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

Optimal. Leaf size=271 \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{4/3} b^{11/3}}+\frac{x^2 (b e-2 a f)}{2 b^3}+\frac{f x^5}{5 b^2} \]

[Out]

((b*e - 2*a*f)*x^2)/(2*b^3) + (f*x^5)/(5*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a*b^3*(a + b*x^3)
) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a
^(4/3)*b^(11/3)) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^(11/3)) +
((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(11/
3))

________________________________________________________________________________________

Rubi [A]  time = 0.289389, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {1828, 1594, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{4/3} b^{11/3}}+\frac{x^2 (b e-2 a f)}{2 b^3}+\frac{f x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*e - 2*a*f)*x^2)/(2*b^3) + (f*x^5)/(5*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a*b^3*(a + b*x^3)
) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a
^(4/3)*b^(11/3)) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^(11/3)) +
((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(11/
3))

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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{x \left (c+d x^3+e x^6+f x^9\right )}{\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 b^3 \left (a+b x^3\right )}-\frac{\int \frac{-b \left (b^3 c+2 a b^2 d-2 a^2 b e+2 a^3 f\right ) x-3 a b^2 (b e-a f) x^4-3 a b^3 f x^7}{a+b x^3} \, dx}{3 a b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\int \frac{x \left (-b \left (b^3 c+2 a b^2 d-2 a^2 b e+2 a^3 f\right )-3 a b^2 (b e-a f) x^3-3 a b^3 f x^6\right )}{a+b x^3} \, dx}{3 a b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\int \left (-3 a b (b e-2 a f) x-3 a b^2 f x^4+\frac{\left (-b^4 c-2 a b^3 d+5 a^2 b^2 e-8 a^3 b f\right ) x}{a+b x^3}\right ) \, dx}{3 a b^4}\\ &=\frac{(b e-2 a f) x^2}{2 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac{(b e-2 a f) x^2}{2 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} b^{10/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 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^{4/3} b^{10/3}}\\ &=\frac{(b e-2 a f) x^2}{2 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{11/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 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^{4/3} b^{11/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 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 b^{10/3}}\\ &=\frac{(b e-2 a f) x^2}{2 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{11/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{11/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 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^{4/3} b^{11/3}}\\ &=\frac{(b e-2 a f) x^2}{2 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a b^3 \left (a+b x^3\right )}-\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 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^{4/3} b^{11/3}}-\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{11/3}}+\frac{\left (b^3 c+2 a b^2 d-5 a^2 b e+8 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{11/3}}\\ \end{align*}

Mathematica [A]  time = 0.152679, size = 255, normalized size = 0.94 \[ \frac{\frac{30 b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}+\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^2 b e+8 a^3 f+2 a b^2 d+b^3 c\right )}{a^{4/3}}+45 b^{2/3} x^2 (b e-2 a f)+18 b^{5/3} f x^5}{90 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*b^(2/3)*(b*e - 2*a*f)*x^2 + 18*b^(5/3)*f*x^5 + (30*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a
+ b*x^3)) - (10*Sqrt[3]*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])
/a^(4/3) - (10*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 2*a*b
^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(90*b^(11/3))

________________________________________________________________________________________

Maple [B]  time = 0.009, size = 495, 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(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

1/5*f*x^5/b^2-1/b^3*x^2*a*f+1/2/b^2*x^2*e-1/3/b^3*x^2*a^2/(b*x^3+a)*f+1/3/b^2*x^2*a/(b*x^3+a)*e-1/3*x^2*d/(b*x
^3+a)/b+1/3*x^2/a/(b*x^3+a)*c-8/9/b^4*a^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f+5/9/b^3*a/(1/b*a)^(1/3)*ln(x+(1/
b*a)^(1/3))*e-2/9/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*d-1/9/b/a/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c+4/9/b^4*
a^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f-5/18/b^3*a/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b
*a)^(2/3))*e+1/9/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d+1/18/b/a/(1/b*a)^(1/3)*ln(x^2-(1/b*
a)^(1/3)*x+(1/b*a)^(2/3))*c+8/9/b^4*a^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*f-5/9/
b^3*a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*e+2/9/b^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1
/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*d+1/9/b/a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c

________________________________________________________________________________________

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(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.48168, size = 1935, normalized size = 7.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[1/90*(18*a^2*b^4*f*x^8 + 9*(5*a^2*b^4*e - 8*a^3*b^3*f)*x^5 + 15*(2*a*b^5*c - 2*a^2*b^4*d + 5*a^3*b^3*e - 8*a^
4*b^2*f)*x^2 + 15*sqrt(1/3)*(a^2*b^4*c + 2*a^3*b^3*d - 5*a^4*b^2*e + 8*a^5*b*f + (a*b^5*c + 2*a^2*b^4*d - 5*a^
3*b^3*e + 8*a^4*b^2*f)*x^3)*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)) + 5*(a*b^3*c + 2*a^2*b^2*d
- 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b
^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 10*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2
*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^6*x^3 + a^3*b^5), 1/90*(18*a^2*b^4*f
*x^8 + 9*(5*a^2*b^4*e - 8*a^3*b^3*f)*x^5 + 15*(2*a*b^5*c - 2*a^2*b^4*d + 5*a^3*b^3*e - 8*a^4*b^2*f)*x^2 + 30*s
qrt(1/3)*(a^2*b^4*c + 2*a^3*b^3*d - 5*a^4*b^2*e + 8*a^5*b*f + (a*b^5*c + 2*a^2*b^4*d - 5*a^3*b^3*e + 8*a^4*b^2
*f)*x^3)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + 5*(a*b
^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*l
og(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 10*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c +
2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^6*x^3 + a^3*b^5)]

________________________________________________________________________________________

Sympy [A]  time = 16.8514, size = 461, normalized size = 1.7 \begin{align*} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{11} + 512 a^{9} f^{3} - 960 a^{8} b e f^{2} + 384 a^{7} b^{2} d f^{2} + 600 a^{7} b^{2} e^{2} f + 192 a^{6} b^{3} c f^{2} - 480 a^{6} b^{3} d e f - 125 a^{6} b^{3} e^{3} - 240 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f + 150 a^{5} b^{4} d e^{2} + 96 a^{4} b^{5} c d f + 75 a^{4} b^{5} c e^{2} - 60 a^{4} b^{5} d^{2} e + 24 a^{3} b^{6} c^{2} f - 60 a^{3} b^{6} c d e + 8 a^{3} b^{6} d^{3} - 15 a^{2} b^{7} c^{2} e + 12 a^{2} b^{7} c d^{2} + 6 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{3} b^{7}}{64 a^{6} f^{2} - 80 a^{5} b e f + 32 a^{4} b^{2} d f + 25 a^{4} b^{2} e^{2} + 16 a^{3} b^{3} c f - 20 a^{3} b^{3} d e - 10 a^{2} b^{4} c e + 4 a^{2} b^{4} d^{2} + 4 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{5}}{5 b^{2}} - \frac{x^{2} \left (2 a f - b e\right )}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

-x**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*a**2*b**3 + 3*a*b**4*x**3) + RootSum(729*_t**3*a**4*b**11 + 5
12*a**9*f**3 - 960*a**8*b*e*f**2 + 384*a**7*b**2*d*f**2 + 600*a**7*b**2*e**2*f + 192*a**6*b**3*c*f**2 - 480*a*
*6*b**3*d*e*f - 125*a**6*b**3*e**3 - 240*a**5*b**4*c*e*f + 96*a**5*b**4*d**2*f + 150*a**5*b**4*d*e**2 + 96*a**
4*b**5*c*d*f + 75*a**4*b**5*c*e**2 - 60*a**4*b**5*d**2*e + 24*a**3*b**6*c**2*f - 60*a**3*b**6*c*d*e + 8*a**3*b
**6*d**3 - 15*a**2*b**7*c**2*e + 12*a**2*b**7*c*d**2 + 6*a*b**8*c**2*d + b**9*c**3, Lambda(_t, _t*log(81*_t**2
*a**3*b**7/(64*a**6*f**2 - 80*a**5*b*e*f + 32*a**4*b**2*d*f + 25*a**4*b**2*e**2 + 16*a**3*b**3*c*f - 20*a**3*b
**3*d*e - 10*a**2*b**4*c*e + 4*a**2*b**4*d**2 + 4*a*b**5*c*d + b**6*c**2) + x))) + f*x**5/(5*b**2) - x**2*(2*a
*f - b*e)/(2*b**3)

________________________________________________________________________________________

Giac [A]  time = 1.07596, size = 494, normalized size = 1.82 \begin{align*} -\frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 8 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, 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^{2} b^{3}} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 5 \, \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^{2} b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 5 \, \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^{2} b^{5}} + \frac{2 \, b^{8} f x^{5} - 10 \, a b^{7} f x^{2} + 5 \, b^{8} x^{2} e}{10 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*(b^3*c*(-a/b)^(1/3) + 2*a*b^2*d*(-a/b)^(1/3) + 8*a^3*f*(-a/b)^(1/3) - 5*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3
)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/3*(b^3*c*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*
a*b^3) - 1/9*sqrt(3)*((-a*b^2)^(2/3)*b^3*c + 2*(-a*b^2)^(2/3)*a*b^2*d + 8*(-a*b^2)^(2/3)*a^3*f - 5*(-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^2*b^5) + 1/18*((-a*b^2)^(2/3)*b^3*c + 2*
(-a*b^2)^(2/3)*a*b^2*d + 8*(-a*b^2)^(2/3)*a^3*f - 5*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^
(2/3))/(a^2*b^5) + 1/10*(2*b^8*f*x^5 - 10*a*b^7*f*x^2 + 5*b^8*x^2*e)/b^10