Integrand size = 30, antiderivative size = 244 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{11/3} \sqrt [3]{b}} \]
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Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1848, 206, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\frac {b c-a d}{5 a^2 x^5}-\frac {a^2 e-a b d+b^2 c}{2 a^3 x^2}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac {c}{8 a x^8} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1848
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{a x^9}+\frac {-b c+a d}{a^2 x^6}+\frac {b^2 c-a b d+a^2 e}{a^3 x^3}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 \left (a+b x^3\right )}\right ) \, dx \\ & = -\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{a^3} \\ & = -\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{11/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-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}{3 a^{11/3}} \\ & = -\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{11/3} \sqrt [3]{b}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{10/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-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}{6 a^{11/3} \sqrt [3]{b}} \\ & = -\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-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 )}{a^{11/3} \sqrt [3]{b}} \\ & = -\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{11/3} \sqrt [3]{b}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\frac {-\frac {15 a^{8/3} c}{x^8}+\frac {24 a^{5/3} (b c-a d)}{x^5}-\frac {60 a^{2/3} \left (b^2 c-a b d+a^2 e\right )}{x^2}+\frac {40 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {40 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {20 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{120 a^{11/3}} \]
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Time = 1.55 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {c}{8 a \,x^{8}}-\frac {a d -b c}{5 a^{2} x^{5}}-\frac {a^{2} e -a b d +b^{2} c}{2 a^{3} x^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{3}}\) | \(170\) |
risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{2 a^{3}}-\frac {\left (a d -b c \right ) x^{3}}{5 a^{2}}-\frac {c}{8 a}}{x^{8}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b \,\textit {\_Z}^{3}-a^{9} f^{3}+3 a^{8} b e \,f^{2}-3 a^{7} b^{2} d \,f^{2}-3 a^{7} b^{2} e^{2} f +3 a^{6} b^{3} c \,f^{2}+6 a^{6} b^{3} d e f +a^{6} b^{3} e^{3}-6 a^{5} b^{4} c e f -3 a^{5} b^{4} d^{2} f -3 a^{5} b^{4} d \,e^{2}+6 a^{4} b^{5} c d f +3 a^{4} b^{5} c \,e^{2}+3 a^{4} b^{5} d^{2} e -3 a^{3} b^{6} c^{2} f -6 a^{3} b^{6} c d e -a^{3} b^{6} d^{3}+3 a^{2} b^{7} c^{2} e +3 a^{2} b^{7} c \,d^{2}-3 a \,b^{8} c^{2} d +c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b +3 a^{9} f^{3}-9 a^{8} b e \,f^{2}+9 a^{7} b^{2} d \,f^{2}+9 a^{7} b^{2} e^{2} f -9 a^{6} b^{3} c \,f^{2}-18 a^{6} b^{3} d e f -3 a^{6} b^{3} e^{3}+18 a^{5} b^{4} c e f +9 a^{5} b^{4} d^{2} f +9 a^{5} b^{4} d \,e^{2}-18 a^{4} b^{5} c d f -9 a^{4} b^{5} c \,e^{2}-9 a^{4} b^{5} d^{2} e +9 a^{3} b^{6} c^{2} f +18 a^{3} b^{6} c d e +3 a^{3} b^{6} d^{3}-9 a^{2} b^{7} c^{2} e -9 a^{2} b^{7} c \,d^{2}+9 a \,b^{8} c^{2} d -3 c^{3} b^{9}\right ) x +\left (-a^{10} f^{2}+2 a^{9} b e f -2 a^{8} b^{2} d f -a^{8} b^{2} e^{2}+2 a^{7} b^{3} c f +2 a^{7} b^{3} d e -2 a^{6} b^{4} c e -a^{6} b^{4} d^{2}+2 a^{5} b^{5} c d -a^{4} b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{3}\) | \(628\) |
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Time = 0.31 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.44 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\left [-\frac {60 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 20 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 60 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}, -\frac {120 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 20 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 60 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}\right ] \]
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Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {20 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 8 \, {\left (a b c - a^{2} d\right )} x^{3} + 5 \, a^{2} c}{40 \, a^{3} x^{8}} \]
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Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{4}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4} b} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{4} b} - \frac {20 \, b^{2} c x^{6} - 20 \, a b d x^{6} + 20 \, a^{2} e x^{6} - 8 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{3} x^{8}} \]
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Time = 9.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {\frac {c}{8\,a}+\frac {x^3\,\left (a\,d-b\,c\right )}{5\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{2\,a^3}}{x^8}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}} \]
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