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

Optimal. Leaf size=244 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{11/3} \sqrt [3]{b}}-\frac{a^2 e-a b d+b^2 c}{2 a^3 x^2}+\frac{b c-a d}{5 a^2 x^5}-\frac{c}{8 a x^8} \]

[Out]

-c/(8*a*x^8) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(2*a^3*x^2) + ((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^(11/3)*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^(11/3)*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^(11/3)*b^(1/3))

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Rubi [A]  time = 0.174033, antiderivative size = 244, 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} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{11/3} \sqrt [3]{b}}-\frac{a^2 e-a b d+b^2 c}{2 a^3 x^2}+\frac{b c-a d}{5 a^2 x^5}-\frac{c}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(8*a*x^8) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(2*a^3*x^2) + ((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^(11/3)*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^(11/3)*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^(11/3)*b^(1/3))

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^9 \left (a+b x^3\right )} \, dx &=\int \left (\frac{c}{a x^9}+\frac{-b c+a d}{a^2 x^6}+\frac{b^2 c-a b d+a^2 e}{a^3 x^3}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac{c}{8 a x^8}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{2 a^3 x^2}+\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^3}\\ &=-\frac{c}{8 a x^8}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{2 a^3 x^2}-\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^{11/3}}-\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^{11/3}}\\ &=-\frac{c}{8 a x^8}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{2 a^3 x^2}-\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^{11/3} \sqrt [3]{b}}-\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^{10/3}}+\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^{11/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a x^8}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{2 a^3 x^2}-\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^{11/3} \sqrt [3]{b}}+\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^{11/3} \sqrt [3]{b}}-\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^{11/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a x^8}+\frac{b c-a d}{5 a^2 x^5}-\frac{b^2 c-a b d+a^2 e}{2 a^3 x^2}+\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^{11/3} \sqrt [3]{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^{11/3} \sqrt [3]{b}}+\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^{11/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.111757, size = 231, normalized size = 0.95 \[ \frac{\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt [3]{b}}-\frac{60 a^{2/3} \left (a^2 e-a b d+b^2 c\right )}{x^2}+\frac{24 a^{5/3} (b c-a d)}{x^5}-\frac{15 a^{8/3} c}{x^8}}{120 a^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-15*a^(8/3)*c)/x^8 + (24*a^(5/3)*(b*c - a*d))/x^5 - (60*a^(2/3)*(b^2*c - a*b*d + a^2*e))/x^2 + (40*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]])/b^(1/3) + (40*(-(b^3*c) + a*b^
2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(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])/b^(1/3))/(120*a^(11/3))

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Maple [B]  time = 0.008, size = 441, 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),x)

[Out]

1/3/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*f-1/3/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*e+1/3/a^2*b/(1/b*a)^(2/3)*ln
(x+(1/b*a)^(1/3))*d-1/3/a^3*b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c-1/6/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x
+(1/b*a)^(2/3))*f+1/6/a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*e-1/6/a^2*b/(1/b*a)^(2/3)*ln(x^2-(
1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/6/a^3*b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+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))*f-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))*e+1/3/a^2*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^3*b^2/
(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c-1/8*c/a/x^8-1/5/a/x^5*d+1/5/a^2/x^5*b*c-1/2/
a/x^2*e+1/2/a^2/x^2*b*d-1/2/a^3/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37799, size = 1353, normalized size = 5.55 \begin{align*} \left [-\frac{60 \, \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^{8} \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 ) - 20 \,{\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^{8} \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 ) + 40 \,{\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^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 60 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}, -\frac{120 \, \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^{8} \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 ) - 20 \,{\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^{8} \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 ) + 40 \,{\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^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 60 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}\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^9/(b*x^3+a),x, algorithm="fricas")

[Out]

[-1/120*(60*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*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*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2
*b)^(1/3)*a) + 40*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x + (a^2*b)^(2/3)) + 60*(a^2*b
^3*c - a^3*b^2*d + a^4*b*e)*x^6 + 15*a^4*b*c - 24*(a^3*b^2*c - a^4*b*d)*x^3)/(a^5*b*x^8), -1/120*(120*sqrt(1/3
)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*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*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b
*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x
+ (a^2*b)^(2/3)) + 60*(a^2*b^3*c - a^3*b^2*d + a^4*b*e)*x^6 + 15*a^4*b*c - 24*(a^3*b^2*c - a^4*b*d)*x^3)/(a^5
*b*x^8)]

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Sympy [A]  time = 52.7116, size = 348, normalized size = 1.43 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{11} b - 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^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} - \frac{5 a^{2} c + x^{6} \left (20 a^{2} e - 20 a b d + 20 b^{2} c\right ) + x^{3} \left (8 a^{2} d - 8 a b c\right )}{40 a^{3} 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),x)

[Out]

RootSum(27*_t**3*a**11*b - 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**4/(a
**3*f - a**2*b*e + a*b**2*d - b**3*c) + x))) - (5*a**2*c + x**6*(20*a**2*e - 20*a*b*d + 20*b**2*c) + x**3*(8*a
**2*d - 8*a*b*c))/(40*a**3*x**8)

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Giac [A]  time = 1.07419, size = 401, normalized size = 1.64 \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^{4}} - \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^{4} b} - \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^{4} b} - \frac{20 \, b^{2} c x^{6} - 20 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 8 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{3} 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),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^4 - 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)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b) - 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^4*b) - 1/40*(20*b^2*c*x^6 - 20*a*b*d*x
^6 + 20*a^2*x^6*e - 8*a*b*c*x^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^3*x^8)