\(\int \frac {c+d x^3+e x^6+f x^9}{x^9 (a+b x^3)^3} \, dx\) [300]

Optimal result

Integrand size = 30, antiderivative size = 341 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}} \]

[Out]

-1/8*c/a^3/x^8+1/5*(-a*d+3*b*c)/a^4/x^5+1/2*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^2-1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c
)*x/a^4/(b*x^3+a)^2-1/18*(-5*a^3*f+11*a^2*b*e-17*a*b^2*d+23*b^3*c)*x/a^5/(b*x^3+a)-1/27*(-5*a^3*f+20*a^2*b*e-4
4*a*b^2*d+77*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(17/3)/b^(1/3)+1/54*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3*c)*ln(a
^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)/b^(1/3)+1/27*(-5*a^3*f+20*a^2*b*e-44*a*b^2*d+77*b^3*c)*arctan(1
/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(17/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 341, 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, 206, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\frac {3 b c-a d}{5 a^4 x^5}-\frac {c}{8 a^3 x^8}-\frac {a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt {3} a^{17/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac {x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

[In]

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

[Out]

-1/8*c/(a^3*x^8) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(2*a^5*x^2) - ((b^3*c - a*b^2*d + a
^2*b*e - a^3*f)*x)/(6*a^4*(a + b*x^3)^2) - ((23*b^3*c - 17*a*b^2*d + 11*a^2*b*e - 5*a^3*f)*x)/(18*a^5*(a + b*x
^3)) + ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*S
qrt[3]*a^(17/3)*b^(1/3)) - ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(17
/3)*b^(1/3)) + ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/(54*a^(17/3)*b^(1/3))

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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 {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \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 {5 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )^2} \, dx}{6 a b^3} \\ & = -\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\int \frac {18 b^6 c-18 b^6 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^6 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-\frac {2 b^6 \left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )} \, dx}{18 a^2 b^6} \\ & = -\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^6 c}{a x^9}+\frac {18 b^6 (-3 b c+a d)}{a^2 x^6}+\frac {18 b^6 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^3}+\frac {2 b^6 \left (-77 b^3 c+44 a b^2 d-20 a^2 b e+5 a^3 f\right )}{a^3 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^6} \\ & = -\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{9 a^5} \\ & = -\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{17/3}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {2 \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^{17/3}} \\ & = -\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{16/3}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 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}{54 a^{17/3} \sqrt [3]{b}} \\ & = -\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\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^{17/3} \sqrt [3]{b}} \\ & = -\frac {c}{8 a^3 x^8}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac {\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 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^{17/3} \sqrt [3]{b}}-\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac {\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {135 a^{8/3} c}{x^8}-\frac {216 a^{5/3} (-3 b c+a d)}{x^5}-\frac {540 a^{2/3} \left (6 b^2 c-3 a b d+a^2 e\right )}{x^2}+\frac {180 a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{\left (a+b x^3\right )^2}+\frac {60 a^{2/3} \left (-23 b^3 c+17 a b^2 d-11 a^2 b e+5 a^3 f\right ) x}{a+b x^3}+\frac {40 \sqrt {3} \left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {40 \left (-77 b^3 c+44 a b^2 d-20 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {20 \left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{1080 a^{17/3}} \]

[In]

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

[Out]

((-135*a^(8/3)*c)/x^8 - (216*a^(5/3)*(-3*b*c + a*d))/x^5 - (540*a^(2/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x^2 + (18
0*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3)^2 + (60*a^(2/3)*(-23*b^3*c + 17*a*b^2*d - 11*a
^2*b*e + 5*a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b
^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-77*b^3*c + 44*a*b^2*d - 20*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/
3)*x])/b^(1/3) + (20*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/b^(1/3))/(1080*a^(17/3))

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {c}{8 a^{3} x^{8}}-\frac {a d -3 b c}{5 a^{4} x^{5}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{2 a^{5} x^{2}}+\frac {\frac {\left (\frac {5}{18} a^{3} b f -\frac {11}{18} a^{2} e \,b^{2}+\frac {17}{18} a \,b^{3} d -\frac {23}{18} b^{4} c \right ) x^{4}+\frac {a \left (4 f \,a^{3}-7 a^{2} b e +10 a \,b^{2} d -13 b^{3} c \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 f \,a^{3}-20 a^{2} b e +44 a \,b^{2} d -77 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}}{a^{5}}\)	\(252\)
risch	\(\frac {\frac {b \left (5 f \,a^{3}-20 a^{2} b e +44 a \,b^{2} d -77 b^{3} c \right ) x^{12}}{18 a^{5}}+\frac {4 \left (5 f \,a^{3}-20 a^{2} b e +44 a \,b^{2} d -77 b^{3} c \right ) x^{9}}{45 a^{4}}-\frac {\left (20 a^{2} e -44 a b d +77 b^{2} c \right ) x^{6}}{40 a^{3}}-\frac {\left (4 a d -7 b c \right ) x^{3}}{20 a^{2}}-\frac {c}{8 a}}{x^{8} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{17} b \,\textit {\_Z}^{3}-125 a^{9} f^{3}+1500 a^{8} b e \,f^{2}-3300 a^{7} b^{2} d \,f^{2}-6000 a^{7} b^{2} e^{2} f +5775 a^{6} b^{3} c \,f^{2}+26400 a^{6} b^{3} d e f +8000 a^{6} b^{3} e^{3}-46200 a^{5} b^{4} c e f -29040 a^{5} b^{4} d^{2} f -52800 a^{5} b^{4} d \,e^{2}+101640 a^{4} b^{5} c d f +92400 a^{4} b^{5} c \,e^{2}+116160 a^{4} b^{5} d^{2} e -88935 a^{3} b^{6} c^{2} f -406560 a^{3} b^{6} c d e -85184 a^{3} b^{6} d^{3}+355740 a^{2} b^{7} c^{2} e +447216 a^{2} b^{7} c \,d^{2}-782628 a \,b^{8} c^{2} d +456533 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{17} b +375 a^{9} f^{3}-4500 a^{8} b e \,f^{2}+9900 a^{7} b^{2} d \,f^{2}+18000 a^{7} b^{2} e^{2} f -17325 a^{6} b^{3} c \,f^{2}-79200 a^{6} b^{3} d e f -24000 a^{6} b^{3} e^{3}+138600 a^{5} b^{4} c e f +87120 a^{5} b^{4} d^{2} f +158400 a^{5} b^{4} d \,e^{2}-304920 a^{4} b^{5} c d f -277200 a^{4} b^{5} c \,e^{2}-348480 a^{4} b^{5} d^{2} e +266805 a^{3} b^{6} c^{2} f +1219680 a^{3} b^{6} c d e +255552 a^{3} b^{6} d^{3}-1067220 a^{2} b^{7} c^{2} e -1341648 a^{2} b^{7} c \,d^{2}+2347884 a \,b^{8} c^{2} d -1369599 c^{3} b^{9}\right ) x +\left (-25 a^{12} f^{2}+200 a^{11} b e f -440 a^{10} b^{2} d f -400 a^{10} b^{2} e^{2}+770 a^{9} b^{3} c f +1760 a^{9} b^{3} d e -3080 a^{8} b^{4} c e -1936 a^{8} b^{4} d^{2}+6776 a^{7} b^{5} c d -5929 a^{6} b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{27}\)	\(713\)

[In]

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

[Out]

-1/8*c/a^3/x^8-1/5*(a*d-3*b*c)/a^4/x^5-1/2*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^2+1/a^5*(((5/18*a^3*b*f-11/18*a^2*e*b
^2+17/18*a*b^3*d-23/18*b^4*c)*x^4+1/9*a*(4*a^3*f-7*a^2*b*e+10*a*b^2*d-13*b^3*c)*x)/(b*x^3+a)^2+1/9*(5*a^3*f-20
*a^2*b*e+44*a*b^2*d-77*b^3*c)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b
)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (294) = 588\).

Time = 0.28 (sec) , antiderivative size = 1317, normalized size of antiderivative = 3.86 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d
+ 20*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135*a^6*b*c + 27*(77*a^4*b^3*c - 44*a^5*b^2*d + 20*a^6*b*e)*x^6 - 54*(7*a^5
*b^2*c - 4*a^6*b*d)*x^3 + 60*sqrt(1/3)*((77*a*b^6*c - 44*a^2*b^5*d + 20*a^3*b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*
a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^11 + (77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*
a^6*b*f)*x^8)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^
2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 20*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3
*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^
3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((77*
b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*
f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/
(a^7*b^3*x^14 + 2*a^8*b^2*x^11 + a^9*b*x^8), -1/1080*(60*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b
^2*f)*x^12 + 96*(77*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^9 + 135*a^6*b*c + 27*(77*a^4*b^3*c
- 44*a^5*b^2*d + 20*a^6*b*e)*x^6 - 54*(7*a^5*b^2*c - 4*a^6*b*d)*x^3 + 120*sqrt(1/3)*((77*a*b^6*c - 44*a^2*b^5*
d + 20*a^3*b^4*e - 5*a^4*b^3*f)*x^14 + 2*(77*a^2*b^5*c - 44*a^3*b^4*d + 20*a^4*b^3*e - 5*a^5*b^2*f)*x^11 + (77
*a^3*b^4*c - 44*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^8)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(
2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 20*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f
)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4
*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*((77*b^5*c - 44*a*b^4
*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^
2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^7*b^3*x^14 +
2*a^8*b^2*x^11 + a^9*b*x^8)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {20 \, {\left (77 \, b^{4} c - 44 \, a b^{3} d + 20 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 32 \, {\left (77 \, a b^{3} c - 44 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} + 9 \, {\left (77 \, a^{2} b^{2} c - 44 \, a^{3} b d + 20 \, a^{4} e\right )} x^{6} + 45 \, a^{4} c - 18 \, {\left (7 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}}{360 \, {\left (a^{5} b^{2} x^{14} + 2 \, a^{6} b x^{11} + a^{7} x^{8}\right )}} - \frac {\sqrt {3} {\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{5} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/360*(20*(77*b^4*c - 44*a*b^3*d + 20*a^2*b^2*e - 5*a^3*b*f)*x^12 + 32*(77*a*b^3*c - 44*a^2*b^2*d + 20*a^3*b*
e - 5*a^4*f)*x^9 + 9*(77*a^2*b^2*c - 44*a^3*b*d + 20*a^4*e)*x^6 + 45*a^4*c - 18*(7*a^3*b*c - 4*a^4*d)*x^3)/(a^
5*b^2*x^14 + 2*a^6*b*x^11 + a^7*x^8) - 1/27*sqrt(3)*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*arctan(1/3*
sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^5*b*(a/b)^(2/3)) + 1/54*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^
3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*b*(a/b)^(2/3)) - 1/27*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5
*a^3*f)*log(x + (a/b)^(1/3))/(a^5*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=\frac {{\left (77 \, b^{3} c - 44 \, a b^{2} d + 20 \, a^{2} b e - 5 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{6}} - \frac {\sqrt {3} {\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} b} - \frac {{\left (77 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} b} - \frac {23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} + 11 \, a^{2} b^{2} e x^{4} - 5 \, a^{3} b f x^{4} + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x + 14 \, a^{3} b e x - 8 \, a^{4} f x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{5}} - \frac {120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} e x^{6} - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \]

[In]

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

[Out]

1/27*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^6 - 1/27*sqrt(3)
*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3)*a*b^2*d + 20*(-a*b^2)^(1/3)*a^2*b*e - 5*(-a*b^2)^(1/3)*a^3*f)*ar
ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^6*b) - 1/54*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3
)*a*b^2*d + 20*(-a*b^2)^(1/3)*a^2*b*e - 5*(-a*b^2)^(1/3)*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*
b) - 1/18*(23*b^4*c*x^4 - 17*a*b^3*d*x^4 + 11*a^2*b^2*e*x^4 - 5*a^3*b*f*x^4 + 26*a*b^3*c*x - 20*a^2*b^2*d*x +
14*a^3*b*e*x - 8*a^4*f*x)/((b*x^3 + a)^2*a^5) - 1/40*(120*b^2*c*x^6 - 60*a*b*d*x^6 + 20*a^2*e*x^6 - 24*a*b*c*x
^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^5*x^8)

Mupad [B] (verification not implemented)

Time = 9.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {c}{8\,a}+\frac {4\,x^9\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{45\,a^4}+\frac {x^3\,\left (4\,a\,d-7\,b\,c\right )}{20\,a^2}+\frac {x^6\,\left (20\,e\,a^2-44\,d\,a\,b+77\,c\,b^2\right )}{40\,a^3}+\frac {b\,x^{12}\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{18\,a^5}}{a^2\,x^8+2\,a\,b\,x^{11}+b^2\,x^{14}}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-5\,f\,a^3+20\,e\,a^2\,b-44\,d\,a\,b^2+77\,c\,b^3\right )}{27\,a^{17/3}\,b^{1/3}} \]

[In]

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

[Out]

(log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*
a^2*b*e))/(27*a^(17/3)*b^(1/3)) - (log(b^(1/3)*x + a^(1/3))*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(2
7*a^(17/3)*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)*(77*b^3*c - 5*a^
3*f - 44*a*b^2*d + 20*a^2*b*e))/(27*a^(17/3)*b^(1/3)) - (c/(8*a) + (4*x^9*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 2
0*a^2*b*e))/(45*a^4) + (x^3*(4*a*d - 7*b*c))/(20*a^2) + (x^6*(77*b^2*c + 20*a^2*e - 44*a*b*d))/(40*a^3) + (b*x
^12*(77*b^3*c - 5*a^3*f - 44*a*b^2*d + 20*a^2*b*e))/(18*a^5))/(a^2*x^8 + b^2*x^14 + 2*a*b*x^11)