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

Optimal. Leaf size=265 \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{3 \sqrt{3} a^{7/3} b^{8/3}}-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.253483, antiderivative size = 265, 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, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{3 \sqrt{3} a^{7/3} b^{8/3}}-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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^2 \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^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+b \left (\frac{b^3 c}{a}-b^2 d-2 a b e+2 a^2 f\right ) x^3-3 a b^2 f x^6}{x^2 \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^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^2}-3 a b f x+\frac{b \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) x}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^2 b^2}\\ &=-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3} b^{7/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\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}{9 a^{7/3} b^{7/3}}\\ &=-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 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^{7/3} b^{8/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 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^2 b^{7/3}}\\ &=-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{8/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 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^{7/3} b^{8/3}}\\ &=-\frac{c}{a^2 x}+\frac{f x^2}{2 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 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^{7/3} b^{8/3}}+\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac{\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.152978, size = 255, normalized size = 0.96 \[ \frac{1}{18} \left (\frac{6 x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^2 b e+5 a^3 f-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}-\frac{18 c}{a^2 x}+\frac{9 f x^2}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.012, size = 474, 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^2/(b*x^3+a)^2,x)

[Out]

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

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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^2/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48022, size = 1877, normalized size = 7.08 \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^2/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 26.6082, size = 457, normalized size = 1.72 \begin{align*} \frac{- 3 a b^{2} c + x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - 4 b^{3} c\right )}{3 a^{3} b^{2} x + 3 a^{2} b^{3} x^{4}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} b^{8} - 125 a^{9} f^{3} + 150 a^{8} b e f^{2} + 75 a^{7} b^{2} d f^{2} - 60 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 60 a^{6} b^{3} d e f + 8 a^{6} b^{3} e^{3} + 240 a^{5} b^{4} c e f - 15 a^{5} b^{4} d^{2} f + 12 a^{5} b^{4} d e^{2} + 120 a^{4} b^{5} c d f - 48 a^{4} b^{5} c e^{2} + 6 a^{4} b^{5} d^{2} e - 240 a^{3} b^{6} c^{2} f - 48 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e - 12 a^{2} b^{7} c d^{2} + 48 a b^{8} c^{2} d - 64 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5} b^{5}}{25 a^{6} f^{2} - 20 a^{5} b e f - 10 a^{4} b^{2} d f + 4 a^{4} b^{2} e^{2} + 40 a^{3} b^{3} c f + 4 a^{3} b^{3} d e - 16 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 8 a b^{5} c d + 16 b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{2}}{2 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**2/(b*x**3+a)**2,x)

[Out]

(-3*a*b**2*c + x**3*(a**3*f - a**2*b*e + a*b**2*d - 4*b**3*c))/(3*a**3*b**2*x + 3*a**2*b**3*x**4) + RootSum(72
9*_t**3*a**7*b**8 - 125*a**9*f**3 + 150*a**8*b*e*f**2 + 75*a**7*b**2*d*f**2 - 60*a**7*b**2*e**2*f - 300*a**6*b
**3*c*f**2 - 60*a**6*b**3*d*e*f + 8*a**6*b**3*e**3 + 240*a**5*b**4*c*e*f - 15*a**5*b**4*d**2*f + 12*a**5*b**4*
d*e**2 + 120*a**4*b**5*c*d*f - 48*a**4*b**5*c*e**2 + 6*a**4*b**5*d**2*e - 240*a**3*b**6*c**2*f - 48*a**3*b**6*
c*d*e + a**3*b**6*d**3 + 96*a**2*b**7*c**2*e - 12*a**2*b**7*c*d**2 + 48*a*b**8*c**2*d - 64*b**9*c**3, Lambda(_
t, _t*log(81*_t**2*a**5*b**5/(25*a**6*f**2 - 20*a**5*b*e*f - 10*a**4*b**2*d*f + 4*a**4*b**2*e**2 + 40*a**3*b**
3*c*f + 4*a**3*b**3*d*e - 16*a**2*b**4*c*e + a**2*b**4*d**2 - 8*a*b**5*c*d + 16*b**6*c**2) + x))) + f*x**2/(2*
b**2)

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Giac [A]  time = 1.07628, size = 477, normalized size = 1.8 \begin{align*} \frac{f x^{2}}{2 \, b^{2}} + \frac{{\left (4 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{2} b \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 )}{9 \, a^{3} b^{2}} - \frac{4 \, b^{3} c x^{3} - a b^{2} d x^{3} - a^{3} f x^{3} + a^{2} b x^{3} e + 3 \, a b^{2} c}{3 \,{\left (b x^{4} + a x\right )} a^{2} b^{2}} + \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 2 \, \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 )}{9 \, a^{3} b^{4}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 2 \, \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 )}{18 \, a^{3} b^{4}} \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^2/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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