 65 \, a^{3} b^{3} d + 104 \, a^{4} b^{2} e - 152 \, a^{5} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 1820 \, {\left (35 \, a^{3} b^{3} c - 65 \, a^{4} b^{2} d + 104 \, a^{5} b e - 152 \, a^{6} f + {\left (35 \, a b^{5} c - 65 \, a^{2} b^{4} d + 104 \, a^{3} b^{3} e - 152 \, a^{4} b^{2} f\right )} x^{6} + 2 \, {\left (35 \, a^{2} b^{4} c - 65 \, a^{3} b^{3} d + 104 \, a^{4} b^{2} e - 152 \, a^{5} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 5460 \, {\left (35 \, a^{3} b^{3} c - 65 \, a^{4} b^{2} d + 104 \, a^{5} b e - 152 \, a^{6} f\right )} x}{49140 \, {\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} \]

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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giac [A]  time = 0.20, size = 500, normalized size = 1.20 \[ \frac {\sqrt {3} {\left (35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8}} - \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d - 152 \, a^{5} f + 104 \, a^{4} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{7}} + \frac {{\left (35 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8}} - \frac {19 \, a^{2} b^{4} c x^{4} - 25 \, a^{3} b^{3} d x^{4} - 37 \, a^{5} b f x^{4} + 31 \, a^{4} b^{2} x^{4} e + 16 \, a^{3} b^{3} c x - 22 \, a^{4} b^{2} d x - 34 \, a^{6} f x + 28 \, a^{5} b x e}{18 \, {\left (b x^{3} + a\right )}^{2} b^{7}} + \frac {140 \, b^{36} f x^{13} - 546 \, a b^{35} f x^{10} + 182 \, b^{36} x^{10} e + 260 \, b^{36} d x^{7} + 1560 \, a^{2} b^{34} f x^{7} - 780 \, a b^{35} x^{7} e + 455 \, b^{36} c x^{4} - 1365 \, a b^{35} d x^{4} - 4550 \, a^{3} b^{33} f x^{4} + 2730 \, a^{2} b^{34} x^{4} e - 5460 \, a b^{35} c x + 10920 \, a^{2} b^{34} d x + 27300 \, a^{4} b^{32} f x - 18200 \, a^{3} b^{33} x e}{1820 \, b^{39}} \]

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

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maple [A]  time = 0.07, size = 706, normalized size = 1.70 \[ \frac {f \,x^{13}}{13 b^{3}}-\frac {3 a f \,x^{10}}{10 b^{4}}+\frac {e \,x^{10}}{10 b^{3}}+\frac {6 a^{2} f \,x^{7}}{7 b^{5}}-\frac {3 a e \,x^{7}}{7 b^{4}}+\frac {d \,x^{7}}{7 b^{3}}+\frac {37 a^{5} f \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{6}}-\frac {31 a^{4} e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{5}}+\frac {25 a^{3} d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {19 a^{2} c \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {5 a^{3} f \,x^{4}}{2 b^{6}}+\frac {3 a^{2} e \,x^{4}}{2 b^{5}}-\frac {3 a d \,x^{4}}{4 b^{4}}+\frac {c \,x^{4}}{4 b^{3}}+\frac {17 a^{6} f x}{9 \left (b \,x^{3}+a \right )^{2} b^{7}}-\frac {14 a^{5} e x}{9 \left (b \,x^{3}+a \right )^{2} b^{6}}+\frac {11 a^{4} d x}{9 \left (b \,x^{3}+a \right )^{2} b^{5}}-\frac {8 a^{3} c x}{9 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {152 \sqrt {3}\, a^{5} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{8}}-\frac {152 a^{5} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{8}}+\frac {76 a^{5} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{8}}+\frac {104 \sqrt {3}\, a^{4} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}+\frac {104 a^{4} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}-\frac {52 a^{4} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}+\frac {15 a^{4} f x}{b^{7}}-\frac {65 \sqrt {3}\, a^{3} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {65 a^{3} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}+\frac {65 a^{3} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {10 a^{3} e x}{b^{6}}+\frac {35 \sqrt {3}\, a^{2} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {35 a^{2} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {35 a^{2} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {6 a^{2} d x}{b^{5}}-\frac {3 a c x}{b^{4}} \]

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]

35/27*a^2/b^5*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-65/27*a^3/b^6*d/(a/b)^(2/3)*3^(1/2
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-8/9*a^3/b^4/(b*x^3+a)^2*c*x+76/27*a^5/b^8*f/(a/b)^(2/3)*ln(x^2-(a/b)
^(1/3)*x+(a/b)^(2/3))+104/27*a^4/b^7*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+3/2/b^5*x^4*a^2*e-3/4/b^4*x^4*a*d-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+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-152/27*a^5/b^8*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+104/27*a^4/b^7*e/(a/b)^(
2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/10/b^3*x^10*e+1/7/b^3*x^7*d+1/4/b^3*x^4*c+35/27*a^2/b^5
*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-35/54*a^2/b^5*c/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-65/27*a^3/b^6*d
/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+65/54*a^3/b^6*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-52/27*a^4/b^7*e/(
a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+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+2
5/18*a^3/b^4/(b*x^3+a)^2*x^4*d-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-152/27*a^5/b^8*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+1/13*f*x^13/b^3

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maxima [A]  time = 3.06, size = 424, normalized size = 1.02 \[ -\frac {{\left (19 \, a^{2} b^{4} c - 25 \, a^{3} b^{3} d + 31 \, a^{4} b^{2} e - 37 \, a^{5} b f\right )} x^{4} + 2 \, {\left (8 \, a^{3} b^{3} c - 11 \, a^{4} b^{2} d + 14 \, a^{5} b e - 17 \, a^{6} f\right )} x}{18 \, {\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} + \frac {140 \, b^{4} f x^{13} + 182 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{10} + 260 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{7} + 455 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{4} - 1820 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x}{1820 \, b^{7}} + \frac {\sqrt {3} {\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 104 \, a^{4} b e - 152 \, a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{8} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

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]

-1/18*((19*a^2*b^4*c - 25*a^3*b^3*d + 31*a^4*b^2*e - 37*a^5*b*f)*x^4 + 2*(8*a^3*b^3*c - 11*a^4*b^2*d + 14*a^5*
b*e - 17*a^6*f)*x)/(b^9*x^6 + 2*a*b^8*x^3 + a^2*b^7) + 1/1820*(140*b^4*f*x^13 + 182*(b^4*e - 3*a*b^3*f)*x^10 +
260*(b^4*d - 3*a*b^3*e + 6*a^2*b^2*f)*x^7 + 455*(b^4*c - 3*a*b^3*d + 6*a^2*b^2*e - 10*a^3*b*f)*x^4 - 1820*(3*
a*b^3*c - 6*a^2*b^2*d + 10*a^3*b*e - 15*a^4*f)*x)/b^7 + 1/27*sqrt(3)*(35*a^2*b^3*c - 65*a^3*b^2*d + 104*a^4*b*
e - 152*a^5*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^8*(a/b)^(2/3)) - 1/54*(35*a^2*b^3*c - 65
*a^3*b^2*d + 104*a^4*b*e - 152*a^5*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^8*(a/b)^(2/3)) + 1/27*(35*a^2*
b^3*c - 65*a^3*b^2*d + 104*a^4*b*e - 152*a^5*f)*log(x + (a/b)^(1/3))/(b^8*(a/b)^(2/3))

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mupad [B]  time = 5.24, size = 575, normalized size = 1.38 \[ x^{10}\,\left (\frac {e}{10\,b^3}-\frac {3\,a\,f}{10\,b^4}\right )+x^4\,\left (\frac {c}{4\,b^3}-\frac {a^3\,f}{4\,b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{4\,b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{4\,b}\right )+\frac {x\,\left (\frac {17\,f\,a^6}{9}-\frac {14\,e\,a^5\,b}{9}+\frac {11\,d\,a^4\,b^2}{9}-\frac {8\,c\,a^3\,b^3}{9}\right )-x^4\,\left (-\frac {37\,f\,a^5\,b}{18}+\frac {31\,e\,a^4\,b^2}{18}-\frac {25\,d\,a^3\,b^3}{18}+\frac {19\,c\,a^2\,b^4}{18}\right )}{a^2\,b^7+2\,a\,b^8\,x^3+b^9\,x^6}-x\,\left (\frac {3\,a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {3\,a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^3}\right )-x^7\,\left (\frac {3\,a^2\,f}{7\,b^5}-\frac {d}{7\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{7\,b}\right )+\frac {f\,x^{13}}{13\,b^3}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-152\,f\,a^3+104\,e\,a^2\,b-65\,d\,a\,b^2+35\,c\,b^3\right )}{27\,b^{22/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^10*(e/(10*b^3) - (3*a*f)/(10*b^4)) + x^4*(c/(4*b^3) - (a^3*f)/(4*b^6) - (3*a^2*(e/b^3 - (3*a*f)/b^4))/(4*b^2
) + (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/(4*b)) + (x*((17*a^6*f)/9 - (8*a^3*b^3*c)/9
+ (11*a^4*b^2*d)/9 - (14*a^5*b*e)/9) - x^4*((19*a^2*b^4*c)/18 - (25*a^3*b^3*d)/18 + (31*a^4*b^2*e)/18 - (37*a^
5*b*f)/18))/(a^2*b^7 + b^9*x^6 + 2*a*b^8*x^3) - x*((3*a*(c/b^3 - (a^3*f)/b^6 - (3*a^2*(e/b^3 - (3*a*f)/b^4))/b
^2 + (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b))/b - (3*a^2*((3*a^2*f)/b^5 - d/b^3 + (3*
a*(e/b^3 - (3*a*f)/b^4))/b))/b^2 + (a^3*(e/b^3 - (3*a*f)/b^4))/b^3) - x^7*((3*a^2*f)/(7*b^5) - d/(7*b^3) + (3*
a*(e/b^3 - (3*a*f)/b^4))/(7*b)) + (f*x^13)/(13*b^3) + (a^(4/3)*log(b^(1/3)*x + a^(1/3))*(35*b^3*c - 152*a^3*f
- 65*a*b^2*d + 104*a^2*b*e))/(27*b^(22/3)) + (a^(4/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2
)*1i)/2 - 1/2)*(35*b^3*c - 152*a^3*f - 65*a*b^2*d + 104*a^2*b*e))/(27*b^(22/3)) - (a^(4/3)*log(3^(1/2)*a^(1/3)
*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(35*b^3*c - 152*a^3*f - 65*a*b^2*d + 104*a^2*b*e))/(27*b^(
22/3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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