Integrand size = 30, antiderivative size = 265 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{8/3}}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{8/3}} \]
-c/a^2/x+1/2*f*x^2/b^2-1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/a^2/b^2/(b*x ^3+a)+1/9*(5*a^3*f-2*a^2*b*e-a*b^2*d+4*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(7/3 )/b^(8/3)-1/18*(5*a^3*f-2*a^2*b*e-a*b^2*d+4*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1 /3)*x+b^(2/3)*x^2)/a^(7/3)/b^(8/3)+1/9*(5*a^3*f-2*a^2*b*e-a*b^2*d+4*b^3*c) *arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(7/3)/b^(8/3)*3^(1/2)
Time = 0.18 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {1}{18} \left (-\frac {18 c}{a^2 x}+\frac {9 f x^2}{b^2}+\frac {6 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{a^2 b^2 \left (a+b x^3\right )}+\frac {2 \sqrt {3} \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{7/3} b^{8/3}}+\frac {2 \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3} b^{8/3}}\right ) \]
((-18*c)/(a^2*x) + (9*f*x^2)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3* f)*x^2)/(a^2*b^2*(a + b*x^3)) + (2*Sqrt[3]*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(7/3)*b^(8/3)) + (2*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/( a^(7/3)*b^(8/3)) - ((4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(7/3)*b^(8/3)))/18
Time = 0.51 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2368, 25, 1812, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {3 a b^2 f x^6-b \left (\frac {c b^3}{a}-d b^2-2 a e b+2 a^2 f\right ) x^3+3 b^3 c}{x^2 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a b^2 f x^6-b \left (\frac {c b^3}{a}-d b^2-2 a e b+2 a^2 f\right ) x^3+3 b^3 c}{x^2 \left (b x^3+a\right )}dx}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \frac {\int \left (\frac {3 c b^3}{a x^2}+3 a f x b-\frac {\left (5 f a^3-2 b e a^2-b^2 d a+4 b^3 c\right ) x b}{a \left (b x^3+a\right )}\right )dx}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{3 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{6 a^{4/3}}-\frac {3 b^3 c}{a x}+\frac {3}{2} a b f x^2}{3 a b^3}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}\) |
-1/3*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a^2*b^2*(a + b*x^3)) + ((- 3*b^3*c)/(a*x) + (3*a*b*f*x^2)/2 + (b^(1/3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^ (4/3)) + (b^(1/3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)) - (b^(1/3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3* f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)))/(3*a*b^3)
3.3.67.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[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}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(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]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 1.58 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {f \,x^{2}}{2 b^{2}}-\frac {c}{a^{2} x}-\frac {\frac {\left (-\frac {1}{3} f \,a^{3}+\frac {1}{3} a^{2} b e -\frac {1}{3} a \,b^{2} d +\frac {1}{3} b^{3} c \right ) x^{2}}{b \,x^{3}+a}+\left (\frac {5}{3} f \,a^{3}-\frac {1}{3} a \,b^{2} d +\frac {4}{3} b^{3} c -\frac {2}{3} a^{2} b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b^{2}}\) | \(187\) |
| risch | \(\frac {f \,x^{2}}{2 b^{2}}+\frac {\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -4 b^{3} c \right ) x^{3}}{3 a^{2}}-\frac {b^{2} c}{a}}{b^{2} x \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} b^{2} \textit {\_Z}^{3}-125 a^{9} f^{3}+150 a^{8} b e \,f^{2}+75 a^{7} b^{2} d \,f^{2}-60 a^{7} b^{2} e^{2} f -300 a^{6} b^{3} c \,f^{2}-60 a^{6} b^{3} d e f +8 a^{6} b^{3} e^{3}+240 a^{5} b^{4} c e f -15 a^{5} b^{4} d^{2} f +12 a^{5} b^{4} d \,e^{2}+120 a^{4} b^{5} c d f -48 a^{4} b^{5} c \,e^{2}+6 a^{4} b^{5} d^{2} e -240 a^{3} b^{6} c^{2} f -48 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}+96 a^{2} b^{7} c^{2} e -12 a^{2} b^{7} c \,d^{2}+48 a \,b^{8} c^{2} d -64 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{7} b^{2}+375 a^{9} f^{3}-450 a^{8} b e \,f^{2}-225 a^{7} b^{2} d \,f^{2}+180 a^{7} b^{2} e^{2} f +900 a^{6} b^{3} c \,f^{2}+180 a^{6} b^{3} d e f -24 a^{6} b^{3} e^{3}-720 a^{5} b^{4} c e f +45 a^{5} b^{4} d^{2} f -36 a^{5} b^{4} d \,e^{2}-360 a^{4} b^{5} c d f +144 a^{4} b^{5} c \,e^{2}-18 a^{4} b^{5} d^{2} e +720 a^{3} b^{6} c^{2} f +144 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}-288 a^{2} b^{7} c^{2} e +36 a^{2} b^{7} c \,d^{2}-144 a \,b^{8} c^{2} d +192 c^{3} b^{9}\right ) x +\left (-5 a^{8} b f +2 a^{7} b^{2} e +a^{6} b^{3} d -4 a^{5} b^{4} c \right ) \textit {\_R}^{2}\right )}{9 b^{2}}\) | \(589\) |
1/2*f*x^2/b^2-c/a^2/x-1/a^2/b^2*((-1/3*f*a^3+1/3*a^2*b*e-1/3*a*b^2*d+1/3*b ^3*c)*x^2/(b*x^3+a)+(5/3*f*a^3-1/3*a*b^2*d+4/3*b^3*c-2/3*a^2*b*e)*(-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))) )
Time = 0.33 (sec) , antiderivative size = 860, normalized size of antiderivative = 3.25 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\left [\frac {9 \, a^{3} b^{3} f x^{6} - 18 \, a^{2} b^{4} c - 3 \, {\left (8 \, a b^{5} c - 2 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e - 5 \, a^{4} b^{2} f\right )} x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, a b^{5} c - a^{2} b^{4} d - 2 \, a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} + {\left (4 \, a^{2} b^{4} c - a^{3} b^{3} d - 2 \, a^{4} b^{2} e + 5 \, a^{5} b f\right )} x\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{5} x^{4} + a^{4} b^{4} x\right )}}, \frac {9 \, a^{3} b^{3} f x^{6} - 18 \, a^{2} b^{4} c - 3 \, {\left (8 \, a b^{5} c - 2 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e - 5 \, a^{4} b^{2} f\right )} x^{3} + 6 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, a b^{5} c - a^{2} b^{4} d - 2 \, a^{3} b^{3} e + 5 \, a^{4} b^{2} f\right )} x^{4} + {\left (4 \, a^{2} b^{4} c - a^{3} b^{3} d - 2 \, a^{4} b^{2} e + 5 \, a^{5} b f\right )} x\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} + {\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{5} x^{4} + a^{4} b^{4} x\right )}}\right ] \]
[1/18*(9*a^3*b^3*f*x^6 - 18*a^2*b^4*c - 3*(8*a*b^5*c - 2*a^2*b^4*d + 2*a^3 *b^3*e - 5*a^4*b^2*f)*x^3 + 3*sqrt(1/3)*((4*a*b^5*c - a^2*b^4*d - 2*a^3*b^ 3*e + 5*a^4*b^2*f)*x^4 + (4*a^2*b^4*c - a^3*b^3*d - 2*a^4*b^2*e + 5*a^5*b* f)*x)*sqrt(-(a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b - 3*sqrt(1/3)*(a*b*x + 2 *(a*b^2)^(2/3)*x^2 - (a*b^2)^(1/3)*a)*sqrt(-(a*b^2)^(1/3)/a) - 3*(a*b^2)^( 2/3)*x)/(b*x^3 + a)) - ((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*(a*b^2)^(2/3)*log(b^2*x ^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) + 2*((4*b^4*c - a*b^3*d - 2*a^2*b^ 2*e + 5*a^3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*(a *b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^3*b^5*x^4 + a^4*b^4*x), 1/18*(9*a ^3*b^3*f*x^6 - 18*a^2*b^4*c - 3*(8*a*b^5*c - 2*a^2*b^4*d + 2*a^3*b^3*e - 5 *a^4*b^2*f)*x^3 + 6*sqrt(1/3)*((4*a*b^5*c - a^2*b^4*d - 2*a^3*b^3*e + 5*a^ 4*b^2*f)*x^4 + (4*a^2*b^4*c - a^3*b^3*d - 2*a^4*b^2*e + 5*a^5*b*f)*x)*sqrt ((a*b^2)^(1/3)/a)*arctan(-sqrt(1/3)*(2*b*x - (a*b^2)^(1/3))*sqrt((a*b^2)^( 1/3)/a)/b) - ((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a*b^3 *c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*(a*b^2)^(2/3)*log(b^2*x^2 - (a*b^ 2)^(1/3)*b*x + (a*b^2)^(2/3)) + 2*((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^ 3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*(a*b^2)^(2/3 )*log(b*x + (a*b^2)^(1/3)))/(a^3*b^5*x^4 + a^4*b^4*x)]
Time = 63.97 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.72 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {- 3 a b^{2} c + x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - 4 b^{3} c\right )}{3 a^{3} b^{2} x + 3 a^{2} b^{3} x^{4}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} b^{8} - 125 a^{9} f^{3} + 150 a^{8} b e f^{2} + 75 a^{7} b^{2} d f^{2} - 60 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 60 a^{6} b^{3} d e f + 8 a^{6} b^{3} e^{3} + 240 a^{5} b^{4} c e f - 15 a^{5} b^{4} d^{2} f + 12 a^{5} b^{4} d e^{2} + 120 a^{4} b^{5} c d f - 48 a^{4} b^{5} c e^{2} + 6 a^{4} b^{5} d^{2} e - 240 a^{3} b^{6} c^{2} f - 48 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e - 12 a^{2} b^{7} c d^{2} + 48 a b^{8} c^{2} d - 64 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{5} b^{5}}{25 a^{6} f^{2} - 20 a^{5} b e f - 10 a^{4} b^{2} d f + 4 a^{4} b^{2} e^{2} + 40 a^{3} b^{3} c f + 4 a^{3} b^{3} d e - 16 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 8 a b^{5} c d + 16 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{2}}{2 b^{2}} \]
(-3*a*b**2*c + x**3*(a**3*f - a**2*b*e + a*b**2*d - 4*b**3*c))/(3*a**3*b** 2*x + 3*a**2*b**3*x**4) + RootSum(729*_t**3*a**7*b**8 - 125*a**9*f**3 + 15 0*a**8*b*e*f**2 + 75*a**7*b**2*d*f**2 - 60*a**7*b**2*e**2*f - 300*a**6*b** 3*c*f**2 - 60*a**6*b**3*d*e*f + 8*a**6*b**3*e**3 + 240*a**5*b**4*c*e*f - 1 5*a**5*b**4*d**2*f + 12*a**5*b**4*d*e**2 + 120*a**4*b**5*c*d*f - 48*a**4*b **5*c*e**2 + 6*a**4*b**5*d**2*e - 240*a**3*b**6*c**2*f - 48*a**3*b**6*c*d* e + a**3*b**6*d**3 + 96*a**2*b**7*c**2*e - 12*a**2*b**7*c*d**2 + 48*a*b**8 *c**2*d - 64*b**9*c**3, Lambda(_t, _t*log(81*_t**2*a**5*b**5/(25*a**6*f**2 - 20*a**5*b*e*f - 10*a**4*b**2*d*f + 4*a**4*b**2*e**2 + 40*a**3*b**3*c*f + 4*a**3*b**3*d*e - 16*a**2*b**4*c*e + a**2*b**4*d**2 - 8*a*b**5*c*d + 16* b**6*c**2) + x))) + f*x**2/(2*b**2)
Time = 0.30 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f x^{2}}{2 \, b^{2}} - \frac {3 \, a b^{2} c + {\left (4 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x\right )}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/2*f*x^2/b^2 - 1/3*(3*a*b^2*c + (4*b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3 )/(a^2*b^3*x^4 + a^3*b^2*x) - 1/9*sqrt(3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^3*(a/ b)^(1/3)) - 1/18*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*log(x^2 - x*(a/ b)^(1/3) + (a/b)^(2/3))/(a^2*b^3*(a/b)^(1/3)) + 1/9*(4*b^3*c - a*b^2*d - 2 *a^2*b*e + 5*a^3*f)*log(x + (a/b)^(1/3))/(a^2*b^3*(a/b)^(1/3))
Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f x^{2}}{2 \, b^{2}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac {4 \, b^{3} c x^{3} - a b^{2} d x^{3} + a^{2} b e x^{3} - a^{3} f x^{3} + 3 \, a b^{2} c}{3 \, {\left (b x^{4} + a x\right )} a^{2} b^{2}} \]
1/2*f*x^2/b^2 - 1/9*sqrt(3)*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*arct an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/3)*a^2*b^2) + 1/18*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^2*b^2) + 1/9*(4*b^3*c*(-a/b)^(1/3) - a* b^2*d*(-a/b)^(1/3) - 2*a^2*b*e*(-a/b)^(1/3) + 5*a^3*f*(-a/b)^(1/3))*(-a/b) ^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b^2) - 1/3*(4*b^3*c*x^3 - a*b^2*d*x ^3 + a^2*b*e*x^3 - a^3*f*x^3 + 3*a*b^2*c)/((b*x^4 + a*x)*a^2*b^2)
Time = 9.51 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f\,x^2}{2\,b^2}-\frac {\frac {x^3\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{3\,a^2}+\frac {b^2\,c}{a}}{b^3\,x^4+a\,b^2\,x}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}} \]
(f*x^2)/(2*b^2) - ((x^3*(4*b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^2) + ( b^2*c)/a)/(b^3*x^4 + a*b^2*x) + (log(b^(1/3)*x + a^(1/3))*(4*b^3*c + 5*a^3 *f - a*b^2*d - 2*a^2*b*e))/(9*a^(7/3)*b^(8/3)) - (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(4*b^3*c + 5*a^3*f - a*b^2* d - 2*a^2*b*e))/(9*a^(7/3)*b^(8/3)) + (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)* x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(4*b^3*c + 5*a^3*f - a*b^2*d - 2*a^2*b *e))/(9*a^(7/3)*b^(8/3))