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

Optimal. Leaf size=375 \[ \frac{a x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{b^6}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^4)/(4*b^5) + ((b*e - 3*a*f
)*x^7)/(7*b^4) + (f*x^10)/(10*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^6*(a + b*x^3)^2) + (a*(1
3*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(18*b^6*(a + b*x^3)) + (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65*a
^2*b*e - 104*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(19/3)) - (a^(1/3)*(14*b^3
*c - 35*a*b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(19/3)) + (a^(1/3)*(14*b^3*c - 35*a*
b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(19/3))

________________________________________________________________________________________

Rubi [A]  time = 0.606352, antiderivative size = 375, 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} \[ \frac{a x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{b^6}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^9*(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)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^4)/(4*b^5) + ((b*e - 3*a*f
)*x^7)/(7*b^4) + (f*x^10)/(10*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^6*(a + b*x^3)^2) + (a*(1
3*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(18*b^6*(a + b*x^3)) + (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65*a
^2*b*e - 104*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*b^(19/3)) - (a^(1/3)*(14*b^3
*c - 35*a*b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(19/3)) + (a^(1/3)*(14*b^3*c - 35*a*
b^2*d + 65*a^2*b*e - 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(19/3))

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^9 \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}{6 b^6 \left (a+b x^3\right )^2}-\frac{\int \frac{-a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+6 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a 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^2 d-a b e+a^2 f\right ) x^9-6 a b^4 (b e-a f) x^{12}-6 a b^5 f x^{15}}{\left (a+b x^3\right )^2} \, dx}{6 a b^6}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\int \frac{-2 a^3 b^5 \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )+18 a^2 b^6 \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^7 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+18 a^2 b^8 (b e-2 a f) x^9+18 a^2 b^9 f x^{12}}{a+b x^3} \, dx}{18 a^2 b^{11}}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\int \left (18 a^2 b^5 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right )+18 a^2 b^6 \left (b^2 d-3 a b e+6 a^2 f\right ) x^3+18 a^2 b^7 (b e-3 a f) x^6+18 a^2 b^8 f x^9+\frac{2 \left (-14 a^3 b^8 c+35 a^4 b^7 d-65 a^5 b^6 e+104 a^6 b^5 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^{11}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\left (a \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac{1}{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}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^6}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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^{19/3}}-\frac{\left (a^{2/3} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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^{19/3}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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^{19/3}}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}}\\ \end{align*}

Mathematica [A]  time = 0.37398, size = 362, normalized size = 0.97 \[ \frac{\frac{210 a \sqrt [3]{b} x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{a+b x^3}+\frac{630 a^2 \sqrt [3]{b} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )+3780 \sqrt [3]{b} x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )+945 b^{4/3} x^4 \left (6 a^2 f-3 a b e+b^2 d\right )+540 b^{7/3} x^7 (b e-3 a f)+378 b^{10/3} f x^{10}}{3780 b^{19/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3780*b^(1/3)*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x + 945*b^(4/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^4 + 540
*b^(7/3)*(b*e - 3*a*f)*x^7 + 378*b^(10/3)*f*x^10 + (630*a^2*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/
(a + b*x^3)^2 + (210*a*b^(1/3)*(13*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(a + b*x^3) - 140*Sqrt[3]*a^
(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1
/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-14*b^3*c + 35*a*
b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*b^(19/3))

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 651, 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^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-3/7/b^4*x^7*a*f+3/2/b^5*x^4*a^2*f-3/4/b^4*x^4*a*e-10/b^6*a^3*f*x+6/b^5*a^2*e*x-3/b^4*a*d*x-19/18*a^2/b^3/(b*x
^3+a)^2*x^4*d+1/10*f*x^10/b^3-14/9*a^5/b^6/(b*x^3+a)^2*f*x+11/9*a^4/b^5/(b*x^3+a)^2*e*x-8/9*a^3/b^4/(b*x^3+a)^
2*d*x+5/9*a^2/b^3/(b*x^3+a)^2*c*x+104/27*a^4/b^7*f/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-65/27*a^3/b^6*e/(1/b*a)^(
2/3)*ln(x+(1/b*a)^(1/3))+65/54*a^3/b^6*e/(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*d/(
1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-35/54*a^2/b^5*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-14/27*a/b
^4*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-31/18*a^4/b^5/(b*x^3+a)^2*x^4*f+25/18*a^3/b^4/(b*x^3+a)^2*x^4*e-52/27*a
^4/b^7*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/7/b^3*x^7*e+1/4/b^3*x^4*d+1/b^3*c*x+7/27*a/b^4*
c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+13/18*a/b^2/(b*x^3+a)^2*x^4*c+104/27*a^4/b^7*f/(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*e/(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*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-14/27
*a/b^4*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^9*(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 = 1.44258, size = 1424, normalized size = 3.8 \begin{align*} \frac{378 \, b^{5} f x^{16} + 108 \,{\left (5 \, b^{5} e - 8 \, a b^{4} f\right )} x^{13} + 27 \,{\left (35 \, b^{5} d - 65 \, a b^{4} e + 104 \, a^{2} b^{3} f\right )} x^{10} + 270 \,{\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{7} + 735 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{4} - 140 \, \sqrt{3}{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 ) + 70 \,{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 ) - 140 \,{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 ) + 420 \,{\left (14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f\right )} x}{3780 \,{\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^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/3780*(378*b^5*f*x^16 + 108*(5*b^5*e - 8*a*b^4*f)*x^13 + 27*(35*b^5*d - 65*a*b^4*e + 104*a^2*b^3*f)*x^10 + 27
0*(14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^7 + 735*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b^2*e -
104*a^4*b*f)*x^4 - 140*sqrt(3)*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^6 + 14*a^2*b^3*c - 3
5*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(
1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) + 70*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a
^3*b^2*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^2*b^3*d + 65*a^3*b
^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*((14*b^5*c - 35*a*b^4*d + 65
*a^2*b^3*e - 104*a^3*b^2*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^
2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(14*a^2*b^3*c - 35*a^3*b^2*d
+ 65*a^4*b*e - 104*a^5*f)*x)/(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**9*(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 = 1.08754, size = 598, normalized size = 1.59 \begin{align*} -\frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{1}{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^{7}} + \frac{{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d - 104 \, a^{4} f + 65 \, a^{3} 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^{6}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{1}{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^{7}} + \frac{13 \, a b^{4} c x^{4} - 19 \, a^{2} b^{3} d x^{4} - 31 \, a^{4} b f x^{4} + 25 \, a^{3} b^{2} x^{4} e + 10 \, a^{2} b^{3} c x - 16 \, a^{3} b^{2} d x - 28 \, a^{5} f x + 22 \, a^{4} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{14 \, b^{27} f x^{10} - 60 \, a b^{26} f x^{7} + 20 \, b^{27} x^{7} e + 35 \, b^{27} d x^{4} + 210 \, a^{2} b^{25} f x^{4} - 105 \, a b^{26} x^{4} e + 140 \, b^{27} c x - 420 \, a b^{26} d x - 1400 \, a^{3} b^{24} f x + 840 \, a^{2} b^{25} x e}{140 \, b^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 35*(-a*b^2)^(1/3)*a*b^2*d - 104*(-a*b^2)^(1/3)*a^3*f + 65*(-a*b^2)^(1
/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^7 + 1/27*(14*a*b^3*c - 35*a^2*b^2*d - 104
*a^4*f + 65*a^3*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^6) - 1/54*(14*(-a*b^2)^(1/3)*b^3*c - 35*(-a*
b^2)^(1/3)*a*b^2*d - 104*(-a*b^2)^(1/3)*a^3*f + 65*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(
2/3))/b^7 + 1/18*(13*a*b^4*c*x^4 - 19*a^2*b^3*d*x^4 - 31*a^4*b*f*x^4 + 25*a^3*b^2*x^4*e + 10*a^2*b^3*c*x - 16*
a^3*b^2*d*x - 28*a^5*f*x + 22*a^4*b*x*e)/((b*x^3 + a)^2*b^6) + 1/140*(14*b^27*f*x^10 - 60*a*b^26*f*x^7 + 20*b^
27*x^7*e + 35*b^27*d*x^4 + 210*a^2*b^25*f*x^4 - 105*a*b^26*x^4*e + 140*b^27*c*x - 420*a*b^26*d*x - 1400*a^3*b^
24*f*x + 840*a^2*b^25*x*e)/b^30