3.272 \(\int \frac{c+d x^3+e x^6+f x^9}{x^9 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=297 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8} \]

[Out]

-c/(8*a^2*x^8) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(2*a^4*x^2) - ((b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x)/(3*a^4*(a + b*x^3)) + ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(14/3)*b^(1/3)) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*Log[a^(
1/3) + b^(1/3)*x])/(9*a^(14/3)*b^(1/3)) + ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(18*a^(14/3)*b^(1/3))

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Rubi [A]  time = 0.369952, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1834, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^2),x]

[Out]

-c/(8*a^2*x^8) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(2*a^4*x^2) - ((b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x)/(3*a^4*(a + b*x^3)) + ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(14/3)*b^(1/3)) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*Log[a^(
1/3) + b^(1/3)*x])/(9*a^(14/3)*b^(1/3)) + ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(18*a^(14/3)*b^(1/3))

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 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{c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^9}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^6}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^3}-\frac{b^3 \left (-11 b^3 c+8 a b^2 d-5 a^2 b e+2 a^3 f\right )}{a^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 a^4}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{14/3}}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\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}{9 a^{14/3}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 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}{6 a^{13/3}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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}{18 a^{14/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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 )}{18 a^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 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 )}{3 a^{14/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}+\frac{\left (11 b^3 c-8 a b^2 d+5 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 )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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 )}{18 a^{14/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.178107, size = 280, normalized size = 0.94 \[ \frac{\frac{120 a^{2/3} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+2 a^3 f+8 a b^2 d-11 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}-\frac{180 a^{2/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^2}-\frac{72 a^{5/3} (a d-2 b c)}{x^5}-\frac{45 a^{8/3} c}{x^8}}{360 a^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^2),x]

[Out]

((-45*a^(8/3)*c)/x^8 - (72*a^(5/3)*(-2*b*c + a*d))/x^5 - (180*a^(2/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^2 + (120*
a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e
- 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-11*b^3*c + 8*a*b^2*d - 5*a^2*b*e + 2*a
^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(11*b^3*c - 8*a*b^2*d + 5*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^(1/3))/(360*a^(14/3))

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Maple [B]  time = 0.014, size = 520, 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^9/(b*x^3+a)^2,x)

[Out]

1/3/a*x/(b*x^3+a)*f-1/3/a^2*x/(b*x^3+a)*b*e+1/3/a^3*x/(b*x^3+a)*b^2*d-1/3/a^4*x/(b*x^3+a)*b^3*c-5/9/a^2*e/(1/b
*a)^(2/3)*ln(x+(1/b*a)^(1/3))+5/18/a^2*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-5/9/a^2*e/(1/b*a)
^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+8/9/a^3*b*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-4/9/a^3
*b*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+8/9/a^3*b*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*
(2/(1/b*a)^(1/3)*x-1))-11/9/a^4*b^2*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+11/18/a^4*b^2*c/(1/b*a)^(2/3)*ln(x^2-(
1/b*a)^(1/3)*x+(1/b*a)^(2/3))-11/9/a^4*b^2*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+2
/9/a*f/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-1/9/a*f/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/a
*f/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/8*c/a^2/x^8-1/5/a^2/x^5*d+2/5/a^3/x^5*b
*c-1/2/a^2/x^2*e+1/a^3/x^2*b*d-3/2/a^4/x^2*b^2*c

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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^9/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55454, size = 2182, normalized size = 7.35 \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^9/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[-1/360*(60*(11*a^2*b^4*c - 8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*b*f)*x^9 + 45*a^5*b*c + 36*(11*a^3*b^3*c - 8*a^4
*b^2*d + 5*a^5*b*e)*x^6 - 9*(11*a^4*b^2*c - 8*a^5*b*d)*x^3 + 60*sqrt(1/3)*((11*a*b^5*c - 8*a^2*b^4*d + 5*a^3*b
^3*e - 2*a^4*b^2*f)*x^11 + (11*a^2*b^4*c - 8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*b*f)*x^8)*sqrt(-(a^2*b)^(1/3)/b)*
log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(
-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 20*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*
a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((11
*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*(a^
2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^2*x^11 + a^7*b*x^8), -1/360*(60*(11*a^2*b^4*c - 8*a^3*b^3*d + 5*
a^4*b^2*e - 2*a^5*b*f)*x^9 + 45*a^5*b*c + 36*(11*a^3*b^3*c - 8*a^4*b^2*d + 5*a^5*b*e)*x^6 - 9*(11*a^4*b^2*c -
8*a^5*b*d)*x^3 + 120*sqrt(1/3)*((11*a*b^5*c - 8*a^2*b^4*d + 5*a^3*b^3*e - 2*a^4*b^2*f)*x^11 + (11*a^2*b^4*c -
8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*b*f)*x^8)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b
)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 20*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*
c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4
0*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^
8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^2*x^11 + a^7*b*x^8)]

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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**9/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.0871, size = 468, normalized size = 1.58 \begin{align*} \frac{{\left (11 \, b^{3} c - 8 \, a b^{2} d - 2 \, a^{3} f + 5 \, a^{2} 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 )}{9 \, a^{5}} - \frac{\sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \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 )}{9 \, a^{5} b} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{4}} - \frac{{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \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 )}{18 \, a^{5} b} - \frac{60 \, b^{2} c x^{6} - 40 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 16 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{4} x^{8}} \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^9/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*(11*b^3*c - 8*a*b^2*d - 2*a^3*f + 5*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^5 - 1/9*sqrt(3)*(11
*(-a*b^2)^(1/3)*b^3*c - 8*(-a*b^2)^(1/3)*a*b^2*d - 2*(-a*b^2)^(1/3)*a^3*f + 5*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1
/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) - 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b*x*e)/((b*x^
3 + a)*a^4) - 1/18*(11*(-a*b^2)^(1/3)*b^3*c - 8*(-a*b^2)^(1/3)*a*b^2*d - 2*(-a*b^2)^(1/3)*a^3*f + 5*(-a*b^2)^(
1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/40*(60*b^2*c*x^6 - 40*a*b*d*x^6 + 20*a^2*x^
6*e - 16*a*b*c*x^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^4*x^8)