\(\int \frac {c+d x^3+e x^6+f x^9}{x^9 (a+b x^3)^2} \, dx\) [272]

Optimal result

Integrand size = 30, antiderivative size = 297 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=-\frac {c}{8 a^2 x^8}+\frac {2 b c-a d}{5 a^3 x^5}-\frac {3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}+\frac {\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{14/3} \sqrt [3]{b}}-\frac {\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac {\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{14/3} \sqrt [3]{b}} \]

[Out]

-1/8*c/a^2/x^8+1/5*(-a*d+2*b*c)/a^3/x^5+1/2*(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x^2-1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c
)*x/a^4/(b*x^3+a)-1/9*(-2*a^3*f+5*a^2*b*e-8*a*b^2*d+11*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(14/3)/b^(1/3)+1/18*(-2*
a^3*f+5*a^2*b*e-8*a*b^2*d+11*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)/b^(1/3)+1/9*(-2*a^3*f+5
*a^2*b*e-8*a*b^2*d+11*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3)/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=\frac {2 b c-a d}{5 a^3 x^5}-\frac {c}{8 a^2 x^8}-\frac {a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 \sqrt {3} a^{14/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {45 a^{8/3} c}{x^8}-\frac {72 a^{5/3} (-2 b c+a d)}{x^5}-\frac {180 a^{2/3} \left (3 b^2 c-2 a b d+a^2 e\right )}{x^2}+\frac {120 a^{2/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a+b x^3}+\frac {40 \sqrt {3} \left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {40 \left (-11 b^3 c+8 a b^2 d-5 a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {20 \left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{360 a^{14/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.72

method	result	size
default	\(-\frac {c}{8 a^{2} x^{8}}-\frac {a d -2 b c}{5 a^{3} x^{5}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{2 a^{4} x^{2}}+\frac {\frac {\left (\frac {1}{3} f \,a^{3}-\frac {1}{3} a^{2} b e +\frac {1}{3} a \,b^{2} d -\frac {1}{3} b^{3} c \right ) x}{b \,x^{3}+a}+\frac {\left (2 f \,a^{3}-5 a^{2} b e +8 a \,b^{2} d -11 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}}{a^{4}}\)	\(214\)
risch	\(\frac {\frac {\left (2 f \,a^{3}-5 a^{2} b e +8 a \,b^{2} d -11 b^{3} c \right ) x^{9}}{6 a^{4}}-\frac {\left (5 a^{2} e -8 a b d +11 b^{2} c \right ) x^{6}}{10 a^{3}}-\frac {\left (8 a d -11 b c \right ) x^{3}}{40 a^{2}}-\frac {c}{8 a}}{x^{8} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} b \,\textit {\_Z}^{3}-8 a^{9} f^{3}+60 a^{8} b e \,f^{2}-96 a^{7} b^{2} d \,f^{2}-150 a^{7} b^{2} e^{2} f +132 a^{6} b^{3} c \,f^{2}+480 a^{6} b^{3} d e f +125 a^{6} b^{3} e^{3}-660 a^{5} b^{4} c e f -384 a^{5} b^{4} d^{2} f -600 a^{5} b^{4} d \,e^{2}+1056 a^{4} b^{5} c d f +825 a^{4} b^{5} c \,e^{2}+960 a^{4} b^{5} d^{2} e -726 a^{3} b^{6} c^{2} f -2640 a^{3} b^{6} c d e -512 a^{3} b^{6} d^{3}+1815 a^{2} b^{7} c^{2} e +2112 a^{2} b^{7} c \,d^{2}-2904 a \,b^{8} c^{2} d +1331 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{14} b +24 a^{9} f^{3}-180 a^{8} b e \,f^{2}+288 a^{7} b^{2} d \,f^{2}+450 a^{7} b^{2} e^{2} f -396 a^{6} b^{3} c \,f^{2}-1440 a^{6} b^{3} d e f -375 a^{6} b^{3} e^{3}+1980 a^{5} b^{4} c e f +1152 a^{5} b^{4} d^{2} f +1800 a^{5} b^{4} d \,e^{2}-3168 a^{4} b^{5} c d f -2475 a^{4} b^{5} c \,e^{2}-2880 a^{4} b^{5} d^{2} e +2178 a^{3} b^{6} c^{2} f +7920 a^{3} b^{6} c d e +1536 a^{3} b^{6} d^{3}-5445 a^{2} b^{7} c^{2} e -6336 a^{2} b^{7} c \,d^{2}+8712 a \,b^{8} c^{2} d -3993 c^{3} b^{9}\right ) x +\left (-4 a^{11} f^{2}+20 a^{10} b e f -32 a^{9} b^{2} d f -25 a^{9} b^{2} e^{2}+44 a^{8} b^{3} c f +80 a^{8} b^{3} d e -110 a^{7} b^{4} c e -64 a^{7} b^{4} d^{2}+176 a^{6} b^{5} c d -121 a^{5} b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{9}\)	\(677\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^9/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/8*c/a^2/x^8-1/5*(a*d-2*b*c)/a^3/x^5-1/2*(a^2*e-2*a*b*d+3*b^2*c)/a^4/x^2+1/a^4*((1/3*f*a^3-1/3*a^2*b*e+1/3*a
*b^2*d-1/3*b^3*c)*x/(b*x^3+a)+1/3*(2*a^3*f-5*a^2*b*e+8*a*b^2*d-11*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 [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 959, normalized size of antiderivative = 3.23 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/360*(60*(11*a^2*b^4*c - 8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*b*f)*x^9 + 45*a^5*b*c + 36*(11*a^3*b^3*c - 8*a^4
*b^2*d + 5*a^5*b*e)*x^6 - 9*(11*a^4*b^2*c - 8*a^5*b*d)*x^3 + 60*sqrt(1/3)*((11*a*b^5*c - 8*a^2*b^4*d + 5*a^3*b
^3*e - 2*a^4*b^2*f)*x^11 + (11*a^2*b^4*c - 8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*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*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*
a^2*b^2*d + 5*a^3*b*e - 2*a^4*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*((11
*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*(a^
2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^2*x^11 + a^7*b*x^8), -1/360*(60*(11*a^2*b^4*c - 8*a^3*b^3*d + 5*
a^4*b^2*e - 2*a^5*b*f)*x^9 + 45*a^5*b*c + 36*(11*a^3*b^3*c - 8*a^4*b^2*d + 5*a^5*b*e)*x^6 - 9*(11*a^4*b^2*c -
8*a^5*b*d)*x^3 + 120*sqrt(1/3)*((11*a*b^5*c - 8*a^2*b^4*d + 5*a^3*b^3*e - 2*a^4*b^2*f)*x^11 + (11*a^2*b^4*c -
8*a^3*b^3*d + 5*a^4*b^2*e - 2*a^5*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*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*
c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*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) + 4
0*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^
8)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^2*x^11 + a^7*b*x^8)]

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**9/(b*x**3+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=-\frac {20 \, {\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} x^{9} + 12 \, {\left (11 \, a b^{2} c - 8 \, a^{2} b d + 5 \, a^{3} e\right )} x^{6} + 15 \, a^{3} c - 3 \, {\left (11 \, a^{2} b c - 8 \, a^{3} d\right )} x^{3}}{120 \, {\left (a^{4} b x^{11} + a^{5} x^{8}\right )}} - \frac {\sqrt {3} {\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-1/120*(20*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^9 + 12*(11*a*b^2*c - 8*a^2*b*d + 5*a^3*e)*x^6 + 15*a
^3*c - 3*(11*a^2*b*c - 8*a^3*d)*x^3)/(a^4*b*x^11 + a^5*x^8) - 1/9*sqrt(3)*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e -
2*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b*(a/b)^(2/3)) + 1/18*(11*b^3*c - 8*a*b^2*d
+ 5*a^2*b*e - 2*a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b*(a/b)^(2/3)) - 1/9*(11*b^3*c - 8*a*b^2*d
+ 5*a^2*b*e - 2*a^3*f)*log(x + (a/b)^(1/3))/(a^4*b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=\frac {{\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5}} - \frac {\sqrt {3} {\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{5} b} - \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{3 \, {\left (b x^{3} + a\right )} a^{4}} - \frac {{\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{5} b} - \frac {60 \, b^{2} c x^{6} - 40 \, a b d x^{6} + 20 \, a^{2} e x^{6} - 16 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{4} x^{8}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )^2} \, dx=-\frac {\frac {c}{8\,a}+\frac {x^9\,\left (-2\,f\,a^3+5\,e\,a^2\,b-8\,d\,a\,b^2+11\,c\,b^3\right )}{6\,a^4}+\frac {x^3\,\left (8\,a\,d-11\,b\,c\right )}{40\,a^2}+\frac {x^6\,\left (5\,e\,a^2-8\,d\,a\,b+11\,c\,b^2\right )}{10\,a^3}}{b\,x^{11}+a\,x^8}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-2\,f\,a^3+5\,e\,a^2\,b-8\,d\,a\,b^2+11\,c\,b^3\right )}{9\,a^{14/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-2\,f\,a^3+5\,e\,a^2\,b-8\,d\,a\,b^2+11\,c\,b^3\right )}{9\,a^{14/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-2\,f\,a^3+5\,e\,a^2\,b-8\,d\,a\,b^2+11\,c\,b^3\right )}{9\,a^{14/3}\,b^{1/3}} \]

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^2),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)*(11*b^3*c - 2*a^3*f - 8*a*b^2*d + 5*a^
2*b*e))/(9*a^(14/3)*b^(1/3)) - (log(b^(1/3)*x + a^(1/3))*(11*b^3*c - 2*a^3*f - 8*a*b^2*d + 5*a^2*b*e))/(9*a^(1
4/3)*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)*(11*b^3*c - 2*a^3*f -
8*a*b^2*d + 5*a^2*b*e))/(9*a^(14/3)*b^(1/3)) - (c/(8*a) + (x^9*(11*b^3*c - 2*a^3*f - 8*a*b^2*d + 5*a^2*b*e))/(
6*a^4) + (x^3*(8*a*d - 11*b*c))/(40*a^2) + (x^6*(11*b^2*c + 5*a^2*e - 8*a*b*d))/(10*a^3))/(a*x^8 + b*x^11)