Integrand size = 24, antiderivative size = 573 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^4} \]
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Time = 0.88 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\frac {3 b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}} \]
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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}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^5} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\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,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\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+e \sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\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+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+e \sqrt {x}\right )}{2 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\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+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+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+e \sqrt {x}\right )}{d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^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}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+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+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+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+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+e \sqrt {x}\right )}{2 d^2} \\ & = -\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\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+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+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+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+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+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+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+e \sqrt {x}\right )}{2 d^2} \\ & = -\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+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+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+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+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+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+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+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+e \sqrt {x}\right )}{d^4} \\ & = -\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\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+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {2 b d^3 e n \sqrt {x} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-3 b d^2 e^2 n x \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+6 b d e^3 n x^{3/2} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+6 b d^4 n \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-6 b e^4 n x^2 \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+2 d^4 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+3 b e^4 n x^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)-2 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (-3 \left (d^4-e^4 x^2\right ) \log ^2\left (d+e \sqrt {x}\right )+e^2 x \left (-d^2+5 d e \sqrt {x}+11 e^2 x \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-\log \left (d+e \sqrt {x}\right ) \left (2 d^3 e \sqrt {x}-3 d^2 e^2 x+6 d e^3 x^{3/2}+11 e^4 x^2+6 e^4 x^2 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-6 e^4 x^2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )+b^3 n^3 \left (d^2 e^2 x \left (2-3 \log \left (d+e \sqrt {x}\right )\right ) \log \left (d+e \sqrt {x}\right )+2 d^3 e \sqrt {x} \log ^2\left (d+e \sqrt {x}\right )+2 d^4 \log ^3\left (d+e \sqrt {x}\right )+2 d e^3 x^{3/2} \left (1-5 \log \left (d+e \sqrt {x}\right )+3 \log ^2\left (d+e \sqrt {x}\right )\right )+12 e^4 x^2 \left (-\log \left (d+e \sqrt {x}\right )+\log \left (-\frac {e \sqrt {x}}{d}\right )\right )+11 e^4 x^2 \left (\log \left (d+e \sqrt {x}\right ) \left (\log \left (d+e \sqrt {x}\right )-2 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )-2 e^4 x^2 \left (\log ^2\left (d+e \sqrt {x}\right ) \left (\log \left (d+e \sqrt {x}\right )-3 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-6 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )+6 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )\right )}{4 d^4 x^2} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^3} \,d x \]
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