Integrand size = 30, antiderivative size = 280 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}-\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{14/3}}+\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{14/3}}-\frac {b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{14/3}} \]
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Time = 0.13 (sec) , antiderivative size = 280, 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} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\frac {b c-a d}{8 a^2 x^8}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{14/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{14/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{14/3}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^4 x^2}-\frac {c}{11 a x^{11}} \]
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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^{12}}+\frac {-b c+a d}{a^2 x^9}+\frac {b^2 c-a b d+a^2 e}{a^3 x^6}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^3}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx \\ & = -\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{a^4} \\ & = -\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{14/3}}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-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}{3 a^{14/3}} \\ & = -\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {b^{2/3} \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^{14/3}}-\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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}{6 a^{14/3}}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{13/3}} \\ & = -\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}+\frac {b^{2/3} \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^{14/3}}-\frac {b^{2/3} \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^{14/3}}+\frac {\left (b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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 )}{a^{14/3}} \\ & = -\frac {c}{11 a x^{11}}+\frac {b c-a d}{8 a^2 x^8}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{2 a^4 x^2}-\frac {b^{2/3} \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^{14/3}}+\frac {b^{2/3} \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^{14/3}}-\frac {b^{2/3} \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^{14/3}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\frac {-\frac {120 a^{11/3} c}{x^{11}}+\frac {165 a^{8/3} (b c-a d)}{x^8}-\frac {264 a^{5/3} \left (b^2 c-a b d+a^2 e\right )}{x^5}+\frac {660 a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{x^2}-440 \sqrt {3} b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+440 b^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+220 b^{2/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1320 a^{14/3}} \]
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Time = 1.53 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {c}{11 a \,x^{11}}-\frac {a d -b c}{8 a^{2} x^{8}}-\frac {a^{2} e -a b d +b^{2} c}{5 a^{3} x^{5}}-\frac {f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c}{2 a^{4} x^{2}}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) b}{a^{4}}\) | \(205\) |
risch | \(\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{9}}{2 a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{5 a^{3}}-\frac {\left (a d -b c \right ) x^{3}}{8 a^{2}}-\frac {c}{11 a}}{x^{11}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} \textit {\_Z}^{3}+a^{9} b^{2} f^{3}-3 a^{8} b^{3} e \,f^{2}+3 a^{7} b^{4} d \,f^{2}+3 a^{7} b^{4} e^{2} f -3 a^{6} b^{5} c \,f^{2}-6 a^{6} b^{5} d e f -a^{6} b^{5} e^{3}+6 a^{5} b^{6} c e f +3 a^{5} b^{6} d^{2} f +3 a^{5} b^{6} d \,e^{2}-6 a^{4} b^{7} c d f -3 a^{4} b^{7} c \,e^{2}-3 a^{4} b^{7} d^{2} e +3 a^{3} b^{8} c^{2} f +6 a^{3} b^{8} c d e +a^{3} b^{8} d^{3}-3 a^{2} b^{9} c^{2} e -3 a^{2} b^{9} c \,d^{2}+3 a \,b^{10} c^{2} d -b^{11} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{14}-3 a^{9} b^{2} f^{3}+9 a^{8} b^{3} e \,f^{2}-9 a^{7} b^{4} d \,f^{2}-9 a^{7} b^{4} e^{2} f +9 a^{6} b^{5} c \,f^{2}+18 a^{6} b^{5} d e f +3 a^{6} b^{5} e^{3}-18 a^{5} b^{6} c e f -9 a^{5} b^{6} d^{2} f -9 a^{5} b^{6} d \,e^{2}+18 a^{4} b^{7} c d f +9 a^{4} b^{7} c \,e^{2}+9 a^{4} b^{7} d^{2} e -9 a^{3} b^{8} c^{2} f -18 a^{3} b^{8} c d e -3 a^{3} b^{8} d^{3}+9 a^{2} b^{9} c^{2} e +9 a^{2} b^{9} c \,d^{2}-9 a \,b^{10} c^{2} d +3 b^{11} c^{3}\right ) x +\left (-a^{11} b \,f^{2}+2 a^{10} b^{2} e f -2 a^{9} b^{3} d f -a^{9} b^{3} e^{2}+2 a^{8} b^{4} c f +2 a^{8} b^{4} d e -2 a^{7} b^{5} c e -a^{7} b^{5} d^{2}+2 a^{6} b^{6} c d -a^{5} b^{7} c^{2}\right ) \textit {\_R} \right )\right )}{3}\) | \(672\) |
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Time = 0.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=-\frac {440 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 440 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 660 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} + 264 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} + 120 \, a^{3} c - 165 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{1320 \, a^{4} x^{11}} \]
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Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 88 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 40 \, a^{3} c + 55 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{440 \, a^{4} x^{11}} \]
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Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{5}} - \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{5}} + \frac {220 \, b^{3} c x^{9} - 220 \, a b^{2} d x^{9} + 220 \, a^{2} b e x^{9} - 220 \, a^{3} f x^{9} - 88 \, a b^{2} c x^{6} + 88 \, a^{2} b d x^{6} - 88 \, a^{3} e x^{6} + 55 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{4} x^{11}} \]
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Time = 9.20 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx=\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{2\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{8\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{5\,a^3}}{x^{11}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{14/3}} \]
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