3.246 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)} \, dx\)

Optimal. Leaf size=277 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{13/3}}-\frac{a^2 e-a b d+b^2 c}{4 a^3 x^4}+\frac{b c-a d}{7 a^2 x^7}-\frac{c}{10 a x^{10}} \]

[Out]

-c/(10*a*x^10) + (b*c - a*d)/(7*a^2*x^7) - (b^2*c - a*b*d + a^2*e)/(4*a^3*x^4) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(a^4*x) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(Sqrt[3]*a^(13/3)) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(13/3)) +
(b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(13/3))

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Rubi [A]  time = 0.222478, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {1834, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{13/3}}-\frac{a^2 e-a b d+b^2 c}{4 a^3 x^4}+\frac{b c-a d}{7 a^2 x^7}-\frac{c}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

-c/(10*a*x^10) + (b*c - a*d)/(7*a^2*x^7) - (b^2*c - a*b*d + a^2*e)/(4*a^3*x^4) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(a^4*x) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(Sqrt[3]*a^(13/3)) - (b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(13/3)) +
(b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(13/3))

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^{11} \left (a+b x^3\right )} \, dx &=\int \left (\frac{c}{a x^{11}}+\frac{-b c+a d}{a^2 x^8}+\frac{b^2 c-a b d+a^2 e}{a^3 x^5}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac{c}{10 a x^{10}}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac{\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{a^4}\\ &=-\frac{c}{10 a x^{10}}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac{\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{13/3}}+\frac{\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\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}{3 a^{13/3}}\\ &=-\frac{c}{10 a x^{10}}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac{\left (\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-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}{6 a^{13/3}}+\frac{\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^4}\\ &=-\frac{c}{10 a x^{10}}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{13/3}}+\frac{\left (\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-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 )}{a^{13/3}}\\ &=-\frac{c}{10 a x^{10}}+\frac{b c-a d}{7 a^2 x^7}-\frac{b^2 c-a b d+a^2 e}{4 a^3 x^4}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}-\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{13/3}}-\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{13/3}}+\frac{\sqrt [3]{b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.112328, size = 266, normalized size = 0.96 \[ \frac{70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+\frac{420 \sqrt [3]{a} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )-\frac{105 a^{4/3} \left (a^2 e-a b d+b^2 c\right )}{x^4}+\frac{60 a^{7/3} (b c-a d)}{x^7}-\frac{42 a^{10/3} c}{x^{10}}}{420 a^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

((-42*a^(10/3)*c)/x^10 + (60*a^(7/3)*(b*c - a*d))/x^7 - (105*a^(4/3)*(b^2*c - a*b*d + a^2*e))/x^4 + (420*a^(1/
3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/x - 140*Sqrt[3]*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x
] + 70*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*a^(13/
3))

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Maple [B]  time = 0.009, size = 491, 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^11/(b*x^3+a),x)

[Out]

1/3/a/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f-1/3*b/a^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*e+1/3*b^2/a^3/(1/b*a)^(1
/3)*ln(x+(1/b*a)^(1/3))*d-1/3*b^3/a^4/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c-1/6/a/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(
1/3)*x+(1/b*a)^(2/3))*f+1/6*b/a^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e-1/6*b^2/a^3/(1/b*a)^(1
/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/6*b^3/a^4/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c-
1/3/a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*f+1/3*b/a^2*3^(1/2)/(1/b*a)^(1/3)*arctan
(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e-1/3*b^2/a^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-
1))*d+1/3*b^3/a^4*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-1/10*c/a/x^10-1/7/a/x^7*d+
1/7/a^2/x^7*b*c-1/4/a/x^4*e+1/4/a^2/x^4*b*d-1/4/a^3/x^4*b^2*c-1/a/x*f+1/a^2/x*b*e-1/a^3/x*b^2*d+1/a^4/x*b^3*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^11/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40998, size = 597, normalized size = 2.16 \begin{align*} \frac{140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 70 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 105 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 42 \, a^{3} c + 60 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}}{420 \, a^{4} x^{10}} \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^11/(b*x^3+a),x, algorithm="fricas")

[Out]

1/420*(140*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3
*sqrt(3)) + 70*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3
)) - 140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) + 420*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^9 - 105*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 42*a^3*c + 60*(a^2*b*c - a^3*d)*x^3)/(a^4*x^10)

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Sympy [A]  time = 142.622, size = 473, normalized size = 1.71 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{13} - a^{9} b f^{3} + 3 a^{8} b^{2} e f^{2} - 3 a^{7} b^{3} d f^{2} - 3 a^{7} b^{3} e^{2} f + 3 a^{6} b^{4} c f^{2} + 6 a^{6} b^{4} d e f + a^{6} b^{4} e^{3} - 6 a^{5} b^{5} c e f - 3 a^{5} b^{5} d^{2} f - 3 a^{5} b^{5} d e^{2} + 6 a^{4} b^{6} c d f + 3 a^{4} b^{6} c e^{2} + 3 a^{4} b^{6} d^{2} e - 3 a^{3} b^{7} c^{2} f - 6 a^{3} b^{7} c d e - a^{3} b^{7} d^{3} + 3 a^{2} b^{8} c^{2} e + 3 a^{2} b^{8} c d^{2} - 3 a b^{9} c^{2} d + b^{10} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{9}}{a^{6} b f^{2} - 2 a^{5} b^{2} e f + 2 a^{4} b^{3} d f + a^{4} b^{3} e^{2} - 2 a^{3} b^{4} c f - 2 a^{3} b^{4} d e + 2 a^{2} b^{5} c e + a^{2} b^{5} d^{2} - 2 a b^{6} c d + b^{7} c^{2}} + x \right )} \right )\right )} - \frac{14 a^{3} c + x^{9} \left (140 a^{3} f - 140 a^{2} b e + 140 a b^{2} d - 140 b^{3} c\right ) + x^{6} \left (35 a^{3} e - 35 a^{2} b d + 35 a b^{2} c\right ) + x^{3} \left (20 a^{3} d - 20 a^{2} b c\right )}{140 a^{4} x^{10}} \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**11/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**13 - a**9*b*f**3 + 3*a**8*b**2*e*f**2 - 3*a**7*b**3*d*f**2 - 3*a**7*b**3*e**2*f + 3*a**6*b
**4*c*f**2 + 6*a**6*b**4*d*e*f + a**6*b**4*e**3 - 6*a**5*b**5*c*e*f - 3*a**5*b**5*d**2*f - 3*a**5*b**5*d*e**2
+ 6*a**4*b**6*c*d*f + 3*a**4*b**6*c*e**2 + 3*a**4*b**6*d**2*e - 3*a**3*b**7*c**2*f - 6*a**3*b**7*c*d*e - a**3*
b**7*d**3 + 3*a**2*b**8*c**2*e + 3*a**2*b**8*c*d**2 - 3*a*b**9*c**2*d + b**10*c**3, Lambda(_t, _t*log(9*_t**2*
a**9/(a**6*b*f**2 - 2*a**5*b**2*e*f + 2*a**4*b**3*d*f + a**4*b**3*e**2 - 2*a**3*b**4*c*f - 2*a**3*b**4*d*e + 2
*a**2*b**5*c*e + a**2*b**5*d**2 - 2*a*b**6*c*d + b**7*c**2) + x))) - (14*a**3*c + x**9*(140*a**3*f - 140*a**2*
b*e + 140*a*b**2*d - 140*b**3*c) + x**6*(35*a**3*e - 35*a**2*b*d + 35*a*b**2*c) + x**3*(20*a**3*d - 20*a**2*b*
c))/(140*a**4*x**10)

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Giac [A]  time = 1.08536, size = 508, normalized size = 1.83 \begin{align*} -\frac{{\left (b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{2} \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 )}{3 \, a^{5}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{3 \, a^{5} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{6 \, a^{5} b} + \frac{140 \, b^{3} c x^{9} - 140 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 140 \, a^{2} b x^{9} e - 35 \, a b^{2} c x^{6} + 35 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 20 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{4} x^{10}} \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^11/(b*x^3+a),x, algorithm="giac")

[Out]

-1/3*(b^4*c*(-a/b)^(1/3) - a*b^3*d*(-a/b)^(1/3) - a^3*b*f*(-a/b)^(1/3) + a^2*b^2*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*
log(abs(x - (-a/b)^(1/3)))/a^5 - 1/3*sqrt(3)*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a
^3*f + (-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^5*b) + 1/6*((-a*b^2)^(
2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^5*b) + 1/140*(140*b^3*c*x^9 - 140*a*b^2*d*x^9 - 140*a^3*f*x^9 + 140*a^2*b*x^9*e - 35*a*b^2*
c*x^6 + 35*a^2*b*d*x^6 - 35*a^3*x^6*e + 20*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^4*x^10)