Integrand size = 30, antiderivative size = 227 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=-\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 b}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{8/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{8/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} b^{8/3}} \]
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Time = 0.13 (sec) , antiderivative size = 227, 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, 298, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{4/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{4/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{4/3} b^{8/3}}+\frac {x^2 (b e-a f)}{2 b^2}-\frac {c}{a x}+\frac {f x^5}{5 b} \]
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Rule 31
Rule 210
Rule 298
Rule 631
Rule 642
Rule 648
Rule 1848
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{a x^2}+\frac {(b e-a f) x}{b^2}+\frac {f x^4}{b}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a b^2 \left (a+b x^3\right )}\right ) \, dx \\ & = -\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 b}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{a b^2} \\ & = -\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 b}+\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^{4/3} b^{7/3}}-\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} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3} b^{7/3}} \\ & = -\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 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^{4/3} b^{8/3}}-\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^{4/3} b^{8/3}}-\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 b^{7/3}} \\ & = -\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 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^{4/3} b^{8/3}}-\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^{4/3} b^{8/3}}-\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^{4/3} b^{8/3}} \\ & = -\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 b}+\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^{4/3} b^{8/3}}+\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^{4/3} b^{8/3}}-\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^{4/3} b^{8/3}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\frac {-30 \sqrt [3]{a} b^{8/3} c+15 a^{4/3} b^{2/3} (b e-a f) x^3+6 a^{4/3} b^{5/3} f x^6+10 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{30 a^{4/3} b^{8/3} x} \]
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Time = 1.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {-\frac {b f \,x^{5}}{5}+\frac {\left (a f -b e \right ) x^{2}}{2}}{b^{2}}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a \,b^{2}}-\frac {c}{a x}\) | \(159\) |
risch | \(\frac {f \,x^{5}}{5 b}-\frac {x^{2} a f}{2 b^{2}}+\frac {e \,x^{2}}{2 b}-\frac {c}{a x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} f^{3}-3 a^{8} b e \,f^{2}+3 a^{7} b^{2} d \,f^{2}+3 a^{7} b^{2} e^{2} f -3 a^{6} b^{3} c \,f^{2}-6 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+6 a^{5} b^{4} c e f +3 a^{5} b^{4} d^{2} f +3 a^{5} b^{4} d \,e^{2}-6 a^{4} b^{5} c d f -3 a^{4} b^{5} c \,e^{2}-3 a^{4} b^{5} d^{2} e +3 a^{3} b^{6} c^{2} f +6 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}-3 a^{2} b^{7} c^{2} e -3 a^{2} b^{7} c \,d^{2}+3 a \,b^{8} c^{2} d -c^{3} b^{9}+\textit {\_Z}^{3} b^{2} a^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (-3 a^{9} f^{3}+9 a^{8} b e \,f^{2}-9 a^{7} b^{2} d \,f^{2}-9 a^{7} b^{2} e^{2} f +9 a^{6} b^{3} c \,f^{2}+18 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-18 a^{5} b^{4} c e f -9 a^{5} b^{4} d^{2} f -9 a^{5} b^{4} d \,e^{2}+18 a^{4} b^{5} c d f +9 a^{4} b^{5} c \,e^{2}+9 a^{4} b^{5} d^{2} e -9 a^{3} b^{6} c^{2} f -18 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}+9 a^{2} b^{7} c^{2} e +9 a^{2} b^{7} c \,d^{2}-9 a \,b^{8} c^{2} d +3 c^{3} b^{9}-4 \textit {\_R}^{3} a^{4} b^{2}\right ) x +\left (a^{6} b f -a^{5} b^{2} e +a^{4} b^{3} d -a^{3} b^{4} c \right ) \textit {\_R}^{2}\right )}{3 b^{2}}\) | \(556\) |
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Time = 0.31 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.47 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\left [\frac {6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 15 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - 5 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{30 \, a^{2} b^{4} x}, \frac {6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 30 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - 5 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{30 \, a^{2} b^{4} x}\right ] \]
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Time = 1.22 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.80 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=x^{2} \left (- \frac {a f}{2 b^{2}} + \frac {e}{2 b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{4} b^{8} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{3} b^{5}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{5}}{5 b} - \frac {c}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\frac {2 \, b f x^{5} + 5 \, {\left (b e - a f\right )} x^{2}}{10 \, b^{2}} - \frac {c}{a x} - \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} - \frac {c}{a x} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} + \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2} b^{2}} + \frac {2 \, b^{4} f x^{5} + 5 \, b^{4} e x^{2} - 5 \, a b^{3} f x^{2}}{10 \, b^{5}} \]
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Time = 9.50 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=x^2\,\left (\frac {e}{2\,b}-\frac {a\,f}{2\,b^2}\right )-\frac {c}{a\,x}+\frac {f\,x^5}{5\,b}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}} \]
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