Integrand size = 25, antiderivative size = 143 \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \]
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Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {129, 499, 455, 65, 212, 2163, 2170, 211} \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \]
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Rule 65
Rule 129
Rule 211
Rule 212
Rule 455
Rule 499
Rule 2163
Rule 2170
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {2 \sqrt [3]{c} d^{2/3}-2 d x-\frac {d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c d}+\frac {\text {Subst}\left (\int \frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c^{2/3} \sqrt [3]{d}}-\frac {\left (3 \sqrt [3]{d}\right ) \text {Subst}\left (\int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 \sqrt [3]{c}} \\ & = \frac {\text {Subst}\left (\int \frac {1}{9-c x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{\sqrt {c+d x}}\right )}{2 \sqrt [3]{c} d^{2/3}}-\frac {\sqrt [3]{d} \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x\right )}{4 \sqrt [3]{c}}+\frac {d^{4/3} \text {Subst}\left (\int \frac {1}{-\frac {2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt {c+d x}}\right )}{c^{4/3}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt [3]{c} d^{2/3}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \left (1+\frac {\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\frac {3 x^{2/3} \sqrt {\frac {c+d x}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x}{c},\frac {d x}{8 c}\right )}{16 c \sqrt {c+d x}} \]
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\[\int \frac {1}{x^{\frac {1}{3}} \left (-d x +8 c \right ) \sqrt {d x +c}}d x\]
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Timed out. \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=- \int \frac {1}{- 8 c \sqrt [3]{x} \sqrt {c + d x} + d x^{\frac {4}{3}} \sqrt {c + d x}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int { -\frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int { -\frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int \frac {1}{x^{1/3}\,\left (8\,c-d\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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