\(\int \frac {c+d x^3+e x^6+f x^9}{x^{14} (a+b x^3)^3} \, dx\) [303]

Optimal result

Integrand size = 30, antiderivative size = 424 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{22/3}}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}} \]

[Out]

-1/13*c/a^3/x^13+1/10*(-a*d+3*b*c)/a^4/x^10+1/7*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^7+1/4*(-a^3*f+3*a^2*b*e-6*a*b^2
*d+10*b^3*c)/a^6/x^4-b*(-3*a^3*f+6*a^2*b*e-10*a*b^2*d+15*b^3*c)/a^7/x-1/6*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x
^2/a^6/(b*x^3+a)^2-1/9*b^2*(-8*a^3*f+11*a^2*b*e-14*a*b^2*d+17*b^3*c)*x^2/a^7/(b*x^3+a)+1/27*b^(4/3)*(-35*a^3*f
+65*a^2*b*e-104*a*b^2*d+152*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(22/3)-1/54*b^(4/3)*(-35*a^3*f+65*a^2*b*e-104*a*b^2
*d+152*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(22/3)+1/27*b^(4/3)*(-35*a^3*f+65*a^2*b*e-104*a*b^2*
d+152*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(22/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 424, 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} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {c}{13 a^3 x^{13}}-\frac {a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt {3} a^{22/3}}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}-\frac {b^2 x^2 \left (-8 a^3 f+11 a^2 b e-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac {b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-1/13*c/(a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b
^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*
c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) - (b^2*(17*b^3*c - 14*a*b^2*d + 11*a^2*b*e - 8*a^3*f
)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)
*Log[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) - (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Log[a^(2
/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))

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^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \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 {6 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac {4 b^5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )^2} \, dx}{6 a b^3} \\ & = -\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\int \frac {18 b^8 c-18 b^8 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^8 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-18 b^8 \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 {18 b^9 \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) x^{12}}{a^4}-\frac {2 b^{10} \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )} \, dx}{18 a^2 b^8} \\ & = -\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^8 c}{a x^{14}}+\frac {18 b^8 (-3 b c+a d)}{a^2 x^{11}}+\frac {18 b^8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^8}+\frac {18 b^8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^5}-\frac {18 b^9 \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^5 x^2}+\frac {2 b^{10} \left (-152 b^3 c+104 a b^2 d-65 a^2 b e+35 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^8} \\ & = -\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}-\frac {\left (b^2 \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{9 a^7} \\ & = -\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{22/3}}-\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}} \\ & = -\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}}-\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^7} \\ & = -\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}}-\frac {\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}} \\ & = -\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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^{22/3}}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}+\frac {b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^7 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}+\frac {b^2 \left (-17 b^3 c+14 a b^2 d-11 a^2 b e+8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{22/3}}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}+\frac {b^{4/3} \left (-152 b^3 c+104 a b^2 d-65 a^2 b e+35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}} \]

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]

[Out]

-1/13*c/(a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b
^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b
^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) + (b^2*(-17*b^3*c + 14*a*b^2*d - 11*a^2*b*e + 8*
a^3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*ArcTan[(1 - (2*b^
(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) + (b^(4/3)*(-152*b^3*c + 104*a*b^2*d - 65*a^2*b*e + 35*a^3*f)*Log[a^(2
/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {c}{13 a^{3} x^{13}}-\frac {a d -3 b c}{10 a^{4} x^{10}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{7 a^{5} x^{7}}-\frac {f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{4 a^{6} x^{4}}+\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right )}{a^{7} x}+\frac {b^{2} \left (\frac {\frac {b \left (8 f \,a^{3}-11 a^{2} b e +14 a \,b^{2} d -17 b^{3} c \right ) x^{5}}{9}+\left (\frac {19}{18} a^{4} f -\frac {25}{18} a^{3} b e +\frac {31}{18} a^{2} b^{2} d -\frac {37}{18} a \,b^{3} c \right ) x^{2}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {35}{9} f \,a^{3}-\frac {65}{9} a^{2} b e +\frac {104}{9} a \,b^{2} d -\frac {152}{9} b^{3} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{a^{7}}\)	\(325\)
risch	\(\frac {-\frac {c}{13 a}-\frac {\left (13 a d -19 b c \right ) x^{3}}{130 a^{2}}-\frac {\left (65 a^{2} e -104 a b d +152 b^{2} c \right ) x^{6}}{455 a^{3}}-\frac {\left (35 f \,a^{3}-65 a^{2} b e +104 a \,b^{2} d -152 b^{3} c \right ) x^{9}}{140 a^{4}}+\frac {b \left (35 f \,a^{3}-65 a^{2} b e +104 a \,b^{2} d -152 b^{3} c \right ) x^{12}}{14 a^{5}}+\frac {7 b^{2} \left (35 f \,a^{3}-65 a^{2} b e +104 a \,b^{2} d -152 b^{3} c \right ) x^{15}}{36 a^{6}}+\frac {b^{3} \left (35 f \,a^{3}-65 a^{2} b e +104 a \,b^{2} d -152 b^{3} c \right ) x^{18}}{9 a^{7}}}{x^{13} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{22} \textit {\_Z}^{3}+42875 a^{9} b^{4} f^{3}-238875 a^{8} b^{5} e \,f^{2}+382200 a^{7} b^{6} d \,f^{2}+443625 a^{7} b^{6} e^{2} f -558600 a^{6} b^{7} c \,f^{2}-1419600 a^{6} b^{7} d e f -274625 a^{6} b^{7} e^{3}+2074800 a^{5} b^{8} c e f +1135680 a^{5} b^{8} d^{2} f +1318200 a^{5} b^{8} d \,e^{2}-3319680 a^{4} b^{9} c d f -1926600 a^{4} b^{9} c \,e^{2}-2109120 a^{4} b^{9} d^{2} e +2425920 a^{3} b^{10} c^{2} f +6165120 a^{3} b^{10} c d e +1124864 a^{3} b^{10} d^{3}-4505280 a^{2} b^{11} c^{2} e -4932096 a^{2} b^{11} c \,d^{2}+7208448 a \,b^{12} c^{2} d -3511808 b^{13} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{22}-128625 a^{9} b^{4} f^{3}+716625 a^{8} b^{5} e \,f^{2}-1146600 a^{7} b^{6} d \,f^{2}-1330875 a^{7} b^{6} e^{2} f +1675800 a^{6} b^{7} c \,f^{2}+4258800 a^{6} b^{7} d e f +823875 a^{6} b^{7} e^{3}-6224400 a^{5} b^{8} c e f -3407040 a^{5} b^{8} d^{2} f -3954600 a^{5} b^{8} d \,e^{2}+9959040 a^{4} b^{9} c d f +5779800 a^{4} b^{9} c \,e^{2}+6327360 a^{4} b^{9} d^{2} e -7277760 a^{3} b^{10} c^{2} f -18495360 a^{3} b^{10} c d e -3374592 a^{3} b^{10} d^{3}[phone] a^{2} b^{11} c^{2} e [phone] a^{2} b^{11} c \,d^{2}-21625344 a \,b^{12} c^{2} d [phone] b^{13} c^{3}\right ) x +\left (35 a^{18} b f -65 a^{17} b^{2} e +104 a^{16} b^{3} d -152 a^{15} b^{4} c \right ) \textit {\_R}^{2}\right )\right )}{27}\)	\(734\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/13*c/a^3/x^13-1/10*(a*d-3*b*c)/a^4/x^10-1/7*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^7-1/4*(a^3*f-3*a^2*b*e+6*a*b^2*d-
10*b^3*c)/a^6/x^4+b*(3*a^3*f-6*a^2*b*e+10*a*b^2*d-15*b^3*c)/a^7/x+b^2/a^7*((1/9*b*(8*a^3*f-11*a^2*b*e+14*a*b^2
*d-17*b^3*c)*x^5+(19/18*a^4*f-25/18*a^3*b*e+31/18*a^2*b^2*d-37/18*a*b^3*c)*x^2)/(b*x^3+a)^2+(35/9*f*a^3-65/9*a
^2*b*e+104/9*a*b^2*d-152/9*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 = 686, normalized size of antiderivative = 1.62 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=-\frac {5460 \, {\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{18} + 9555 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{15} + 3510 \, {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{12} - 351 \, {\left (152 \, a^{3} b^{3} c - 104 \, a^{4} b^{2} d + 65 \, a^{5} b e - 35 \, a^{6} f\right )} x^{9} + 3780 \, a^{6} c + 108 \, {\left (152 \, a^{4} b^{2} c - 104 \, a^{5} b d + 65 \, a^{6} e\right )} x^{6} - 378 \, {\left (19 \, a^{5} b c - 13 \, a^{6} d\right )} x^{3} + 1820 \, \sqrt {3} {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\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 ) - 910 \, {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\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 ) + 1820 \, {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{49140 \, {\left (a^{7} b^{2} x^{19} + 2 \, a^{8} b x^{16} + a^{9} x^{13}\right )}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/49140*(5460*(152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^18 + 9555*(152*a*b^5*c - 104*a^2*b^4*
d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^15 + 3510*(152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^12
- 351*(152*a^3*b^3*c - 104*a^4*b^2*d + 65*a^5*b*e - 35*a^6*f)*x^9 + 3780*a^6*c + 108*(152*a^4*b^2*c - 104*a^5*
b*d + 65*a^6*e)*x^6 - 378*(19*a^5*b*c - 13*a^6*d)*x^3 + 1820*sqrt(3)*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e
- 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 1
04*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3))
- 910*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*
b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*lo
g(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 1820*((152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x
^19 + 2*(152*a*b^5*c - 104*a^2*b^4*d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^16 + (152*a^2*b^4*c - 104*a^3*b^3*d + 65
*a^4*b^2*e - 35*a^5*b*f)*x^13)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^7*b^2*x^19 + 2*a^8*b*x^16 + a^9*x^13
)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**14/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=-\frac {1820 \, {\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{18} + 3185 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{15} + 1170 \, {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{12} - 117 \, {\left (152 \, a^{3} b^{3} c - 104 \, a^{4} b^{2} d + 65 \, a^{5} b e - 35 \, a^{6} f\right )} x^{9} + 1260 \, a^{6} c + 36 \, {\left (152 \, a^{4} b^{2} c - 104 \, a^{5} b d + 65 \, a^{6} e\right )} x^{6} - 126 \, {\left (19 \, a^{5} b c - 13 \, a^{6} d\right )} x^{3}}{16380 \, {\left (a^{7} b^{2} x^{19} + 2 \, a^{8} b x^{16} + a^{9} x^{13}\right )}} - \frac {\sqrt {3} {\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/16380*(1820*(152*b^6*c - 104*a*b^5*d + 65*a^2*b^4*e - 35*a^3*b^3*f)*x^18 + 3185*(152*a*b^5*c - 104*a^2*b^4*
d + 65*a^3*b^3*e - 35*a^4*b^2*f)*x^15 + 1170*(152*a^2*b^4*c - 104*a^3*b^3*d + 65*a^4*b^2*e - 35*a^5*b*f)*x^12
- 117*(152*a^3*b^3*c - 104*a^4*b^2*d + 65*a^5*b*e - 35*a^6*f)*x^9 + 1260*a^6*c + 36*(152*a^4*b^2*c - 104*a^5*b
*d + 65*a^6*e)*x^6 - 126*(19*a^5*b*c - 13*a^6*d)*x^3)/(a^7*b^2*x^19 + 2*a^8*b*x^16 + a^9*x^13) - 1/27*sqrt(3)*
(152*b^4*c - 104*a*b^3*d + 65*a^2*b^2*e - 35*a^3*b*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^7
*(a/b)^(1/3)) - 1/54*(152*b^4*c - 104*a*b^3*d + 65*a^2*b^2*e - 35*a^3*b*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/
3))/(a^7*(a/b)^(1/3)) + 1/27*(152*b^4*c - 104*a*b^3*d + 65*a^2*b^2*e - 35*a^3*b*f)*log(x + (a/b)^(1/3))/(a^7*(
a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.23 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (152 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{8}} + \frac {{\left (152 \, b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 104 \, a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 65 \, a^{2} b^{3} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 35 \, a^{3} b^{2} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{8}} - \frac {{\left (152 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d + 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e - 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{8}} - \frac {34 \, b^{6} c x^{5} - 28 \, a b^{5} d x^{5} + 22 \, a^{2} b^{4} e x^{5} - 16 \, a^{3} b^{3} f x^{5} + 37 \, a b^{5} c x^{2} - 31 \, a^{2} b^{4} d x^{2} + 25 \, a^{3} b^{3} e x^{2} - 19 \, a^{4} b^{2} f x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{7}} - \frac {27300 \, b^{4} c x^{12} - 18200 \, a b^{3} d x^{12} + 10920 \, a^{2} b^{2} e x^{12} - 5460 \, a^{3} b f x^{12} - 4550 \, a b^{3} c x^{9} + 2730 \, a^{2} b^{2} d x^{9} - 1365 \, a^{3} b e x^{9} + 455 \, a^{4} f x^{9} + 1560 \, a^{2} b^{2} c x^{6} - 780 \, a^{3} b d x^{6} + 260 \, a^{4} e x^{6} - 546 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{7} x^{13}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d + 65*(-a*b^2)^(2/3)*a^2*b*e - 35*(-a*b^2)^
(2/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^8 + 1/27*(152*b^5*c*(-a/b)^(1/3) - 104*a*
b^4*d*(-a/b)^(1/3) + 65*a^2*b^3*e*(-a/b)^(1/3) - 35*a^3*b^2*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1
/3)))/a^8 - 1/54*(152*(-a*b^2)^(2/3)*b^3*c - 104*(-a*b^2)^(2/3)*a*b^2*d + 65*(-a*b^2)^(2/3)*a^2*b*e - 35*(-a*b
^2)^(2/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^8 - 1/18*(34*b^6*c*x^5 - 28*a*b^5*d*x^5 + 22*a^2*b
^4*e*x^5 - 16*a^3*b^3*f*x^5 + 37*a*b^5*c*x^2 - 31*a^2*b^4*d*x^2 + 25*a^3*b^3*e*x^2 - 19*a^4*b^2*f*x^2)/((b*x^3
+ a)^2*a^7) - 1/1820*(27300*b^4*c*x^12 - 18200*a*b^3*d*x^12 + 10920*a^2*b^2*e*x^12 - 5460*a^3*b*f*x^12 - 4550
*a*b^3*c*x^9 + 2730*a^2*b^2*d*x^9 - 1365*a^3*b*e*x^9 + 455*a^4*f*x^9 + 1560*a^2*b^2*c*x^6 - 780*a^3*b*d*x^6 +
260*a^4*e*x^6 - 546*a^3*b*c*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^7*x^13)

Mupad [B] (verification not implemented)

Time = 9.45 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx=\frac {b^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}}-\frac {\frac {c}{13\,a}-\frac {x^9\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{140\,a^4}+\frac {x^3\,\left (13\,a\,d-19\,b\,c\right )}{130\,a^2}+\frac {x^6\,\left (65\,e\,a^2-104\,d\,a\,b+152\,c\,b^2\right )}{455\,a^3}+\frac {b\,x^{12}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{14\,a^5}+\frac {7\,b^2\,x^{15}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{36\,a^6}+\frac {b^3\,x^{18}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{9\,a^7}}{a^2\,x^{13}+2\,a\,b\,x^{16}+b^2\,x^{19}}-\frac {b^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}}+\frac {b^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}} \]

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x)

[Out]

(b^(4/3)*log(b^(1/3)*x + a^(1/3))*(152*b^3*c - 35*a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(27*a^(22/3)) - (c/(13*a)
- (x^9*(152*b^3*c - 35*a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(140*a^4) + (x^3*(13*a*d - 19*b*c))/(130*a^2) + (x^
6*(152*b^2*c + 65*a^2*e - 104*a*b*d))/(455*a^3) + (b*x^12*(152*b^3*c - 35*a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(
14*a^5) + (7*b^2*x^15*(152*b^3*c - 35*a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(36*a^6) + (b^3*x^18*(152*b^3*c - 35*
a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(9*a^7))/(a^2*x^13 + b^2*x^19 + 2*a*b*x^16) - (b^(4/3)*log(3^(1/2)*a^(1/3)*
1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(152*b^3*c - 35*a^3*f - 104*a*b^2*d + 65*a^2*b*e))/(27*a^(2
2/3)) + (b^(4/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(152*b^3*c - 35*a^3*f
- 104*a*b^2*d + 65*a^2*b*e))/(27*a^(22/3))