Integrand size = 30, antiderivative size = 242 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} b^{2/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} b^{2/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} b^{2/3}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 242, 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} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\frac {b c-a d}{4 a^2 x^4}-\frac {a^2 e-a b d+b^2 c}{a^3 x}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{10/3} b^{2/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{10/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{10/3} b^{2/3}}-\frac {c}{7 a x^7} \]
[In]
[Out]
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^8}+\frac {-b c+a d}{a^2 x^5}+\frac {b^2 c-a b d+a^2 e}{a^3 x^2}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^3 \left (a+b x^3\right )}\right ) \, dx \\ & = -\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\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^3} \\ & = -\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} \sqrt [3]{b}}-\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^{10/3} \sqrt [3]{b}} \\ & = -\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}-\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^3 \sqrt [3]{b}} \\ & = -\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}} \\ & = -\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\frac {-\frac {12 a^{7/3} c}{x^7}+\frac {21 a^{4/3} (b c-a d)}{x^4}-\frac {84 \sqrt [3]{a} \left (b^2 c-a b d+a^2 e\right )}{x}+\frac {28 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {28 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {14 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{84 a^{10/3}} \]
[In]
[Out]
Time = 1.53 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {c}{7 a \,x^{7}}-\frac {a d -b c}{4 a^{2} x^{4}}-\frac {a^{2} e -a b d +b^{2} c}{a^{3} x}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{3}}\) | \(170\) |
risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{a^{3}}-\frac {\left (a d -b c \right ) x^{3}}{4 a^{2}}-\frac {c}{7 a}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} b^{2} \textit {\_Z}^{3}+a^{9} f^{3}-3 a^{8} b e \,f^{2}+3 a^{7} b^{2} d \,f^{2}+3 a^{7} b^{2} e^{2} f -3 a^{6} b^{3} c \,f^{2}-6 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+6 a^{5} b^{4} c e f +3 a^{5} b^{4} d^{2} f +3 a^{5} b^{4} d \,e^{2}-6 a^{4} b^{5} c d f -3 a^{4} b^{5} c \,e^{2}-3 a^{4} b^{5} d^{2} e +3 a^{3} b^{6} c^{2} f +6 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}-3 a^{2} b^{7} c^{2} e -3 a^{2} b^{7} c \,d^{2}+3 a \,b^{8} c^{2} d -c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b^{2}-3 a^{9} f^{3}+9 a^{8} b e \,f^{2}-9 a^{7} b^{2} d \,f^{2}-9 a^{7} b^{2} e^{2} f +9 a^{6} b^{3} c \,f^{2}+18 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-18 a^{5} b^{4} c e f -9 a^{5} b^{4} d^{2} f -9 a^{5} b^{4} d \,e^{2}+18 a^{4} b^{5} c d f +9 a^{4} b^{5} c \,e^{2}+9 a^{4} b^{5} d^{2} e -9 a^{3} b^{6} c^{2} f -18 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}+9 a^{2} b^{7} c^{2} e +9 a^{2} b^{7} c \,d^{2}-9 a \,b^{8} c^{2} d +3 c^{3} b^{9}\right ) x +\left (a^{10} b f -a^{9} b^{2} e +a^{8} b^{3} d -a^{7} b^{4} c \right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(567\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.52 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\left [-\frac {42 \, \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^{7} \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 ) + 14 \, {\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^{7} \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 ) - 28 \, {\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}, -\frac {84 \, \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^{7} \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 ) + 14 \, {\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^{7} \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 ) - 28 \, {\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}\right ] \]
[In]
[Out]
Time = 43.35 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.79 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{10} b^{2} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{7} b}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {- 4 a^{2} c + x^{6} \left (- 28 a^{2} e + 28 a b d - 28 b^{2} c\right ) + x^{3} \left (- 7 a^{2} d + 7 a b c\right )}{28 a^{3} x^{7}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {28 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 7 \, {\left (a b c - a^{2} d\right )} x^{3} + 4 \, a^{2} c}{28 \, a^{3} x^{7}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} + \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{4}} - \frac {28 \, b^{2} c x^{6} - 28 \, a b d x^{6} + 28 \, a^{2} e x^{6} - 7 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{3} x^{7}} \]
[In]
[Out]
Time = 9.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}}-\frac {\frac {c}{7\,a}+\frac {x^3\,\left (a\,d-b\,c\right )}{4\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{a^3}}{x^7}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}} \]
[In]
[Out]