Optimal. Leaf size=408 \[ -\frac {2 b e^6 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {2 b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{3 d^6}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {77 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {b^2 e^2 n^2}{30 d^2 x^2} \]
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Rubi [A] time = 1.03, antiderivative size = 432, normalized size of antiderivative = 1.06, number of steps used = 26, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac {2 b^2 e^6 n^2 \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )}{3 d^6}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{6 d^2 x^2}+\frac {e^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 d^6}-\frac {2 b e^6 n \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6}-\frac {2 b e^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^6 \sqrt {x}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d^4 x}-\frac {2 b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}+\frac {b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac {b^2 e^2 n^2}{30 d^2 x^2}+\frac {77 b^2 e^5 n^2}{90 d^5 \sqrt {x}}-\frac {47 b^2 e^4 n^2}{180 d^4 x}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt {x}\right )}{90 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2454
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^4} \, dx &=2 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 {\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 (2 b e^5 n\right ) \operatorname {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 )}{3 d^6}+\frac {\left (2 b e^6 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{3 d^6}-\frac {\left (b^2 e^4 n^2\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, 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 {e^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}-\frac {2 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{3 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {\left (2 b^2 e^6 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\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 {e^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 x^3}-\frac {2 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{3 d^6}+\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 (1+\frac {e \sqrt {x}}{d}\right )}{3 d^6}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 538, normalized size = 1.32 \[ -\frac {60 a^2 d^6-60 a^2 e^6 x^3+120 a b d^6 \log \left (c \left (d+e \sqrt {x}\right )^n\right )-120 a b e^6 x^3 \log \left (c \left (d+e \sqrt {x}\right )^n\right )+24 a b d^5 e n \sqrt {x}-30 a b d^4 e^2 n x+40 a b d^3 e^3 n x^{3/2}-60 a b d^2 e^4 n x^2+120 a b e^6 n x^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+120 a b d e^5 n x^{5/2}+60 b^2 d^6 \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+24 b^2 d^5 e n \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^n\right )-30 b^2 d^4 e^2 n x \log \left (c \left (d+e \sqrt {x}\right )^n\right )+40 b^2 d^3 e^3 n x^{3/2} \log \left (c \left (d+e \sqrt {x}\right )^n\right )-60 b^2 d^2 e^4 n x^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right )-60 b^2 e^6 x^3 \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+120 b^2 e^6 n x^3 \log \left (-\frac {e \sqrt {x}}{d}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+120 b^2 d e^5 n x^{5/2} \log \left (c \left (d+e \sqrt {x}\right )^n\right )+6 b^2 d^4 e^2 n^2 x-18 b^2 d^3 e^3 n^2 x^{3/2}+47 b^2 d^2 e^4 n^2 x^2+120 b^2 e^6 n^2 x^3 \text {Li}_2\left (\frac {\sqrt {x} e}{d}+1\right )+274 b^2 e^6 n^2 x^3 \log \left (d+e \sqrt {x}\right )-154 b^2 d e^5 n^2 x^{5/2}-137 b^2 e^6 n^2 x^3 \log (x)}{180 d^6 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a^{2}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{2} n^{2} \log \left (e \sqrt {x} + d\right )^{2}}{3 \, x^{3}} + \int \frac {{\left (b^{2} e n x + 6 \, {\left (b^{2} e \log \relax (c) + a b e\right )} x + 6 \, {\left (b^{2} d \log \relax (c) + a b d\right )} \sqrt {x}\right )} n \log \left (e \sqrt {x} + d\right ) + 3 \, {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x + 3 \, {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} \sqrt {x}}{3 \, {\left (e x^{5} + d x^{\frac {9}{2}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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