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

Optimal. Leaf size=260 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2} \]

[Out]

-c/(2*a^2*x^2) + (f*x)/b^2 - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^2*b^2*(a + b*x^3)) + ((5*b^3*c - 2*a
*b^2*d - a^2*b*e + 4*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(7/3)) - (
(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(7/3)) + ((5*b^3*c - 2*a*b^2*
d - a^2*b*e + 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(7/3))

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Rubi [A]  time = 0.246217, antiderivative size = 260, 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, 1488, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a^2*x^2) + (f*x)/b^2 - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^2*b^2*(a + b*x^3)) + ((5*b^3*c - 2*a
*b^2*d - a^2*b*e + 4*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(7/3)) - (
(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(7/3)) + ((5*b^3*c - 2*a*b^2*
d - a^2*b*e + 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(7/3))

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 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 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 )^2} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+b \left (\frac{2 b^3 c}{a}-2 b^2 d-a b e+a^2 f\right ) x^3-3 a b^2 f x^6}{x^3 \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^2 b^2 \left (a+b x^3\right )}-\frac{\int \left (-3 a b f-\frac{3 b^3 c}{a x^3}+\frac{b \left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right )}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 a^2 b^2}\\ &=-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3} b^2}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 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^{8/3} b^2}\\ &=-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} b^{7/3}}+\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 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^{8/3} b^{7/3}}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 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^{7/3} b^2}\\ &=-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} b^{7/3}}+\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} b^{7/3}}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 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^{8/3} b^{7/3}}\\ &=-\frac{c}{2 a^2 x^2}+\frac{f x}{b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 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^{8/3} b^{7/3}}-\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} b^{7/3}}+\frac{\left (5 b^3 c-2 a b^2 d-a^2 b e+4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} b^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.152077, size = 250, normalized size = 0.96 \[ \frac{1}{18} \left (\frac{6 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+4 a^3 f-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}-\frac{9 c}{a^2 x^2}+\frac{18 f x}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*c)/(a^2*x^2) + (18*f*x)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a^2*b^2*(a + b*x^3)) + (2*Sqr
t[3]*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(8/3)*b^(7/3))
- (2*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(8/3)*b^(7/3)) + ((5*b^3*c - 2*a*b
^2*d - a^2*b*e + 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(8/3)*b^(7/3)))/18

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Maple [B]  time = 0.011, size = 463, 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^3/(b*x^3+a)^2,x)

[Out]

f*x/b^2+1/3/b^2*a*x/(b*x^3+a)*f-1/3/b*x/(b*x^3+a)*e+1/3/a*x/(b*x^3+a)*d-1/3*b/a^2*x/(b*x^3+a)*c-4/9/b^3*a*f/(1
/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+2/9/b^3*a*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-4/9/b^3*a*f/(1
/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+2/9/b/a*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-1/9/
b/a*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/b/a*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(
2/(1/b*a)^(1/3)*x-1))-5/9/a^2*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+5/18/a^2*c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3
)*x+(1/b*a)^(2/3))-5/9/a^2*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/9/b^2*e/(1/b*a)
^(2/3)*ln(x+(1/b*a)^(1/3))-1/18/b^2*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/9/b^2*e/(1/b*a)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/2*c/a^2/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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4632, size = 1952, normalized size = 7.51 \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^3/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[1/18*(18*a^4*b^2*f*x^6 - 9*a^3*b^3*c - 3*(5*a^2*b^4*c - 2*a^3*b^3*d + 2*a^4*b^2*e - 8*a^5*b*f)*x^3 + 3*sqrt(1
/3)*((5*a*b^5*c - 2*a^2*b^4*d - a^3*b^3*e + 4*a^4*b^2*f)*x^5 + (5*a^2*b^4*c - 2*a^3*b^3*d - a^4*b^2*e + 4*a^5*
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)) + ((5*b^4*c - 2*a*b^3*d - a^2*b^2*e + 4*a^
3*b*f)*x^5 + (5*a*b^3*c - 2*a^2*b^2*d - a^3*b*e + 4*a^4*f)*x^2)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x
- (-a^2*b)^(1/3)*a) - 2*((5*b^4*c - 2*a*b^3*d - a^2*b^2*e + 4*a^3*b*f)*x^5 + (5*a*b^3*c - 2*a^2*b^2*d - a^3*b*
e + 4*a^4*f)*x^2)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)))/(a^4*b^4*x^5 + a^5*b^3*x^2), 1/18*(18*a^4*b^2*f*
x^6 - 9*a^3*b^3*c - 3*(5*a^2*b^4*c - 2*a^3*b^3*d + 2*a^4*b^2*e - 8*a^5*b*f)*x^3 - 6*sqrt(1/3)*((5*a*b^5*c - 2*
a^2*b^4*d - a^3*b^3*e + 4*a^4*b^2*f)*x^5 + (5*a^2*b^4*c - 2*a^3*b^3*d - a^4*b^2*e + 4*a^5*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) + ((5*b^4*
c - 2*a*b^3*d - a^2*b^2*e + 4*a^3*b*f)*x^5 + (5*a*b^3*c - 2*a^2*b^2*d - a^3*b*e + 4*a^4*f)*x^2)*(-a^2*b)^(2/3)
*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) - 2*((5*b^4*c - 2*a*b^3*d - a^2*b^2*e + 4*a^3*b*f)*x^5 + (
5*a*b^3*c - 2*a^2*b^2*d - a^3*b*e + 4*a^4*f)*x^2)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)))/(a^4*b^4*x^5 + a
^5*b^3*x^2)]

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Sympy [A]  time = 58.6862, size = 381, normalized size = 1.47 \begin{align*} \frac{- 3 a b^{2} c + x^{3} \left (2 a^{3} f - 2 a^{2} b e + 2 a b^{2} d - 5 b^{3} c\right )}{6 a^{3} b^{2} x^{2} + 6 a^{2} b^{3} x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} b^{7} + 64 a^{9} f^{3} - 48 a^{8} b e f^{2} - 96 a^{7} b^{2} d f^{2} + 12 a^{7} b^{2} e^{2} f + 240 a^{6} b^{3} c f^{2} + 48 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} - 120 a^{5} b^{4} c e f + 48 a^{5} b^{4} d^{2} f - 6 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f + 15 a^{4} b^{5} c e^{2} - 12 a^{4} b^{5} d^{2} e + 300 a^{3} b^{6} c^{2} f + 60 a^{3} b^{6} c d e - 8 a^{3} b^{6} d^{3} - 75 a^{2} b^{7} c^{2} e + 60 a^{2} b^{7} c d^{2} - 150 a b^{8} c^{2} d + 125 b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{3} b^{2}}{4 a^{3} f - a^{2} b e - 2 a b^{2} d + 5 b^{3} c} + x \right )} \right )\right )} + \frac{f x}{b^{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)**2,x)

[Out]

(-3*a*b**2*c + x**3*(2*a**3*f - 2*a**2*b*e + 2*a*b**2*d - 5*b**3*c))/(6*a**3*b**2*x**2 + 6*a**2*b**3*x**5) + R
ootSum(729*_t**3*a**8*b**7 + 64*a**9*f**3 - 48*a**8*b*e*f**2 - 96*a**7*b**2*d*f**2 + 12*a**7*b**2*e**2*f + 240
*a**6*b**3*c*f**2 + 48*a**6*b**3*d*e*f - a**6*b**3*e**3 - 120*a**5*b**4*c*e*f + 48*a**5*b**4*d**2*f - 6*a**5*b
**4*d*e**2 - 240*a**4*b**5*c*d*f + 15*a**4*b**5*c*e**2 - 12*a**4*b**5*d**2*e + 300*a**3*b**6*c**2*f + 60*a**3*
b**6*c*d*e - 8*a**3*b**6*d**3 - 75*a**2*b**7*c**2*e + 60*a**2*b**7*c*d**2 - 150*a*b**8*c**2*d + 125*b**9*c**3,
Lambda(_t, _t*log(-9*_t*a**3*b**2/(4*a**3*f - a**2*b*e - 2*a*b**2*d + 5*b**3*c) + x))) + f*x/b**2

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Giac [A]  time = 1.08146, size = 417, normalized size = 1.6 \begin{align*} \frac{f x}{b^{2}} + \frac{{\left (5 \, b^{3} c - 2 \, a b^{2} d + 4 \, 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 )}{9 \, a^{3} b^{2}} - \frac{c}{2 \, a^{2} x^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \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 )}{9 \, a^{3} b^{3}} - \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^{2} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \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 )}{18 \, a^{3} 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)^2,x, algorithm="giac")

[Out]

f*x/b^2 + 1/9*(5*b^3*c - 2*a*b^2*d + 4*a^3*f - a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^2) - 1/
2*c/(a^2*x^2) - 1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*b^3*c - 2*(-a*b^2)^(1/3)*a*b^2*d + 4*(-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^3*b^3) - 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^2*b^2) - 1/18*(5*(-a*b^2)^(1/3)*b^3*c - 2*(-a*b^2)^(1/3)*a*b^2*d + 4*(-
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^3*b^3)