3.301 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)^3} \, dx\)

Optimal. Leaf size=381 \[ \frac{b x^2 \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{a^6 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}} \]

[Out]

-c/(10*a^3*x^10) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(4*a^5*x^4) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^5*(a + b*x^3)^2) + (b*(14*b^3
*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*x^2)/(9*a^6*(a + b*x^3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*
e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(19/3)) - (b^(1/3)*(104*b^3*c -
65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(19/3)) + (b^(1/3)*(104*b^3*c - 65*a*b^2*d
+ 35*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(19/3))

________________________________________________________________________________________

Rubi [A]  time = 0.713572, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1834, 292, 31, 634, 617, 204, 628} \[ \frac{b x^2 \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac{b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{a^6 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{9 \sqrt{3} a^{19/3}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]

[Out]

-c/(10*a^3*x^10) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(4*a^5*x^4) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^5*(a + b*x^3)^2) + (b*(14*b^3
*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*x^2)/(9*a^6*(a + b*x^3)) - (b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*
e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(19/3)) - (b^(1/3)*(104*b^3*c -
65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(19/3)) + (b^(1/3)*(104*b^3*c - 65*a*b^2*d
+ 35*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(19/3))

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 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 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^{11} \left (a+b x^3\right )^3} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+6 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{6 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac{4 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac{\int \frac{18 b^7 c-18 b^7 \left (\frac{2 b c}{a}-d\right ) x^3+18 b^7 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^6-18 b^7 \left (\frac{4 b^3 c}{a^3}-\frac{3 b^2 d}{a^2}+\frac{2 b e}{a}-f\right ) x^9+\frac{2 b^8 \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )} \, dx}{18 a^2 b^7}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^7 c}{a x^{11}}+\frac{18 b^7 (-3 b c+a d)}{a^2 x^8}+\frac{18 b^7 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^5}+\frac{18 b^7 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^2}-\frac{2 b^8 \left (-104 b^3 c+65 a b^2 d-35 a^2 b e+14 a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^7}\\ &=-\frac{c}{10 a^3 x^{10}}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac{\left (b \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^6}\\ &=-\frac{c}{10 a^3 x^{10}}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac{\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{19/3}}+\frac{\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\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}{27 a^{19/3}}\\ &=-\frac{c}{10 a^3 x^{10}}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac{\left (\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\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}{54 a^{19/3}}+\frac{\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^6}\\ &=-\frac{c}{10 a^3 x^{10}}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{19/3}}+\frac{\left (\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{19/3}}\\ &=-\frac{c}{10 a^3 x^{10}}+\frac{3 b c-a d}{7 a^4 x^7}-\frac{6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{19/3}}-\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac{\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{19/3}}\\ \end{align*}

Mathematica [A]  time = 0.392011, size = 366, normalized size = 0.96 \[ \frac{-\frac{630 a^{4/3} b x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}-\frac{420 \sqrt [3]{a} b x^2 \left (-8 a^2 b e+5 a^3 f+11 a b^2 d-14 b^3 c\right )}{a+b x^3}+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )-\frac{3780 \sqrt [3]{a} \left (-3 a^2 b e+a^3 f+6 a b^2 d-10 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^2 b e+14 a^3 f+65 a b^2 d-104 b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )-\frac{945 a^{4/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^4}-\frac{540 a^{7/3} (a d-3 b c)}{x^7}-\frac{378 a^{10/3} c}{x^{10}}}{3780 a^{19/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]

[Out]

((-378*a^(10/3)*c)/x^10 - (540*a^(7/3)*(-3*b*c + a*d))/x^7 - (945*a^(4/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x^4 - (
3780*a^(1/3)*(-10*b^3*c + 6*a*b^2*d - 3*a^2*b*e + a^3*f))/x - (630*a^(4/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a
^3*f)*x^2)/(a + b*x^3)^2 - (420*a^(1/3)*b*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*x^2)/(a + b*x^3) - 14
0*Sqrt[3]*b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]]
+ 140*b^(1/3)*(-104*b^3*c + 65*a*b^2*d - 35*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(104*b^
3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*a^(19/3))

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 659, normalized size = 1.7 \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^11/(b*x^3+a)^3,x)

[Out]

-104/27*b^3/a^6*c/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-6/a^5/x*b^2*d+10/a^6/x*b^3*c-25/18*b^3/a^4/(b*x^3+a)^2*x^2
*d+65/27*b^2/a^5*d/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-65/54*b^2/a^5*d/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b
*a)^(2/3))-1/10*c/a^3/x^10-11/9*b^4/a^5/(b*x^3+a)^2*x^5*d+14/9*b^5/a^6/(b*x^3+a)^2*x^5*c+14/27/a^3*f/(1/b*a)^(
1/3)*ln(x+(1/b*a)^(1/3))-7/27/a^3*f/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+3/7/a^4/x^7*b*c+3/4/a^
4/x^4*b*d-3/2/a^5/x^4*b^2*c+3/a^4/x*b*e-1/7/a^3/x^7*d-1/4/a^3/x^4*e-1/a^3/x*f+35/27*b/a^4*e*3^(1/2)/(1/b*a)^(1
/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-65/27*b^2/a^5*d*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b
*a)^(1/3)*x-1))+104/27*b^3/a^6*c*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^2/a^3/(
b*x^3+a)^2*x^5*f+8/9*b^3/a^4/(b*x^3+a)^2*x^5*e+52/27*b^3/a^6*c/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2
/3))-14/27/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))-35/27*b/a^4*e/(1/b*a)^(1/3)*l
n(x+(1/b*a)^(1/3))+35/54*b/a^4*e/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-13/18*b/a^2/(b*x^3+a)^2*x
^2*f+19/18*b^2/a^3/(b*x^3+a)^2*x^2*e+31/18*b^4/a^5/(b*x^3+a)^2*x^2*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((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61915, size = 1472, normalized size = 3.86 \begin{align*} \frac{420 \,{\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{15} + 735 \,{\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{12} + 270 \,{\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{9} - 27 \,{\left (104 \, a^{3} b^{2} c - 65 \, a^{4} b d + 35 \, a^{5} e\right )} x^{6} - 378 \, a^{5} c + 108 \,{\left (8 \, a^{4} b c - 5 \, a^{5} d\right )} x^{3} + 140 \, \sqrt{3}{\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \,{\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} +{\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 70 \,{\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \,{\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} +{\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 140 \,{\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \,{\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} +{\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right )}{3780 \,{\left (a^{6} b^{2} x^{16} + 2 \, a^{7} b x^{13} + a^{8} x^{10}\right )}} \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^11/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/3780*(420*(104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^15 + 735*(104*a*b^4*c - 65*a^2*b^3*d + 35
*a^3*b^2*e - 14*a^4*b*f)*x^12 + 270*(104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^9 - 27*(104*a^3*b
^2*c - 65*a^4*b*d + 35*a^5*e)*x^6 - 378*a^5*c + 108*(8*a^4*b*c - 5*a^5*d)*x^3 + 140*sqrt(3)*((104*b^5*c - 65*a
*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^16 + 2*(104*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 14*a^4*b*f)*x^13 +
(104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^10)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1
/3*sqrt(3)) + 70*((104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^16 + 2*(104*a*b^4*c - 65*a^2*b^3*d
+ 35*a^3*b^2*e - 14*a^4*b*f)*x^13 + (104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^10)*(b/a)^(1/3)*l
og(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 140*((104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^16
+ 2*(104*a*b^4*c - 65*a^2*b^3*d + 35*a^3*b^2*e - 14*a^4*b*f)*x^13 + (104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*
e - 14*a^5*f)*x^10)*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)))/(a^6*b^2*x^16 + 2*a^7*b*x^13 + a^8*x^10)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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**11/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.09807, size = 656, normalized size = 1.72 \begin{align*} -\frac{{\left (104 \, b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 65 \, a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 35 \, a^{2} b^{2} \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 )}{27 \, a^{7}} - \frac{\sqrt{3}{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \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 )}{27 \, a^{7} b} + \frac{{\left (104 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 35 \, \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 )}{54 \, a^{7} b} + \frac{28 \, b^{5} c x^{5} - 22 \, a b^{4} d x^{5} - 10 \, a^{3} b^{2} f x^{5} + 16 \, a^{2} b^{3} x^{5} e + 31 \, a b^{4} c x^{2} - 25 \, a^{2} b^{3} d x^{2} - 13 \, a^{4} b f x^{2} + 19 \, a^{3} b^{2} x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{1400 \, b^{3} c x^{9} - 840 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 420 \, a^{2} b x^{9} e - 210 \, a b^{2} c x^{6} + 105 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 60 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{6} x^{10}} \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^11/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(104*b^4*c*(-a/b)^(1/3) - 65*a*b^3*d*(-a/b)^(1/3) - 14*a^3*b*f*(-a/b)^(1/3) + 35*a^2*b^2*(-a/b)^(1/3)*e)
*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^7 - 1/27*sqrt(3)*(104*(-a*b^2)^(2/3)*b^3*c - 65*(-a*b^2)^(2/3)*a*b^
2*d - 14*(-a*b^2)^(2/3)*a^3*f + 35*(-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^7*b) + 1/54*(104*(-a*b^2)^(2/3)*b^3*c - 65*(-a*b^2)^(2/3)*a*b^2*d - 14*(-a*b^2)^(2/3)*a^3*f + 35*(-a*b^2
)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^7*b) + 1/18*(28*b^5*c*x^5 - 22*a*b^4*d*x^5 - 10*a
^3*b^2*f*x^5 + 16*a^2*b^3*x^5*e + 31*a*b^4*c*x^2 - 25*a^2*b^3*d*x^2 - 13*a^4*b*f*x^2 + 19*a^3*b^2*x^2*e)/((b*x
^3 + a)^2*a^6) + 1/140*(1400*b^3*c*x^9 - 840*a*b^2*d*x^9 - 140*a^3*f*x^9 + 420*a^2*b*x^9*e - 210*a*b^2*c*x^6 +
105*a^2*b*d*x^6 - 35*a^3*x^6*e + 60*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^6*x^10)