Integrand size = 30, antiderivative size = 369 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{19/3}}+\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}} \]
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Time = 0.30 (sec) , antiderivative size = 369, 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 = {1842, 1901, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}-\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac {a x \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac {x^4 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{4 b^5}-\frac {a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \sqrt {3} b^{19/3}}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{18 b^{19/3}}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^{19/3}}+\frac {x^{10} (b e-2 a f)}{10 b^3}+\frac {f x^{13}}{13 b^2} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1842
Rule 1901
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac {\int \frac {-a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+3 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^9-3 a b^4 (b e-a f) x^{12}-3 a b^5 f x^{15}}{a+b x^3} \, dx}{3 a b^6} \\ & = -\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac {\int \left (3 a^2 \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right )-3 a b \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3-3 a b^2 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6-3 a b^3 (b e-2 a f) x^9-3 a b^4 f x^{12}+\frac {-7 a^3 b^3 c+10 a^4 b^2 d-13 a^5 b e+16 a^6 f}{a+b x^3}\right ) \, dx}{3 a b^6} \\ & = -\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {\left (a^2 \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{3 b^6} \\ & = -\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^6}+\frac {\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\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 b^6} \\ & = -\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac {\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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}{18 b^{19/3}}+\frac {\left (a^{5/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^6} \\ & = -\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}}+\frac {\left (a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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 )}{3 b^{19/3}} \\ & = -\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 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} b^{19/3}}+\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}-\frac {a^{4/3} \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.99 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4}{4 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^7}{7 b^4}+\frac {(b e-2 a f) x^{10}}{10 b^3}+\frac {f x^{13}}{13 b^2}+\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{3 b^6 \left (a+b x^3\right )}+\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{19/3}}-\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{19/3}}+\frac {a^{4/3} \left (-7 b^3 c+10 a b^2 d-13 a^2 b e+16 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{19/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {f \,x^{13}}{13 b^{2}}-\frac {x^{10} a f}{5 b^{3}}+\frac {x^{10} e}{10 b^{2}}+\frac {3 x^{7} a^{2} f}{7 b^{4}}-\frac {2 x^{7} a e}{7 b^{3}}+\frac {d \,x^{7}}{7 b^{2}}-\frac {a^{3} f \,x^{4}}{b^{5}}+\frac {3 a^{2} e \,x^{4}}{4 b^{4}}-\frac {a d \,x^{4}}{2 b^{3}}+\frac {c \,x^{4}}{4 b^{2}}+\frac {5 a^{4} f x}{b^{6}}-\frac {4 a^{3} e x}{b^{5}}+\frac {3 a^{2} d x}{b^{4}}-\frac {2 a c x}{b^{3}}+\frac {\left (\frac {1}{3} f \,a^{5}-\frac {1}{3} a^{4} e b +\frac {1}{3} a^{3} d \,b^{2}-\frac {1}{3} a^{2} c \,b^{3}\right ) x}{b^{6} \left (b \,x^{3}+a \right )}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-16 f \,a^{3}+13 a^{2} b e -10 a \,b^{2} d +7 b^{3} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{7}}\) | \(244\) |
default | \(\frac {\frac {1}{13} f \,x^{13} b^{4}-\frac {1}{5} x^{10} a \,b^{3} f +\frac {1}{10} x^{10} b^{4} e +\frac {3}{7} x^{7} a^{2} b^{2} f -\frac {2}{7} x^{7} a \,b^{3} e +\frac {1}{7} b^{4} d \,x^{7}-a^{3} b f \,x^{4}+\frac {3}{4} a^{2} b^{2} e \,x^{4}-\frac {1}{2} a \,b^{3} d \,x^{4}+\frac {1}{4} b^{4} c \,x^{4}+5 a^{4} f x -4 a^{3} b e x +3 a^{2} b^{2} d x -2 a \,b^{3} c x}{b^{6}}-\frac {a^{2} \left (\frac {\left (-\frac {1}{3} f \,a^{3}+\frac {1}{3} a^{2} b e -\frac {1}{3} a \,b^{2} d +\frac {1}{3} b^{3} c \right ) x}{b \,x^{3}+a}+\frac {\left (16 f \,a^{3}-13 a^{2} b e +10 a \,b^{2} d -7 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}\right )}{b^{6}}\) | \(306\) |
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Time = 0.30 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.32 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {1260 \, b^{5} f x^{16} + 126 \, {\left (13 \, b^{5} e - 16 \, a b^{4} f\right )} x^{13} + 234 \, {\left (10 \, b^{5} d - 13 \, a b^{4} e + 16 \, a^{2} b^{3} f\right )} x^{10} + 585 \, {\left (7 \, b^{5} c - 10 \, a b^{4} d + 13 \, a^{2} b^{3} e - 16 \, a^{3} b^{2} f\right )} x^{7} - 4095 \, {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{4} - 1820 \, \sqrt {3} {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 910 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 1820 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f + {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 5460 \, {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} x}{16380 \, {\left (b^{7} x^{3} + a b^{6}\right )}} \]
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Time = 85.66 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.36 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{10} \left (- \frac {a f}{5 b^{3}} + \frac {e}{10 b^{2}}\right ) + x^{7} \cdot \left (\frac {3 a^{2} f}{7 b^{4}} - \frac {2 a e}{7 b^{3}} + \frac {d}{7 b^{2}}\right ) + x^{4} \left (- \frac {a^{3} f}{b^{5}} + \frac {3 a^{2} e}{4 b^{4}} - \frac {a d}{2 b^{3}} + \frac {c}{4 b^{2}}\right ) + x \left (\frac {5 a^{4} f}{b^{6}} - \frac {4 a^{3} e}{b^{5}} + \frac {3 a^{2} d}{b^{4}} - \frac {2 a c}{b^{3}}\right ) + \frac {x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{3 a b^{6} + 3 b^{7} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{19} + 4096 a^{13} f^{3} - 9984 a^{12} b e f^{2} + 7680 a^{11} b^{2} d f^{2} + 8112 a^{11} b^{2} e^{2} f - 5376 a^{10} b^{3} c f^{2} - 12480 a^{10} b^{3} d e f - 2197 a^{10} b^{3} e^{3} + 8736 a^{9} b^{4} c e f + 4800 a^{9} b^{4} d^{2} f + 5070 a^{9} b^{4} d e^{2} - 6720 a^{8} b^{5} c d f - 3549 a^{8} b^{5} c e^{2} - 3900 a^{8} b^{5} d^{2} e + 2352 a^{7} b^{6} c^{2} f + 5460 a^{7} b^{6} c d e + 1000 a^{7} b^{6} d^{3} - 1911 a^{6} b^{7} c^{2} e - 2100 a^{6} b^{7} c d^{2} + 1470 a^{5} b^{8} c^{2} d - 343 a^{4} b^{9} c^{3}, \left ( t \mapsto t \log {\left (- \frac {9 t b^{6}}{16 a^{4} f - 13 a^{3} b e + 10 a^{2} b^{2} d - 7 a b^{3} c} + x \right )} \right )\right )} + \frac {f x^{13}}{13 b^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3 \, {\left (b^{7} x^{3} + a b^{6}\right )}} + \frac {140 \, b^{4} f x^{13} + 182 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{10} + 260 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{7} + 455 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{4} - 1820 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x}{1820 \, b^{6}} + \frac {\sqrt {3} {\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.20 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} {\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 16 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7}} - \frac {{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{6}} + \frac {{\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d + 13 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e - 16 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7}} - \frac {a^{2} b^{3} c x - a^{3} b^{2} d x + a^{4} b e x - a^{5} f x}{3 \, {\left (b x^{3} + a\right )} b^{6}} + \frac {140 \, b^{24} f x^{13} + 182 \, b^{24} e x^{10} - 364 \, a b^{23} f x^{10} + 260 \, b^{24} d x^{7} - 520 \, a b^{23} e x^{7} + 780 \, a^{2} b^{22} f x^{7} + 455 \, b^{24} c x^{4} - 910 \, a b^{23} d x^{4} + 1365 \, a^{2} b^{22} e x^{4} - 1820 \, a^{3} b^{21} f x^{4} - 3640 \, a b^{23} c x + 5460 \, a^{2} b^{22} d x - 7280 \, a^{3} b^{21} e x + 9100 \, a^{4} b^{20} f x}{1820 \, b^{26}} \]
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Time = 0.35 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.30 \[ \int \frac {x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{10}\,\left (\frac {e}{10\,b^2}-\frac {a\,f}{5\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b^2}\right )-x^7\,\left (\frac {a^2\,f}{7\,b^4}-\frac {d}{7\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{7\,b}\right )+x^4\,\left (\frac {c}{4\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{4\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{2\,b}\right )+\frac {f\,x^{13}}{13\,b^2}+\frac {x\,\left (\frac {f\,a^5}{3}-\frac {e\,a^4\,b}{3}+\frac {d\,a^3\,b^2}{3}-\frac {c\,a^2\,b^3}{3}\right )}{b^7\,x^3+a\,b^6}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-16\,f\,a^3+13\,e\,a^2\,b-10\,d\,a\,b^2+7\,c\,b^3\right )}{9\,b^{19/3}} \]
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