Integrand size = 30, antiderivative size = 265 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=-\frac {c}{a^2 x}+\frac {f x^2}{2 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{8/3}}+\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{8/3}} \]
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Time = 0.17 (sec) , antiderivative size = 265, 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, 1502, 298, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=-\frac {c}{a^2 x}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{3 \sqrt {3} a^{7/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac {f x^2}{2 b^2} \]
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Rule 31
Rule 210
Rule 298
Rule 631
Rule 642
Rule 648
Rule 1502
Rule 1843
Rubi steps \begin{align*} \text {integral}& = -\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 {\int \frac {-3 b^3 c+b \left (\frac {b^3 c}{a}-b^2 d-2 a b e+2 a^2 f\right ) x^3-3 a b^2 f x^6}{x^2 \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^2}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^2}-3 a b f x+\frac {b \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) x}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^3} \\ & = -\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 ) \int \frac {x}{a+b x^3} \, dx}{3 a^2 b^2} \\ & = -\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 ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3} b^{7/3}}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\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}{9 a^{7/3} b^{7/3}} \\ & = -\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 ) \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 ) \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^{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 ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^2 b^{7/3}} \\ & = -\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 ) \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}}-\frac {\left (4 b^3 c-a b^2 d-2 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 )}{3 a^{7/3} b^{8/3}} \\ & = -\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 ) \tan ^{-1}\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}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {1}{18} \left (-\frac {18 c}{a^2 x}+\frac {9 f x^2}{b^2}+\frac {6 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^2}{a^2 b^2 \left (a+b x^3\right )}+\frac {2 \sqrt {3} \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{7/3} b^{8/3}}+\frac {2 \left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3} b^{8/3}}-\frac {\left (4 b^3 c-a b^2 d-2 a^2 b e+5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3} b^{8/3}}\right ) \]
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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 | \(\frac {f \,x^{2}}{2 b^{2}}+\frac {\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -4 b^{3} c \right ) x^{3}}{3 a^{2}}-\frac {b^{2} c}{a}}{b^{2} x \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} b^{2} \textit {\_Z}^{3}-125 a^{9} f^{3}+150 a^{8} b e \,f^{2}+75 a^{7} b^{2} d \,f^{2}-60 a^{7} b^{2} e^{2} f -300 a^{6} b^{3} c \,f^{2}-60 a^{6} b^{3} d e f +8 a^{6} b^{3} e^{3}+240 a^{5} b^{4} c e f -15 a^{5} b^{4} d^{2} f +12 a^{5} b^{4} d \,e^{2}+120 a^{4} b^{5} c d f -48 a^{4} b^{5} c \,e^{2}+6 a^{4} b^{5} d^{2} e -240 a^{3} b^{6} c^{2} f -48 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}+96 a^{2} b^{7} c^{2} e -12 a^{2} b^{7} c \,d^{2}+48 a \,b^{8} c^{2} d -64 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{7} b^{2}+375 a^{9} f^{3}-450 a^{8} b e \,f^{2}-225 a^{7} b^{2} d \,f^{2}+180 a^{7} b^{2} e^{2} f +900 a^{6} b^{3} c \,f^{2}+180 a^{6} b^{3} d e f -24 a^{6} b^{3} e^{3}-720 a^{5} b^{4} c e f +45 a^{5} b^{4} d^{2} f -36 a^{5} b^{4} d \,e^{2}-360 a^{4} b^{5} c d f +144 a^{4} b^{5} c \,e^{2}-18 a^{4} b^{5} d^{2} e +720 a^{3} b^{6} c^{2} f +144 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}-288 a^{2} b^{7} c^{2} e +36 a^{2} b^{7} c \,d^{2}-144 a \,b^{8} c^{2} d +192 c^{3} b^{9}\right ) x +\left (-5 a^{8} b f +2 a^{7} b^{2} e +a^{6} b^{3} d -4 a^{5} b^{4} c \right ) \textit {\_R}^{2}\right )}{9 b^{2}}\) | \(589\) |
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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 ] \]
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Time = 63.97 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.72 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {- 3 a b^{2} c + x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - 4 b^{3} c\right )}{3 a^{3} b^{2} x + 3 a^{2} b^{3} x^{4}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} b^{8} - 125 a^{9} f^{3} + 150 a^{8} b e f^{2} + 75 a^{7} b^{2} d f^{2} - 60 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 60 a^{6} b^{3} d e f + 8 a^{6} b^{3} e^{3} + 240 a^{5} b^{4} c e f - 15 a^{5} b^{4} d^{2} f + 12 a^{5} b^{4} d e^{2} + 120 a^{4} b^{5} c d f - 48 a^{4} b^{5} c e^{2} + 6 a^{4} b^{5} d^{2} e - 240 a^{3} b^{6} c^{2} f - 48 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e - 12 a^{2} b^{7} c d^{2} + 48 a b^{8} c^{2} d - 64 b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{5} b^{5}}{25 a^{6} f^{2} - 20 a^{5} b e f - 10 a^{4} b^{2} d f + 4 a^{4} b^{2} e^{2} + 40 a^{3} b^{3} c f + 4 a^{3} b^{3} d e - 16 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 8 a b^{5} c d + 16 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{2}}{2 b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f x^{2}}{2 \, b^{2}} - \frac {3 \, a b^{2} c + {\left (4 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x\right )}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f x^{2}}{2 \, b^{2}} - \frac {\sqrt {3} {\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c - a b^{2} d - 2 \, a^{2} b e + 5 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2}} + \frac {{\left (4 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac {4 \, b^{3} c x^{3} - a b^{2} d x^{3} + a^{2} b e x^{3} - a^{3} f x^{3} + 3 \, a b^{2} c}{3 \, {\left (b x^{4} + a x\right )} a^{2} b^{2}} \]
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Time = 9.51 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {f\,x^2}{2\,b^2}-\frac {\frac {x^3\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{3\,a^2}+\frac {b^2\,c}{a}}{b^3\,x^4+a\,b^2\,x}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,f\,a^3-2\,e\,a^2\,b-d\,a\,b^2+4\,c\,b^3\right )}{9\,a^{7/3}\,b^{8/3}} \]
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