Integrand size = 22, antiderivative size = 569 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {5 b^2 e^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}-\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {3 b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}+\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{2 d^4}-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4} \]
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Time = 0.89 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=-\frac {3 b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}-\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}-\frac {3 b^2 e^4 n^2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}-\frac {5 b^2 e^3 n^2 \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {3 b e^3 n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3+\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{2 d^4}-\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right )}{d^4}-\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}+\frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3} \]
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Rule 31
Rule 46
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^5} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {1}{2} (3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d}+\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d} \\ & = \frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3+\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2}-\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d} \\ & = -\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^3}+\frac {\left (3 b e^3 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}+\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2} \\ & = \frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {3 b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^3}+\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^3}-\frac {\left (3 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^3}-\frac {\left (3 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^3}-\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {\left (b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2} \\ & = -\frac {5 b^2 e^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}-\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {3 b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {\left (b^3 e^2 n^3\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2}+\frac {\left (b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {\left (3 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\frac {\left (b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4} \\ & = \frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {5 b^2 e^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}-\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {3 b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}+\frac {5 b^3 e^4 n^3 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{2 d^4}-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \text {Li}_3\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4} \\ \end{align*}
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx \]
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\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}d x\]
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\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \]
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\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}\, dx \]
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\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \]
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\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \]
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Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^3 \,d x \]
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