3.263 \(\int \frac{x^4 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=298 \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{14/3}}+\frac{x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^8}{8 b^2} \]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^2)/(2*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^8)/(8*b^2) - ((b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(3*b^4*(a + b*x^3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(14/3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(14/3)) + ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(14/3))

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Rubi [A]  time = 0.462677, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1851, 1836, 1488, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{14/3}}+\frac{x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^8}{8 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^2)/(2*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^8)/(8*b^2) - ((b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(3*b^4*(a + b*x^3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(14/3)) - ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(14/3)) + ((2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(14/3))

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 1851

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /;
FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m
]]

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^4 \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 b^4 \left (a+b x^3\right )}-\frac{\int \frac{-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^4-3 a b^3 (b e-a f) x^7-3 a b^4 f x^{10}}{a+b x^3} \, dx}{3 a b^5}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\int \frac{x \left (-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^3-3 a b^3 (b e-a f) x^6-3 a b^4 f x^9\right )}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\int \frac{x \left (-16 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-24 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^3-24 a b^4 (b e-2 a f) x^6\right )}{a+b x^3} \, dx}{24 a b^6}\\ &=\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\int \left (-24 a b^2 \left (b^2 d-2 a b e+3 a^2 f\right ) x-24 a b^3 (b e-2 a f) x^4+\frac{8 \left (-2 a b^5 c+5 a^2 b^4 d-8 a^3 b^3 e+11 a^4 b^2 f\right ) x}{a+b x^3}\right ) \, dx}{24 a b^6}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 b^4}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{13/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 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 \sqrt [3]{a} b^{13/3}}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 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 \sqrt [3]{a} b^{14/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{13/3}}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{14/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 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 \sqrt [3]{a} b^{14/3}}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac{(b e-2 a f) x^5}{5 b^3}+\frac{f x^8}{8 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 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} \sqrt [3]{a} b^{14/3}}-\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac{\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{14/3}}\\ \end{align*}

Mathematica [A]  time = 0.156751, size = 282, normalized size = 0.95 \[ \frac{-\frac{120 b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-8 a^2 b e+11 a^3 f+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-8 a^2 b e+11 a^3 f+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}+180 b^{2/3} x^2 \left (3 a^2 f-2 a b e+b^2 d\right )+72 b^{5/3} x^5 (b e-2 a f)+45 b^{8/3} f x^8}{360 b^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(180*b^(2/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^2 + 72*b^(5/3)*(b*e - 2*a*f)*x^5 + 45*b^(8/3)*f*x^8 - (120*b^(2/3)*
(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3) + (40*Sqrt[3]*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11*a^3*
f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11*a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/a^(1/3))/(360*b^(14/3))

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Maple [B]  time = 0.009, size = 529, 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^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

1/8*f*x^8/b^2-2/5/b^3*x^5*a*f+1/5/b^2*x^5*e+3/2/b^4*x^2*a^2*f-1/b^3*x^2*a*e+1/2/b^2*x^2*d+1/3/b^4*x^2/(b*x^3+a
)*a^3*f-1/3/b^3*x^2/(b*x^3+a)*a^2*e+1/3/b^2*x^2/(b*x^3+a)*a*d-1/3/b*x^2/(b*x^3+a)*c+11/9/b^5*a^3*f/(1/b*a)^(1/
3)*ln(x+(1/b*a)^(1/3))-11/18/b^5*a^3*f/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-11/9/b^5*a^3*f*3^(1
/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-8/9/b^4*a^2*e/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+4/
9/b^4*a^2*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+8/9/b^4*a^2*e*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3
*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+5/9/b^3*a*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-5/18/b^3*a*d/(1/b*a)^(1/3)*ln(x^
2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-5/9/b^3*a*d*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*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/9/b^2*c/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/b
^2*c*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))

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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^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7054, size = 2094, normalized size = 7.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[1/360*(45*a*b^5*f*x^11 + 9*(8*a*b^5*e - 11*a^2*b^4*f)*x^8 + 36*(5*a*b^5*d - 8*a^2*b^4*e + 11*a^3*b^3*f)*x^5 -
60*(2*a*b^5*c - 5*a^2*b^4*d + 8*a^3*b^3*e - 11*a^4*b^2*f)*x^2 - 60*sqrt(1/3)*(2*a^2*b^4*c - 5*a^3*b^3*d + 8*a
^4*b^2*e - 11*a^5*b*f + (2*a*b^5*c - 5*a^2*b^4*d + 8*a^3*b^3*e - 11*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)) + 20*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f + (2*b^4*c - 5*a*b^3*d + 8*a^2
*b^2*e - 11*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)) - 40*(2*a*b^3*c - 5*a
^2*b^2*d + 8*a^3*b*e - 11*a^4*f + (2*b^4*c - 5*a*b^3*d + 8*a^2*b^2*e - 11*a^3*b*f)*x^3)*(a*b^2)^(2/3)*log(b*x
+ (a*b^2)^(1/3)))/(a*b^7*x^3 + a^2*b^6), 1/360*(45*a*b^5*f*x^11 + 9*(8*a*b^5*e - 11*a^2*b^4*f)*x^8 + 36*(5*a*b
^5*d - 8*a^2*b^4*e + 11*a^3*b^3*f)*x^5 - 60*(2*a*b^5*c - 5*a^2*b^4*d + 8*a^3*b^3*e - 11*a^4*b^2*f)*x^2 - 120*s
qrt(1/3)*(2*a^2*b^4*c - 5*a^3*b^3*d + 8*a^4*b^2*e - 11*a^5*b*f + (2*a*b^5*c - 5*a^2*b^4*d + 8*a^3*b^3*e - 11*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) + 20*(
2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f + (2*b^4*c - 5*a*b^3*d + 8*a^2*b^2*e - 11*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)) - 40*(2*a*b^3*c - 5*a^2*b^2*d + 8*a^3*b*e - 11*a^4*f +
(2*b^4*c - 5*a*b^3*d + 8*a^2*b^2*e - 11*a^3*b*f)*x^3)*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a*b^7*x^3 + a^
2*b^6)]

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Sympy [A]  time = 35.7815, size = 484, normalized size = 1.62 \begin{align*} \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{14} - 1331 a^{9} f^{3} + 2904 a^{8} b e f^{2} - 1815 a^{7} b^{2} d f^{2} - 2112 a^{7} b^{2} e^{2} f + 726 a^{6} b^{3} c f^{2} + 2640 a^{6} b^{3} d e f + 512 a^{6} b^{3} e^{3} - 1056 a^{5} b^{4} c e f - 825 a^{5} b^{4} d^{2} f - 960 a^{5} b^{4} d e^{2} + 660 a^{4} b^{5} c d f + 384 a^{4} b^{5} c e^{2} + 600 a^{4} b^{5} d^{2} e - 132 a^{3} b^{6} c^{2} f - 480 a^{3} b^{6} c d e - 125 a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e + 150 a^{2} b^{7} c d^{2} - 60 a b^{8} c^{2} d + 8 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{9}}{121 a^{6} f^{2} - 176 a^{5} b e f + 110 a^{4} b^{2} d f + 64 a^{4} b^{2} e^{2} - 44 a^{3} b^{3} c f - 80 a^{3} b^{3} d e + 32 a^{2} b^{4} c e + 25 a^{2} b^{4} d^{2} - 20 a b^{5} c d + 4 b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{8}}{8 b^{2}} - \frac{x^{5} \left (2 a f - b e\right )}{5 b^{3}} + \frac{x^{2} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(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*b**4 + 3*b**5*x**3) + RootSum(729*_t**3*a*b**14 - 1331*a**9*
f**3 + 2904*a**8*b*e*f**2 - 1815*a**7*b**2*d*f**2 - 2112*a**7*b**2*e**2*f + 726*a**6*b**3*c*f**2 + 2640*a**6*b
**3*d*e*f + 512*a**6*b**3*e**3 - 1056*a**5*b**4*c*e*f - 825*a**5*b**4*d**2*f - 960*a**5*b**4*d*e**2 + 660*a**4
*b**5*c*d*f + 384*a**4*b**5*c*e**2 + 600*a**4*b**5*d**2*e - 132*a**3*b**6*c**2*f - 480*a**3*b**6*c*d*e - 125*a
**3*b**6*d**3 + 96*a**2*b**7*c**2*e + 150*a**2*b**7*c*d**2 - 60*a*b**8*c**2*d + 8*b**9*c**3, Lambda(_t, _t*log
(81*_t**2*a*b**9/(121*a**6*f**2 - 176*a**5*b*e*f + 110*a**4*b**2*d*f + 64*a**4*b**2*e**2 - 44*a**3*b**3*c*f -
80*a**3*b**3*d*e + 32*a**2*b**4*c*e + 25*a**2*b**4*d**2 - 20*a*b**5*c*d + 4*b**6*c**2) + x))) + f*x**8/(8*b**2
) - x**5*(2*a*f - b*e)/(5*b**3) + x**2*(3*a**2*f - 2*a*b*e + b**2*d)/(2*b**4)

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Giac [A]  time = 1.08374, size = 537, normalized size = 1.8 \begin{align*} -\frac{{\left (2 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 11 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 8 \, 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 b^{4}} - \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 )} b^{4}} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 8 \, \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 b^{6}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 8 \, \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 b^{6}} + \frac{5 \, b^{14} f x^{8} - 16 \, a b^{13} f x^{5} + 8 \, b^{14} x^{5} e + 20 \, b^{14} d x^{2} + 60 \, a^{2} b^{12} f x^{2} - 40 \, a b^{13} x^{2} e}{40 \, b^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*(2*b^3*c*(-a/b)^(1/3) - 5*a*b^2*d*(-a/b)^(1/3) - 11*a^3*f*(-a/b)^(1/3) + 8*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(
1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) - 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)
*b^4) - 1/9*sqrt(3)*(2*(-a*b^2)^(2/3)*b^3*c - 5*(-a*b^2)^(2/3)*a*b^2*d - 11*(-a*b^2)^(2/3)*a^3*f + 8*(-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*b^6) + 1/18*(2*(-a*b^2)^(2/3)*b^3*c -
5*(-a*b^2)^(2/3)*a*b^2*d - 11*(-a*b^2)^(2/3)*a^3*f + 8*(-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/
b)^(2/3))/(a*b^6) + 1/40*(5*b^14*f*x^8 - 16*a*b^13*f*x^5 + 8*b^14*x^5*e + 20*b^14*d*x^2 + 60*a^2*b^12*f*x^2 -
40*a*b^13*x^2*e)/b^16