3.299 \(\int \frac{c+d x^3+e x^6+f x^9}{x^8 (a+b x^3)^3} \, dx\)

Optimal. Leaf size=343 \[ -\frac{x^2 \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^5 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{54 a^{16/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{27 a^{16/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{9 \sqrt{3} a^{16/3} b^{2/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{c}{7 a^3 x^7} \]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(4*a^4*x^4) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x) - ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(6*a^4*(a + b*x^3)^2) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^2)/(9*a^5*(a + b*x^3))
+ ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3
]*a^(16/3)*b^(2/3)) + ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(16/3)*b
^(2/3)) - ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*
a^(16/3)*b^(2/3))

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Rubi [A]  time = 0.56931, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1834, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^5 \left (a+b x^3\right )}-\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{54 a^{16/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{27 a^{16/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{9 \sqrt{3} a^{16/3} b^{2/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{c}{7 a^3 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(4*a^4*x^4) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x) - ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^2)/(6*a^4*(a + b*x^3)^2) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^2)/(9*a^5*(a + b*x^3))
+ ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3
]*a^(16/3)*b^(2/3)) + ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(16/3)*b
^(2/3)) - ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*
a^(16/3)*b^(2/3))

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 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{c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+6 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{4 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^8 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\int \frac{18 b^6 c-18 b^6 \left (\frac{2 b c}{a}-d\right ) x^3+18 b^6 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^6-\frac{2 b^6 \left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^9}{a^3}}{x^8 \left (a+b x^3\right )} \, dx}{18 a^2 b^6}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^6 c}{a x^8}+\frac{18 b^6 (-3 b c+a d)}{a^2 x^5}+\frac{18 b^6 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^2}+\frac{2 b^6 \left (-65 b^3 c+35 a b^2 d-14 a^2 b e+2 a^3 f\right ) x}{a^3 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^6}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^5}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{16/3} \sqrt [3]{b}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\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 a^{16/3} \sqrt [3]{b}}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{16/3} b^{2/3}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{16/3} b^{2/3}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^5 \sqrt [3]{b}}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{16/3} b^{2/3}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{16/3} b^{2/3}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{16/3} b^{2/3}}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{4 a^4 x^4}-\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) x^2}{9 a^5 \left (a+b x^3\right )}+\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 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} a^{16/3} b^{2/3}}+\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{16/3} b^{2/3}}-\frac{\left (65 b^3 c-35 a b^2 d+14 a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{16/3} b^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.23793, size = 328, normalized size = 0.96 \[ \frac{\frac{126 a^{4/3} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{84 \sqrt [3]{a} x^2 \left (-5 a^2 b e+2 a^3 f+8 a b^2 d-11 b^3 c\right )}{a+b x^3}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^2 b e+2 a^3 f+35 a b^2 d-65 b^3 c\right )}{b^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (14 a^2 b e-2 a^3 f-35 a b^2 d+65 b^3 c\right )}{b^{2/3}}-\frac{756 \sqrt [3]{a} \left (a^2 e-3 a b d+6 b^2 c\right )}{x}-\frac{189 a^{4/3} (a d-3 b c)}{x^4}-\frac{108 a^{7/3} c}{x^7}}{756 a^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]

[Out]

((-108*a^(7/3)*c)/x^7 - (189*a^(4/3)*(-3*b*c + a*d))/x^4 - (756*a^(1/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x + (126*
a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3)^2 + (84*a^(1/3)*(-11*b^3*c + 8*a*b^2*d - 5*a^2
*b*e + 2*a^3*f)*x^2)/(a + b*x^3) + (28*Sqrt[3]*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(1 - (2*b
^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3
)*x])/b^(2/3) + (14*(-65*b^3*c + 35*a*b^2*d - 14*a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/b^(2/3))/(756*a^(16/3))

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Maple [B]  time = 0.02, size = 611, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x)

[Out]

2/9/a^2/(b*x^3+a)^2*x^5*b*f-5/9/a^3/(b*x^3+a)^2*x^5*e*b^2-e/a^3/x-1/7*c/a^3/x^7+8/9/a^4/(b*x^3+a)^2*x^5*d*b^3-
11/9/a^5/(b*x^3+a)^2*x^5*c*b^4-13/18/a^2/(b*x^3+a)^2*x^2*b*e+19/18/a^3/(b*x^3+a)^2*x^2*b^2*d-25/18/a^4/(b*x^3+
a)^2*x^2*b^3*c-14/27/a^3*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-35/27/a^4*b*d/(1/b*
a)^(1/3)*ln(x+(1/b*a)^(1/3))+35/54/a^4*b*d/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+65/27/a^5*b^2*c
/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-65/54/a^5*b^2*c/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-2/27/a^
2*f/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/27/a^2*f/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+7/18/
a/(b*x^3+a)^2*x^2*f+14/27/a^3*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-7/27/a^3*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3
)*x+(1/b*a)^(2/3))+3/4/a^4/x^4*b*c+3/a^4/x*b*d-6/a^5/x*b^2*c-1/4/a^3/x^4*d+35/27/a^4*b*d*3^(1/2)/(1/b*a)^(1/3)
*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-65/27/a^5*b^2*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)
^(1/3)*x-1))+2/27/a^2*f*3^(1/2)/b/(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((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.76082, size = 3031, normalized size = 8.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[-1/756*(84*(65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^12 + 147*(65*a^2*b^5*c - 35*a^3*b^4*d +
14*a^4*b^3*e - 2*a^5*b^2*f)*x^9 + 108*a^5*b^2*c + 54*(65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2*e)*x^6 - 27*(1
3*a^4*b^3*c - 7*a^5*b^2*d)*x^3 + 42*sqrt(1/3)*((65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^13 +
2*(65*a^2*b^5*c - 35*a^3*b^4*d + 14*a^4*b^3*e - 2*a^5*b^2*f)*x^10 + (65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2
*e - 2*a^6*b*f)*x^7)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 +
(-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 14*((65*b^5*c - 35*a*b^4*d + 14
*a^2*b^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (65*a^2*b^3*c
- 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3))
- 28*((65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e
- 2*a^4*b*f)*x^10 + (65*a^2*b^3*c - 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*(-a*b^2)^(2/3)*log(b*x - (-a*b^
2)^(1/3)))/(a^6*b^4*x^13 + 2*a^7*b^3*x^10 + a^8*b^2*x^7), -1/756*(84*(65*a*b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e
- 2*a^4*b^3*f)*x^12 + 147*(65*a^2*b^5*c - 35*a^3*b^4*d + 14*a^4*b^3*e - 2*a^5*b^2*f)*x^9 + 108*a^5*b^2*c + 54
*(65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2*e)*x^6 - 27*(13*a^4*b^3*c - 7*a^5*b^2*d)*x^3 + 84*sqrt(1/3)*((65*a*
b^6*c - 35*a^2*b^5*d + 14*a^3*b^4*e - 2*a^4*b^3*f)*x^13 + 2*(65*a^2*b^5*c - 35*a^3*b^4*d + 14*a^4*b^3*e - 2*a^
5*b^2*f)*x^10 + (65*a^3*b^4*c - 35*a^4*b^3*d + 14*a^5*b^2*e - 2*a^6*b*f)*x^7)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(s
qrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + 14*((65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a
^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (65*a^2*b^3*c - 35*a^3*b^2*d
+ 14*a^4*b*e - 2*a^5*f)*x^7)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*((65*b^5*c
- 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^
10 + (65*a^2*b^3*c - 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^6*
b^4*x^13 + 2*a^7*b^3*x^10 + a^8*b^2*x^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((f*x**9+e*x**6+d*x**3+c)/x**8/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.09876, size = 586, normalized size = 1.71 \begin{align*} \frac{{\left (65 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 35 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 14 \, a^{2} 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^{6}} + \frac{\sqrt{3}{\left (65 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 14 \, \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 \, a^{6} b^{2}} - \frac{22 \, b^{4} c x^{5} - 16 \, a b^{3} d x^{5} - 4 \, a^{3} b f x^{5} + 10 \, a^{2} b^{2} x^{5} e + 25 \, a b^{3} c x^{2} - 19 \, a^{2} b^{2} d x^{2} - 7 \, a^{4} f x^{2} + 13 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{5}} - \frac{{\left (65 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 14 \, \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 \, a^{6} b^{2}} - \frac{168 \, b^{2} c x^{6} - 84 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 21 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{5} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^8/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*(65*b^3*c*(-a/b)^(1/3) - 35*a*b^2*d*(-a/b)^(1/3) - 2*a^3*f*(-a/b)^(1/3) + 14*a^2*b*(-a/b)^(1/3)*e)*(-a/b)
^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 + 1/27*sqrt(3)*(65*(-a*b^2)^(2/3)*b^3*c - 35*(-a*b^2)^(2/3)*a*b^2*d - 2*
(-a*b^2)^(2/3)*a^3*f + 14*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b
^2) - 1/18*(22*b^4*c*x^5 - 16*a*b^3*d*x^5 - 4*a^3*b*f*x^5 + 10*a^2*b^2*x^5*e + 25*a*b^3*c*x^2 - 19*a^2*b^2*d*x
^2 - 7*a^4*f*x^2 + 13*a^3*b*x^2*e)/((b*x^3 + a)^2*a^5) - 1/54*(65*(-a*b^2)^(2/3)*b^3*c - 35*(-a*b^2)^(2/3)*a*b
^2*d - 2*(-a*b^2)^(2/3)*a^3*f + 14*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*b^2)
- 1/28*(168*b^2*c*x^6 - 84*a*b*d*x^6 + 28*a^2*x^6*e - 21*a*b*c*x^3 + 7*a^2*d*x^3 + 4*a^2*c)/(a^5*x^7)