Integrand size = 24, antiderivative size = 408 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}+\frac {77 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}-\frac {2 b e^6 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {2 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{3 d^6} \]
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Time = 0.60 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {2 b e^6 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {2 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{3 d^6}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {77 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {b^2 e^2 n^2}{30 d^2 x^2} \]
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^7} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^6 (d+e x)} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {1}{3} (2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt {x}\right )}{3 d}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt {x}\right )}{3 d} \\ & = -\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt {x}\right )}{3 d^2}+\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt {x}\right )}{3 d^2}+\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt {x}\right )}{15 d} \\ & = -\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt {x}\right )}{3 d^3}-\frac {\left (2 b e^3 n\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 )}{3 d^3}+\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \left (-\frac {e^5}{d (d-x)^5}-\frac {e^5}{d^2 (d-x)^4}-\frac {e^5}{d^3 (d-x)^3}-\frac {e^5}{d^4 (d-x)^2}-\frac {e^5}{d^5 (d-x)}-\frac {e^5}{d^5 x}\right ) \, dx,x,d+e \sqrt {x}\right )}{15 d}-\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt {x}\right )}{6 d^2} \\ & = -\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {2 b^2 e^3 n^2}{45 d^3 x^{3/2}}-\frac {b^2 e^4 n^2}{15 d^4 x}+\frac {2 b^2 e^5 n^2}{15 d^5 \sqrt {x}}-\frac {2 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{15 d^6}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {b^2 e^6 n^2 \log (x)}{15 d^6}-\frac {\left (2 b e^3 n\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 )}{3 d^4}+\frac {\left (2 b e^4 n\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 )}{3 d^4}-\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {e^4}{d (d-x)^4}+\frac {e^4}{d^2 (d-x)^3}+\frac {e^4}{d^3 (d-x)^2}+\frac {e^4}{d^4 (d-x)}+\frac {e^4}{d^4 x}\right ) \, dx,x,d+e \sqrt {x}\right )}{6 d^2}+\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{9 d^3} \\ & = -\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {3 b^2 e^4 n^2}{20 d^4 x}+\frac {3 b^2 e^5 n^2}{10 d^5 \sqrt {x}}-\frac {3 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{10 d^6}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {3 b^2 e^6 n^2 \log (x)}{20 d^6}+\frac {\left (2 b e^4 n\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 )}{3 d^5}-\frac {\left (2 b e^5 n\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 )}{3 d^5}+\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+e \sqrt {x}\right )}{9 d^3}-\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{3 d^4} \\ & = -\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}+\frac {47 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {47 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}-\frac {2 b e^6 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 )}{3 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {47 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {\left (b^2 e^4 n^2\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 )}{3 d^4}+\frac {\left (2 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{3 d^6}+\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{3 d^6} \\ & = -\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}+\frac {77 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}-\frac {2 b e^6 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 )}{3 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {2 b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{3 d^6} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}-\frac {b e \left (24 a d^5 n-30 a d^4 e n \sqrt {x}+6 b d^4 e n^2 \sqrt {x}+40 a d^3 e^2 n x-18 b d^3 e^2 n^2 x-60 a d^2 e^3 n x^{3/2}+47 b d^2 e^3 n^2 x^{3/2}+120 a d e^4 n x^2-154 b d e^4 n^2 x^2+2 e^5 n (-60 a+137 b n) x^{5/2} \log \left (d+e \sqrt {x}\right )+24 b d^5 n \log \left (c \left (d+e \sqrt {x}\right )^n\right )-30 b d^4 e n \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^n\right )+40 b d^3 e^2 n x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-60 b d^2 e^3 n x^{3/2} \log \left (c \left (d+e \sqrt {x}\right )^n\right )+120 b d e^4 n x^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right )-60 b e^5 x^{5/2} \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+120 b e^5 n x^{5/2} \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+60 a e^5 n x^{5/2} \log (x)-137 b e^5 n^2 x^{5/2} \log (x)+120 b e^5 n^2 x^{5/2} \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )}{180 d^6 x^{5/2}} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}}{x^{4}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x^{4}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^4} \,d x \]
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