Integrand size = 21, antiderivative size = 239 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac {\left (2 b^{2/3} c+a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{4/3}}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}} \]
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Time = 0.14 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1842, 1868, 1874, 31, 648, 631, 210, 642} \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{2/3} e+2 b^{2/3} c\right )}{9 \sqrt {3} a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2} \]
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Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1842
Rule 1868
Rule 1874
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {\int \frac {-a e-4 b c x-3 b d x^2}{\left (a+b x^3\right )^2} \, dx}{6 a b} \\ & = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}+\frac {\int \frac {2 a e+4 b c x}{a+b x^3} \, dx}{18 a^2 b} \\ & = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}+\frac {\int \frac {\sqrt [3]{a} \left (4 \sqrt [3]{a} b c+4 a \sqrt [3]{b} e\right )+\sqrt [3]{b} \left (4 \sqrt [3]{a} b c-2 a \sqrt [3]{b} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{4/3}}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b} \\ & = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c-a^{2/3} e\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^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c+a^{2/3} e\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b} \\ & = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c+a^{2/3} e\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^{7/3} b^{4/3}} \\ & = -\frac {x \left (a e-b c x-b d x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a d-x (a e+4 b c x)}{18 a^2 b \left (a+b x^3\right )}-\frac {\left (2 b^{2/3} c+a^{2/3} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{4/3}}-\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{4/3}}+\frac {\left (2 b^{2/3} c-a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{4/3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {3 a b^{2/3} \left (4 b^2 c x^5-a^2 (3 d+2 e x)+a b x^2 \left (7 c+e x^2\right )\right )}{\left (a+b x^3\right )^2}-2 \sqrt {3} a^{2/3} \sqrt [3]{b} \left (2 b^{2/3} c+a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (-2 a^{2/3} b c+a^{4/3} \sqrt [3]{b} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\left (2 a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^3 b^{5/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.52 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.40
method | result | size |
risch | \(\frac {\frac {2 b c \,x^{5}}{9 a^{2}}+\frac {e \,x^{4}}{18 a}+\frac {7 c \,x^{2}}{18 a}-\frac {e x}{9 b}-\frac {d}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\frac {2 c \textit {\_R}}{a}+\frac {e}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 b a}\) | \(96\) |
default | \(\frac {\frac {2 b c \,x^{5}}{9 a^{2}}+\frac {e \,x^{4}}{18 a}+\frac {7 c \,x^{2}}{18 a}-\frac {e x}{9 b}-\frac {d}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+2 b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2} b}\) | \(250\) |
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Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 2519, normalized size of antiderivative = 10.54 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Time = 1.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{4} + 162 t a^{3} b^{2} c e - a^{2} e^{3} + 8 b^{2} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {1458 t^{2} a^{5} b^{3} c + 27 t a^{4} b e^{2} + 8 a b c^{2} e}{a^{2} e^{3} + 8 b^{2} c^{3}} \right )} \right )\right )} + \frac {- 3 a^{2} d - 2 a^{2} e x + 7 a b c x^{2} + a b e x^{4} + 4 b^{2} c x^{5}}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} \]
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Time = 0.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {4 \, b^{2} c x^{5} + a b e x^{4} + 7 \, a b c x^{2} - 2 \, a^{2} e x - 3 \, a^{2} d}{18 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {\sqrt {3} {\left (2 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (a e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} - \frac {{\left (a e + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2}} - \frac {{\left (2 \, b c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac {4 \, b^{2} c x^{5} + a b e x^{4} + 7 \, a b c x^{2} - 2 \, a^{2} e x - 3 \, a^{2} d}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b} \]
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Time = 9.03 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97 \[ \int \frac {x \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {7\,c\,x^2}{18\,a}-\frac {d}{6\,b}+\frac {e\,x^4}{18\,a}-\frac {e\,x}{9\,b}+\frac {2\,b\,c\,x^5}{9\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\frac {2\,a\,c\,e+{\mathrm {root}\left (19683\,a^7\,b^4\,z^3+162\,a^3\,b^2\,c\,e\,z+8\,b^2\,c^3-a^2\,e^3,z,k\right )}^2\,a^5\,b^2\,729+4\,b\,c^2\,x+\mathrm {root}\left (19683\,a^7\,b^4\,z^3+162\,a^3\,b^2\,c\,e\,z+8\,b^2\,c^3-a^2\,e^3,z,k\right )\,a^3\,b\,e\,x\,27}{a^4\,81}\right )\,\mathrm {root}\left (19683\,a^7\,b^4\,z^3+162\,a^3\,b^2\,c\,e\,z+8\,b^2\,c^3-a^2\,e^3,z,k\right )\right ) \]
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