Integrand size = 30, antiderivative size = 301 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{7/3}}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}} \]
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Time = 0.23 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1843, 1498, 464, 206, 31, 648, 631, 210, 642} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {c}{2 a^3 x^2}-\frac {x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{18 a^3 b^2 \left (a+b x^3\right )}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{9 \sqrt {3} a^{11/3} b^{7/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{54 a^{11/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{27 a^{11/3} b^{7/3}} \]
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Rule 31
Rule 206
Rule 210
Rule 464
Rule 631
Rule 642
Rule 648
Rule 1498
Rule 1843
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+b \left (\frac {5 b^3 c}{a}-5 b^2 d-a b e+a^2 f\right ) x^3-6 a b^2 f x^6}{x^3 \left (a+b x^3\right )^2} \, dx}{6 a b^3} \\ & = -\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac {\int \frac {18 a b^5 c-2 b^3 \left (11 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) x^3}{x^3 \left (a+b x^3\right )} \, dx}{18 a^3 b^5} \\ & = -\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{9 a^3 b^2} \\ & = -\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3} b^2}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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}{27 a^{11/3} b^2} \\ & = -\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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}{54 a^{11/3} b^{7/3}}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3} b^2} \\ & = -\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 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 )}{9 a^{11/3} b^{7/3}} \\ & = -\frac {c}{2 a^3 x^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac {\left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{18 a^3 b^2 \left (a+b x^3\right )}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} b^{7/3}}-\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} b^{7/3}}+\frac {\left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} b^{7/3}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {27 a^{2/3} c}{x^2}+\frac {9 a^{5/3} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{b^2 \left (a+b x^3\right )^2}-\frac {3 a^{2/3} \left (11 b^3 c-5 a b^2 d-a^2 b e+7 a^3 f\right ) x}{b^2 \left (a+b x^3\right )}+\frac {2 \sqrt {3} \left (20 b^3 c-5 a b^2 d-a^2 b e-2 a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{7/3}}+\frac {2 \left (-20 b^3 c+5 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{7/3}}-\frac {\left (-20 b^3 c+5 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{7/3}}}{54 a^{11/3}} \]
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Time = 1.58 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {c}{2 a^{3} x^{2}}+\frac {\frac {-\frac {\left (7 f \,a^{3}-a^{2} b e -5 a \,b^{2} d +11 b^{3} c \right ) x^{4}}{18 b}-\frac {a \left (2 f \,a^{3}+a^{2} b e -4 a \,b^{2} d +7 b^{3} c \right ) x}{9 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 f \,a^{3}+a^{2} b e +5 a \,b^{2} d -20 b^{3} c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9 b^{2}}}{a^{3}}\) | \(216\) |
risch | \(\frac {-\frac {\left (7 f \,a^{3}-a^{2} b e -5 a \,b^{2} d +20 b^{3} c \right ) x^{6}}{18 a^{3} b}-\frac {\left (2 f \,a^{3}+a^{2} b e -4 a \,b^{2} d +16 b^{3} c \right ) x^{3}}{9 a^{2} b^{2}}-\frac {c}{2 a}}{x^{2} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b^{7} \textit {\_Z}^{3}-8 a^{9} f^{3}-12 a^{8} b e \,f^{2}-60 a^{7} b^{2} d \,f^{2}-6 a^{7} b^{2} e^{2} f +240 a^{6} b^{3} c \,f^{2}-60 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+240 a^{5} b^{4} c e f -150 a^{5} b^{4} d^{2} f -15 a^{5} b^{4} d \,e^{2}+1200 a^{4} b^{5} c d f +60 a^{4} b^{5} c \,e^{2}-75 a^{4} b^{5} d^{2} e -2400 a^{3} b^{6} c^{2} f +600 a^{3} b^{6} c d e -125 a^{3} b^{6} d^{3}-1200 a^{2} b^{7} c^{2} e +1500 a^{2} b^{7} c \,d^{2}-6000 a \,b^{8} c^{2} d +8000 c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b^{7}+24 a^{9} f^{3}+36 a^{8} b e \,f^{2}+180 a^{7} b^{2} d \,f^{2}+18 a^{7} b^{2} e^{2} f -720 a^{6} b^{3} c \,f^{2}+180 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-720 a^{5} b^{4} c e f +450 a^{5} b^{4} d^{2} f +45 a^{5} b^{4} d \,e^{2}-3600 a^{4} b^{5} c d f -180 a^{4} b^{5} c \,e^{2}+225 a^{4} b^{5} d^{2} e +7200 a^{3} b^{6} c^{2} f -1800 a^{3} b^{6} c d e +375 a^{3} b^{6} d^{3}+3600 a^{2} b^{7} c^{2} e -4500 a^{2} b^{7} c \,d^{2}+18000 a \,b^{8} c^{2} d -24000 c^{3} b^{9}\right ) x +\left (-4 a^{10} f^{2} b^{2}-4 e f \,b^{3} a^{9}-20 d f \,b^{4} a^{8}-e^{2} b^{4} a^{8}+80 b^{5} c f \,a^{7}-10 b^{5} d e \,a^{7}+40 b^{6} c e \,a^{6}-25 b^{6} d^{2} a^{6}+200 b^{7} c d \,a^{5}-400 b^{8} c^{2} a^{4}\right ) \textit {\_R} \right )\right )}{27}\) | \(683\) |
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Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (258) = 516\).
Time = 0.29 (sec) , antiderivative size = 1217, normalized size of antiderivative = 4.04 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {{\left (20 \, b^{4} c - 5 \, a b^{3} d - a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{6} + 9 \, a^{2} b^{2} c + 2 \, {\left (16 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{3}}{18 \, {\left (a^{3} b^{4} x^{8} + 2 \, a^{4} b^{3} x^{5} + a^{5} b^{2} x^{2}\right )}} - \frac {\sqrt {3} {\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} b} + \frac {{\left (20 \, b^{3} c - 5 \, a b^{2} d - a^{2} b e - 2 \, a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} b^{2}} - \frac {20 \, b^{4} c x^{6} - 5 \, a b^{3} d x^{6} - a^{2} b^{2} e x^{6} + 7 \, a^{3} b f x^{6} + 32 \, a b^{3} c x^{3} - 8 \, a^{2} b^{2} d x^{3} + 2 \, a^{3} b e x^{3} + 4 \, a^{4} f x^{3} + 9 \, a^{2} b^{2} c}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3} b^{2}} \]
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Time = 10.05 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}}-\frac {\frac {c}{2\,a}+\frac {x^3\,\left (2\,f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^2\,b^2}+\frac {x^6\,\left (7\,f\,a^3-e\,a^2\,b-5\,d\,a\,b^2+20\,c\,b^3\right )}{18\,a^3\,b}}{a^2\,x^2+2\,a\,b\,x^5+b^2\,x^8}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b+5\,d\,a\,b^2-20\,c\,b^3\right )}{27\,a^{11/3}\,b^{7/3}} \]
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