\(\int \frac {c+d x^3+e x^6+f x^9}{x^{11} (a+b x^3)^3} \, dx\) [301]

Optimal result

Integrand size = 30, antiderivative size = 381 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{19/3}}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{19/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1843, 1848, 298, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=\frac {3 b c-a d}{7 a^4 x^7}-\frac {c}{10 a^3 x^{10}}-\frac {a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt {3} a^{19/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac {b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac {b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

[In]

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

[Out]

-1/10*c/(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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

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 631

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 642

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]

Rule 648

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 1843

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)/a)*Coeff[R, x, i]*x^(i - m), {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[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1848

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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] (verified)

Time = 0.44 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=\frac {-\frac {378 a^{10/3} c}{x^{10}}-\frac {540 a^{7/3} (-3 b c+a d)}{x^7}-\frac {945 a^{4/3} \left (6 b^2 c-3 a b d+a^2 e\right )}{x^4}-\frac {3780 \sqrt [3]{a} \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{x}-\frac {630 a^{4/3} b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{\left (a+b x^3\right )^2}-\frac {420 \sqrt [3]{a} b \left (-14 b^3 c+11 a b^2 d-8 a^2 b e+5 a^3 f\right ) x^2}{a+b x^3}-140 \sqrt {3} \sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+140 \sqrt [3]{b} \left (-104 b^3 c+65 a b^2 d-35 a^2 b e+14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+70 \sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3780 a^{19/3}} \]

[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] (verified)

Time = 1.55 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {c}{10 a^{3} x^{10}}-\frac {a d -3 b c}{7 a^{4} x^{7}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{4 a^{5} x^{4}}-\frac {f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{a^{6} x}-\frac {b \left (\frac {\frac {b \left (5 f \,a^{3}-8 a^{2} b e +11 a \,b^{2} d -14 b^{3} c \right ) x^{5}}{9}+\left (\frac {13}{18} a^{4} f -\frac {19}{18} a^{3} b e +\frac {25}{18} a^{2} b^{2} d -\frac {31}{18} a \,b^{3} c \right ) x^{2}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {14}{9} f \,a^{3}-\frac {35}{9} a^{2} b e +\frac {65}{9} a \,b^{2} d -\frac {104}{9} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{a^{6}}\)	\(289\)
risch	\(\frac {-\frac {c}{10 a}-\frac {\left (5 a d -8 b c \right ) x^{3}}{35 a^{2}}-\frac {\left (35 a^{2} e -65 a b d +104 b^{2} c \right ) x^{6}}{140 a^{3}}-\frac {\left (14 f \,a^{3}-35 a^{2} b e +65 a \,b^{2} d -104 b^{3} c \right ) x^{9}}{14 a^{4}}-\frac {7 b \left (14 f \,a^{3}-35 a^{2} b e +65 a \,b^{2} d -104 b^{3} c \right ) x^{12}}{36 a^{5}}-\frac {b^{2} \left (14 f \,a^{3}-35 a^{2} b e +65 a \,b^{2} d -104 b^{3} c \right ) x^{15}}{9 a^{6}}}{x^{10} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{19} \textit {\_Z}^{3}-2744 a^{9} b \,f^{3}+20580 a^{8} b^{2} e \,f^{2}-38220 a^{7} b^{3} d \,f^{2}-51450 a^{7} b^{3} e^{2} f +61152 a^{6} b^{4} c \,f^{2}+191100 a^{6} b^{4} d e f +42875 a^{6} b^{4} e^{3}-305760 a^{5} b^{5} c e f -177450 a^{5} b^{5} d^{2} f -238875 a^{5} b^{5} d \,e^{2}+567840 a^{4} b^{6} c d f +382200 a^{4} b^{6} c \,e^{2}+443625 a^{4} b^{6} d^{2} e -454272 a^{3} b^{7} c^{2} f -1419600 a^{3} b^{7} c d e -274625 a^{3} b^{7} d^{3}+1135680 a^{2} b^{8} c^{2} e +1318200 a^{2} b^{8} c \,d^{2}-2109120 a \,b^{9} c^{2} d +1124864 b^{10} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{19}+8232 a^{9} b \,f^{3}-61740 a^{8} b^{2} e \,f^{2}+114660 a^{7} b^{3} d \,f^{2}+154350 a^{7} b^{3} e^{2} f -183456 a^{6} b^{4} c \,f^{2}-573300 a^{6} b^{4} d e f -128625 a^{6} b^{4} e^{3}+917280 a^{5} b^{5} c e f +532350 a^{5} b^{5} d^{2} f +716625 a^{5} b^{5} d \,e^{2}-1703520 a^{4} b^{6} c d f -1146600 a^{4} b^{6} c \,e^{2}-1330875 a^{4} b^{6} d^{2} e +1362816 a^{3} b^{7} c^{2} f +4258800 a^{3} b^{7} c d e +823875 a^{3} b^{7} d^{3}-3407040 a^{2} b^{8} c^{2} e -3954600 a^{2} b^{8} c \,d^{2}+6327360 a \,b^{9} c^{2} d -3374592 b^{10} c^{3}\right ) x +\left (-14 a^{16} f +35 a^{15} b e -65 a^{14} b^{2} d +104 a^{13} b^{3} c \right ) \textit {\_R}^{2}\right )\right )}{27}\)	\(689\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/10*c/a^3/x^10-1/7*(a*d-3*b*c)/a^4/x^7-1/4*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^4-(a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3
*c)/a^6/x-b/a^6*((1/9*b*(5*a^3*f-8*a^2*b*e+11*a*b^2*d-14*b^3*c)*x^5+(13/18*a^4*f-19/18*a^3*b*e+25/18*a^2*b^2*d
-31/18*a*b^3*c)*x^2)/(b*x^3+a)^2+(14/9*f*a^3-35/9*a^2*b*e+65/9*a*b^2*d-104/9*b^3*c)*(-1/3/b/(a/b)^(1/3)*ln(x+(
a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.63 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=\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 )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=\frac {140 \, {\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{15} + 245 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{12} + 90 \, {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{9} - 9 \, {\left (104 \, a^{3} b^{2} c - 65 \, a^{4} b d + 35 \, a^{5} e\right )} x^{6} - 126 \, a^{5} c + 36 \, {\left (8 \, a^{4} b c - 5 \, a^{5} d\right )} x^{3}}{1260 \, {\left (a^{6} b^{2} x^{16} + 2 \, a^{7} b x^{13} + a^{8} x^{10}\right )}} + \frac {\sqrt {3} {\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/1260*(140*(104*b^5*c - 65*a*b^4*d + 35*a^2*b^3*e - 14*a^3*b^2*f)*x^15 + 245*(104*a*b^4*c - 65*a^2*b^3*d + 35
*a^3*b^2*e - 14*a^4*b*f)*x^12 + 90*(104*a^2*b^3*c - 65*a^3*b^2*d + 35*a^4*b*e - 14*a^5*f)*x^9 - 9*(104*a^3*b^2
*c - 65*a^4*b*d + 35*a^5*e)*x^6 - 126*a^5*c + 36*(8*a^4*b*c - 5*a^5*d)*x^3)/(a^6*b^2*x^16 + 2*a^7*b*x^13 + a^8
*x^10) + 1/27*sqrt(3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/
(a/b)^(1/3))/(a^6*(a/b)^(1/3)) + 1/54*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*log(x^2 - x*(a/b)^(1/3)
+ (a/b)^(2/3))/(a^6*(a/b)^(1/3)) - 1/27*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*log(x + (a/b)^(1/3))
/(a^6*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.26 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=-\frac {{\left (104 \, b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 65 \, a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 35 \, a^{2} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{3} b f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7}} - \frac {\sqrt {3} {\left (104 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7} b} + \frac {{\left (104 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7} b} + \frac {28 \, b^{5} c x^{5} - 22 \, a b^{4} d x^{5} + 16 \, a^{2} b^{3} e x^{5} - 10 \, a^{3} b^{2} f x^{5} + 31 \, a b^{4} c x^{2} - 25 \, a^{2} b^{3} d x^{2} + 19 \, a^{3} b^{2} e x^{2} - 13 \, a^{4} b f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{6}} + \frac {1400 \, b^{3} c x^{9} - 840 \, a b^{2} d x^{9} + 420 \, a^{2} b e x^{9} - 140 \, a^{3} f x^{9} - 210 \, a b^{2} c x^{6} + 105 \, a^{2} b d x^{6} - 35 \, a^{3} e x^{6} + 60 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{6} x^{10}} \]

[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) + 35*a^2*b^2*e*(-a/b)^(1/3) - 14*a^3*b*f*(-a/b)^(1/3))
*(-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 + 35*(-a*b^2)^(2/3)*a^2*b*e - 14*(-a*b^2)^(2/3)*a^3*f)*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 + 35*(-a*b^2)^(2/3)*a^2*b*e - 14*(-a*b
^2)^(2/3)*a^3*f)*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 + 16*a
^2*b^3*e*x^5 - 10*a^3*b^2*f*x^5 + 31*a*b^4*c*x^2 - 25*a^2*b^3*d*x^2 + 19*a^3*b^2*e*x^2 - 13*a^4*b*f*x^2)/((b*x
^3 + a)^2*a^6) + 1/140*(1400*b^3*c*x^9 - 840*a*b^2*d*x^9 + 420*a^2*b*e*x^9 - 140*a^3*f*x^9 - 210*a*b^2*c*x^6 +
105*a^2*b*d*x^6 - 35*a^3*e*x^6 + 60*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^6*x^10)

Mupad [B] (verification not implemented)

Time = 9.39 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^3} \, dx=-\frac {\frac {c}{10\,a}-\frac {x^9\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{14\,a^4}+\frac {x^3\,\left (5\,a\,d-8\,b\,c\right )}{35\,a^2}+\frac {x^6\,\left (35\,e\,a^2-65\,d\,a\,b+104\,c\,b^2\right )}{140\,a^3}-\frac {7\,b\,x^{12}\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{36\,a^5}-\frac {b^2\,x^{15}\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{9\,a^6}}{a^2\,x^{10}+2\,a\,b\,x^{13}+b^2\,x^{16}}-\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}}-\frac {b^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}} \]

[In]

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

[Out]

(b^(1/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(104*b^3*c - 14*a^3*f - 65*a*b
^2*d + 35*a^2*b*e))/(27*a^(19/3)) - (b^(1/3)*log(b^(1/3)*x + a^(1/3))*(104*b^3*c - 14*a^3*f - 65*a*b^2*d + 35*
a^2*b*e))/(27*a^(19/3)) - (c/(10*a) - (x^9*(104*b^3*c - 14*a^3*f - 65*a*b^2*d + 35*a^2*b*e))/(14*a^4) + (x^3*(
5*a*d - 8*b*c))/(35*a^2) + (x^6*(104*b^2*c + 35*a^2*e - 65*a*b*d))/(140*a^3) - (7*b*x^12*(104*b^3*c - 14*a^3*f
- 65*a*b^2*d + 35*a^2*b*e))/(36*a^5) - (b^2*x^15*(104*b^3*c - 14*a^3*f - 65*a*b^2*d + 35*a^2*b*e))/(9*a^6))/(
a^2*x^10 + b^2*x^16 + 2*a*b*x^13) - (b^(1/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 -
1/2)*(104*b^3*c - 14*a^3*f - 65*a*b^2*d + 35*a^2*b*e))/(27*a^(19/3))