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

Optimal. Leaf size=245 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]

[Out]

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

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Rubi [A]  time = 0.215567, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1836, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 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{x \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^8}{8 b}+\frac{\int \frac{x \left (8 b c+8 b d x^3+8 (b e-a f) x^6\right )}{a+b x^3} \, dx}{8 b}\\ &=\frac{f x^8}{8 b}+\frac{\int \left (\frac{8 \left (b^2 d-a b e+a^2 f\right ) x}{b^2}+\frac{8 (b e-a f) x^4}{b}+\frac{8 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^2 \left (a+b x^3\right )}\right ) \, dx}{8 b}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{2 b^3}+\frac{(b e-a f) x^5}{5 b^2}+\frac{f x^8}{8 b}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{b^3}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{2 b^3}+\frac{(b e-a f) x^5}{5 b^2}+\frac{f x^8}{8 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 \sqrt [3]{a} b^{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} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 \sqrt [3]{a} b^{10/3}}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{2 b^3}+\frac{(b e-a f) x^5}{5 b^2}+\frac{f x^8}{8 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 \sqrt [3]{a} b^{11/3}}+\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 \sqrt [3]{a} b^{11/3}}+\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 b^{10/3}}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{2 b^3}+\frac{(b e-a f) x^5}{5 b^2}+\frac{f x^8}{8 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 \sqrt [3]{a} b^{11/3}}+\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 \sqrt [3]{a} b^{11/3}}+\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 )}{\sqrt [3]{a} b^{11/3}}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{2 b^3}+\frac{(b e-a f) x^5}{5 b^2}+\frac{f x^8}{8 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} \sqrt [3]{a} b^{11/3}}-\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 \sqrt [3]{a} b^{11/3}}+\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 \sqrt [3]{a} b^{11/3}}\\ \end{align*}

Mathematica [A]  time = 0.160058, size = 231, normalized size = 0.94 \[ \frac{\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+60 b^{2/3} x^2 \left (a^2 f-a b e+b^2 d\right )+24 b^{5/3} x^5 (b e-a f)+15 b^{8/3} f x^8}{120 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 450, 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),x)

[Out]

1/8*f*x^8/b-1/5/b^2*x^5*a*f+1/5/b*x^5*e+1/2/b^3*x^2*a^2*f-1/2/b^2*x^2*a*e+1/2*d*x^2/b+1/3/b^4/(1/b*a)^(1/3)*ln
(x+(1/b*a)^(1/3))*a^3*f-1/3/b^3/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*a^2*e+1/3/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/
3))*a*d-1/3/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c-1/6/b^4/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*
a^3*f+1/6/b^3/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*a^2*e-1/6/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(
1/3)*x+(1/b*a)^(2/3))*a*d+1/6/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c-1/3/b^4*3^(1/2)/(1/b*a)^
(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*a^3*f+1/3/b^3*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b
*a)^(1/3)*x-1))*a^2*e-1/3/b^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*a*d+1/3/b*3^(1/2
)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*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(x*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51902, size = 1289, normalized size = 5.26 \begin{align*} \left [\frac{15 \, a b^{4} f x^{8} + 24 \,{\left (a b^{4} e - a^{2} b^{3} f\right )} x^{5} + 60 \,{\left (a b^{4} d - a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{2} - 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 )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b - 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) + 20 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} b x + \left (a b^{2}\right )^{\frac{2}{3}}\right ) - 40 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac{1}{3}}\right )}{120 \, a b^{5}}, \frac{15 \, a b^{4} f x^{8} + 24 \,{\left (a b^{4} e - a^{2} b^{3} f\right )} x^{5} + 60 \,{\left (a b^{4} d - a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{2} - 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 )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x - \left (a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) + 20 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} b x + \left (a b^{2}\right )^{\frac{2}{3}}\right ) - 40 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac{1}{3}}\right )}{120 \, a b^{5}}\right ] \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),x, algorithm="fricas")

[Out]

[1/120*(15*a*b^4*f*x^8 + 24*(a*b^4*e - a^2*b^3*f)*x^5 + 60*(a*b^4*d - a^2*b^3*e + a^3*b^2*f)*x^2 - 60*sqrt(1/3
)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*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)) + 20*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*(a*b^2)^(2/3)*log(b^2*x^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) - 40*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a*b^5), 1/120*(15*a*b^4*f*x^8 + 24*(a*b^4*e -
a^2*b^3*f)*x^5 + 60*(a*b^4*d - a^2*b^3*e + a^3*b^2*f)*x^2 - 120*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e -
a^4*b*f)*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) + 20*(b^3*c
- a*b^2*d + a^2*b*e - a^3*f)*(a*b^2)^(2/3)*log(b^2*x^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) - 40*(b^3*c - a*b^
2*d + a^2*b*e - a^3*f)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a*b^5)]

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Sympy [A]  time = 1.47288, size = 422, normalized size = 1.72 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a b^{11} - 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{9 t^{2} a b^{7}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{8}}{8 b} - \frac{x^{5} \left (a f - b e\right )}{5 b^{2}} + \frac{x^{2} \left (a^{2} f - a b e + b^{2} d\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),x)

[Out]

RootSum(27*_t**3*a*b**11 - 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(9*_t**2*a*b*
*7/(a**6*f**2 - 2*a**5*b*e*f + 2*a**4*b**2*d*f + a**4*b**2*e**2 - 2*a**3*b**3*c*f - 2*a**3*b**3*d*e + 2*a**2*b
**4*c*e + a**2*b**4*d**2 - 2*a*b**5*c*d + b**6*c**2) + x))) + f*x**8/(8*b) - x**5*(a*f - b*e)/(5*b**2) + x**2*
(a**2*f - a*b*e + b**2*d)/(2*b**3)

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Giac [A]  time = 1.08358, size = 466, normalized size = 1.9 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{6 \, a b^{5}} - \frac{{\left (b^{8} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{7} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{6} \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 )}{3 \, a b^{8}} + \frac{5 \, b^{7} f x^{8} - 8 \, a b^{6} f x^{5} + 8 \, b^{7} x^{5} e + 20 \, b^{7} d x^{2} + 20 \, a^{2} b^{5} f x^{2} - 20 \, a b^{6} x^{2} e}{40 \, b^{8}} \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),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^5) + 1/6*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^
2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^5) - 1/3*(b
^8*c*(-a/b)^(1/3) - a*b^7*d*(-a/b)^(1/3) - a^3*b^5*f*(-a/b)^(1/3) + a^2*b^6*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(a
bs(x - (-a/b)^(1/3)))/(a*b^8) + 1/40*(5*b^7*f*x^8 - 8*a*b^6*f*x^5 + 8*b^7*x^5*e + 20*b^7*d*x^2 + 20*a^2*b^5*f*
x^2 - 20*a*b^6*x^2*e)/b^8