Integrand size = 23, antiderivative size = 103 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{d} x\right )}{\sqrt {1+d x^3}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (1+\sqrt [3]{d} x\right )^2}{3 \sqrt {1+d x^3}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {1+d x^3}\right )}{18 d^{2/3}} \]
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Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {499, 455, 65, 212, 2163, 2170, 211} \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \left (\sqrt [3]{d} x+1\right )}{\sqrt {d x^3+1}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt {d x^3+1}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}} \]
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Rule 65
Rule 211
Rule 212
Rule 455
Rule 499
Rule 2163
Rule 2170
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {2 d^{2/3}-2 d x-d^{4/3} x^2}{\left (4+2 \sqrt [3]{d} x+d^{2/3} x^2\right ) \sqrt {1+d x^3}} \, dx}{12 d}+\frac {\int \frac {1+\sqrt [3]{d} x}{\left (2-\sqrt [3]{d} x\right ) \sqrt {1+d x^3}} \, dx}{12 \sqrt [3]{d}}-\frac {1}{4} \sqrt [3]{d} \int \frac {x^2}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\frac {\left (1+\sqrt [3]{d} x\right )^2}{\sqrt {1+d x^3}}\right )}{6 d^{2/3}}-\frac {1}{12} \sqrt [3]{d} \text {Subst}\left (\int \frac {1}{(8-d x) \sqrt {1+d x}} \, dx,x,x^3\right )+\frac {1}{3} d^{4/3} \text {Subst}\left (\int \frac {1}{-2 d^2-6 d^2 x^2} \, dx,x,\frac {1+\sqrt [3]{d} x}{\sqrt {1+d x^3}}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{d} x\right )}{\sqrt {1+d x^3}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (1+\sqrt [3]{d} x\right )^2}{3 \sqrt {1+d x^3}}\right )}{18 d^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\sqrt {1+d x^3}\right )}{6 d^{2/3}} \\ & = -\frac {\arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{d} x\right )}{\sqrt {1+d x^3}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (1+\sqrt [3]{d} x\right )^2}{3 \sqrt {1+d x^3}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {1+d x^3}\right )}{18 d^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\frac {1}{16} x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-d x^3,\frac {d x^3}{8}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.09 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.72
method | result | size |
default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8\right )}{\sum }\frac {\left (-d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, d -3 d}{18 d}, \sqrt {\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+1}}\right )}{27 d^{3}}\) | \(383\) |
elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8\right )}{\sum }\frac {\left (-d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, d -3 d}{18 d}, \sqrt {\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+1}}\right )}{27 d^{3}}\) | \(383\) |
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Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (73) = 146\).
Time = 0.42 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.83 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (-\frac {{\left (9 \, \sqrt {3} d^{3} x^{5} - \sqrt {3} {\left (d^{2} x^{6} - 40 \, d x^{3} - 32\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 3 \, \sqrt {3} {\left (5 \, d^{2} x^{4} + 8 \, d x\right )} {\left (d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {d x^{3} + 1} {\left (d^{2}\right )}^{\frac {1}{6}}}{9 \, {\left (d^{4} x^{7} - 7 \, d^{3} x^{4} - 8 \, d^{2} x\right )}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {d^{4} x^{9} + 318 \, d^{3} x^{6} + 1200 \, d^{2} x^{3} + 18 \, {\left (5 \, d^{2} x^{7} + 64 \, d x^{4} + 32 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 6 \, {\left (7 \, d^{3} x^{6} + 152 \, d^{2} x^{3} + {\left (d^{2} x^{7} + 80 \, d x^{4} + 160 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 6 \, {\left (5 \, d^{2} x^{5} + 32 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 64 \, d\right )} \sqrt {d x^{3} + 1} + 18 \, {\left (d^{3} x^{8} + 38 \, d^{2} x^{5} + 64 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 640 \, d}{d^{3} x^{9} - 24 \, d^{2} x^{6} + 192 \, d x^{3} - 512}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {d^{4} x^{9} - 276 \, d^{3} x^{6} - 1608 \, d^{2} x^{3} - 18 \, {\left (d^{2} x^{7} - 52 \, d x^{4} - 80 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - 6 \, {\left (4 \, d^{3} x^{6} + 164 \, d^{2} x^{3} + {\left (d^{2} x^{7} - 28 \, d x^{4} - 272 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - 24 \, {\left (d^{2} x^{5} + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 160 \, d\right )} \sqrt {d x^{3} + 1} + 18 \, {\left (d^{3} x^{8} + 20 \, d^{2} x^{5} - 8 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 1088 \, d}{d^{3} x^{9} - 24 \, d^{2} x^{6} + 192 \, d x^{3} - 512}\right )}{108 \, d^{2}} \]
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\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=- \int \frac {x}{d x^{3} \sqrt {d x^{3} + 1} - 8 \sqrt {d x^{3} + 1}}\, dx \]
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\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + 1} {\left (d x^{3} - 8\right )}} \,d x } \]
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\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + 1} {\left (d x^{3} - 8\right )}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=-\int \frac {x}{\sqrt {d\,x^3+1}\,\left (d\,x^3-8\right )} \,d x \]
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