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

Optimal. Leaf size=224 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{5/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{5/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b e-a f)}{b^2}-\frac{c}{2 a x^2}+\frac{f x^4}{4 b} \]

[Out]

-c/(2*a*x^2) + ((b*e - a*f)*x)/b^2 + (f*x^4)/(4*b) + ((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^(5/3)*b^(7/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) +
b^(1/3)*x])/(3*a^(5/3)*b^(7/3)) + ((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^(5/3)*b^(7/3))

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Rubi [A]  time = 0.169822, antiderivative size = 224, 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, 200, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{5/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{5/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b e-a f)}{b^2}-\frac{c}{2 a x^2}+\frac{f x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a*x^2) + ((b*e - a*f)*x)/b^2 + (f*x^4)/(4*b) + ((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^(5/3)*b^(7/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) +
b^(1/3)*x])/(3*a^(5/3)*b^(7/3)) + ((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^(5/3)*b^(7/3))

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

Mathematica [A]  time = 0.117203, size = 218, normalized size = 0.97 \[ \frac{1}{12} \left (\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{12 x (b e-a f)}{b^2}-\frac{6 c}{a x^2}+\frac{3 f x^4}{b}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*c)/(a*x^2) + (12*(b*e - a*f)*x)/b^2 + (3*f*x^4)/b + (4*Sqrt[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]])/(a^(5/3)*b^(7/3)) + (4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/
3) + b^(1/3)*x])/(a^(5/3)*b^(7/3)) + (2*(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])/(a^(5/3)*b^(7/3)))/12

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Maple [B]  time = 0.005, size = 414, normalized size = 1.9 \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^3/(b*x^3+a),x)

[Out]

1/4*f*x^4/b-1/b^2*a*f*x+e*x/b+1/3/b^3*a^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f-1/3/b^2*a/(1/b*a)^(2/3)*ln(x+(1/
b*a)^(1/3))*e+1/3/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d-1/3/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c-1/6/b^3*a^2/
(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f+1/6/b^2*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(
2/3))*e-1/6/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/6/a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x
+(1/b*a)^(2/3))*c+1/3/b^3*a^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*f-1/3/b^2*a/(1/b
*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(
2/(1/b*a)^(1/3)*x-1))*d-1/3/a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-1/2*c/a/x^2

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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^3/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53733, size = 1280, normalized size = 5.71 \begin{align*} \left [\frac{3 \, a^{3} b^{2} f x^{6} - 6 \, a^{2} b^{3} c - 6 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{2} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) + 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac{2}{3}} x^{2} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) - 4 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac{2}{3}} x^{2} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 12 \,{\left (a^{3} b^{2} e - a^{4} b f\right )} x^{3}}{12 \, a^{3} b^{3} x^{2}}, \frac{3 \, a^{3} b^{2} f x^{6} - 6 \, a^{2} b^{3} c - 12 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{2} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) + 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac{2}{3}} x^{2} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) - 4 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac{2}{3}} x^{2} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 12 \,{\left (a^{3} b^{2} e - a^{4} b f\right )} x^{3}}{12 \, a^{3} b^{3} x^{2}}\right ] \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^3/(b*x^3+a),x, algorithm="fricas")

[Out]

[1/12*(3*a^3*b^2*f*x^6 - 6*a^2*b^3*c - 6*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^2*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)) + 2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*
x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x +
(a^2*b)^(2/3)) + 12*(a^3*b^2*e - a^4*b*f)*x^3)/(a^3*b^3*x^2), 1/12*(3*a^3*b^2*f*x^6 - 6*a^2*b^3*c - 12*sqrt(1
/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^2*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) + 2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*
b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x
+ (a^2*b)^(2/3)) + 12*(a^3*b^2*e - a^4*b*f)*x^3)/(a^3*b^3*x^2)]

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Sympy [A]  time = 3.05793, size = 326, normalized size = 1.46 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{5} b^{7} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2} b^{2}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{4}}{4 b} - \frac{x \left (a f - b e\right )}{b^{2}} - \frac{c}{2 a x^{2}} \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**3/(b*x**3+a),x)

[Out]

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

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Giac [A]  time = 1.07595, size = 378, normalized size = 1.69 \begin{align*} \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + 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 )}{3 \, a^{2} b^{2}} - \frac{c}{2 \, a x^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \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 )}{3 \, a^{2} b^{3}} + \frac{b^{3} f x^{4} - 4 \, a b^{2} f x + 4 \, b^{3} x e}{4 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \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 )}{6 \, a^{2} b^{3}} \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^3/(b*x^3+a),x, algorithm="giac")

[Out]

1/3*(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2) - 1/2*c/(a*x^2) - 1/
3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-a*b^2)^(1/3)*a^3*f + (-a*b^2)^(1/3)*a^2*b*e)*arct
an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^3) + 1/4*(b^3*f*x^4 - 4*a*b^2*f*x + 4*b^3*x*e)/b^4 -
1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-a*b^2)^(1/3)*a^3*f + (-a*b^2)^(1/3)*a^2*b*e)*log(x^2 +
x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^3)