Integrand size = 24, antiderivative size = 285 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2} \]
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Time = 0.17 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}-\frac {12 b^3 d n^3}{e \sqrt {x}} \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {2 \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e}+\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e} \\ & = -\frac {2 \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}+\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {(3 b n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {(6 b d n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = -\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}+\frac {\left (12 b^2 d n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {\left (12 b^3 d n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-4 a^3 e^2+6 a^2 b e^2 n-6 a b^2 e^2 n^2+3 b^3 e^2 n^3-12 a^2 b d e n \sqrt {x}+36 a b^2 d e n^2 \sqrt {x}-42 b^3 d e n^3 \sqrt {x}-8 b^3 d^2 n^3 x \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-4 b^3 e^2 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+12 a^2 b d^2 n x \log \left (e+d \sqrt {x}\right )-36 a b^2 d^2 n^2 x \log \left (e+d \sqrt {x}\right )+42 b^3 d^2 n^3 x \log \left (e+d \sqrt {x}\right )+6 b^2 d^2 n^2 x \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-2 a+3 b n-2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (2 \log \left (e+d \sqrt {x}\right )-\log (x)\right )-6 a^2 b d^2 n x \log (x)+18 a b^2 d^2 n^2 x \log (x)-21 b^3 d^2 n^3 x \log (x)+6 b^2 d^2 n^2 x \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (2 a-3 b n+2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+2 b n \log \left (e+d \sqrt {x}\right )-b n \log (x)\right )+6 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-2 a e+b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n x \log \left (e+d \sqrt {x}\right )-b d^2 n x \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (2 a^2 e+b^2 n^2 \left (e-6 d \sqrt {x}\right )-2 a b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n (-2 a+3 b n) x \log \left (e+d \sqrt {x}\right )+b d^2 n (2 a-3 b n) x \log (x)\right )}{4 e^2 x} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (249) = 498\).
Time = 0.34 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3 \, b^{3} e^{2} n^{3} - 4 \, b^{3} e^{2} \log \left (c\right )^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2} + 4 \, {\left (b^{3} d^{2} n^{3} x - b^{3} e^{2} n^{3}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{3} + 6 \, {\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} d e n^{3} \sqrt {x} - b^{3} e^{2} n^{3} + 2 \, a b^{2} e^{2} n^{2} + {\left (3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2}\right )} x - 2 \, {\left (b^{3} d^{2} n^{2} x - b^{3} e^{2} n^{2}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} - 6 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} \log \left (c\right ) - 6 \, {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n - 2 \, {\left (b^{3} d^{2} n x - b^{3} e^{2} n\right )} \log \left (c\right )^{2} - {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n\right )} x - 2 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n - {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n\right )} x\right )} \log \left (c\right ) - 2 \, {\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \left (c\right ) - 2 \, a b^{2} d e n^{2}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 6 \, {\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \left (c\right )^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \, {\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \left (c\right )\right )} \sqrt {x}}{4 \, e^{2} x} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (249) = 498\).
Time = 0.23 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3}{2} \, a^{2} b e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{3}}{x} + \frac {3}{4} \, {\left (4 \, e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d e \sqrt {x} + 2 \, e^{2} - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{2} x}\right )} a b^{2} + \frac {1}{8} \, {\left (12 \, e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2} + e n {\left (\frac {{\left (8 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{3} - d^{2} x \log \left (x\right )^{3} + 9 \, d^{2} x \log \left (x\right )^{2} - 42 \, d^{2} x \log \left (x\right ) - 12 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )^{2} - 84 \, d e \sqrt {x} + 6 \, e^{2} + 6 \, {\left (d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) + 14 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{3} x} - \frac {6 \, {\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d e \sqrt {x} + 2 \, e^{2} - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{e^{3} x}\right )}\right )} b^{3} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x} - \frac {a^{3}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (249) = 498\).
Time = 0.40 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.91 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {4 \, {\left (\frac {2 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3}}{e \sqrt {x}} - \frac {{\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3}}{e x}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{3} + 6 \, {\left (\frac {{\left (b^{3} n^{3} - 2 \, b^{3} n^{2} \log \left (c\right ) - 2 \, a b^{2} n^{2}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {4 \, {\left (b^{3} d n^{3} - b^{3} d n^{2} \log \left (c\right ) - a b^{2} d n^{2}\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2} - 6 \, {\left (\frac {{\left (b^{3} n^{3} - 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, b^{3} n \log \left (c\right )^{2} - 2 \, a b^{2} n^{2} + 4 \, a b^{2} n \log \left (c\right ) + 2 \, a^{2} b n\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {4 \, {\left (2 \, b^{3} d n^{3} - 2 \, b^{3} d n^{2} \log \left (c\right ) + b^{3} d n \log \left (c\right )^{2} - 2 \, a b^{2} d n^{2} + 2 \, a b^{2} d n \log \left (c\right ) + a^{2} b d n\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right ) + \frac {{\left (3 \, b^{3} n^{3} - 6 \, b^{3} n^{2} \log \left (c\right ) + 6 \, b^{3} n \log \left (c\right )^{2} - 4 \, b^{3} \log \left (c\right )^{3} - 6 \, a b^{2} n^{2} + 12 \, a b^{2} n \log \left (c\right ) - 12 \, a b^{2} \log \left (c\right )^{2} + 6 \, a^{2} b n - 12 \, a^{2} b \log \left (c\right ) - 4 \, a^{3}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {8 \, {\left (6 \, b^{3} d n^{3} - 6 \, b^{3} d n^{2} \log \left (c\right ) + 3 \, b^{3} d n \log \left (c\right )^{2} - b^{3} d \log \left (c\right )^{3} - 6 \, a b^{2} d n^{2} + 6 \, a b^{2} d n \log \left (c\right ) - 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d n - 3 \, a^{2} b d \log \left (c\right ) - a^{3} d\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}}{4 \, e} \]
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Time = 1.82 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}}{\sqrt {x}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^3\,\left (\frac {b^3}{x}-\frac {b^3\,d^2}{e^2}\right )+\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}}{\sqrt {x}}-\frac {3\,b\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2\,x}\right )+{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}}{\sqrt {x}}-\frac {3\,b^2\,\left (2\,a-b\,n\right )}{2\,x}+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}\right )-\frac {a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}}{x}+\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \]
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