Integrand size = 24, antiderivative size = 269 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=\frac {3 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}+\frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \]
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Time = 0.22 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}-\frac {6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {3 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3} \]
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}+(2 b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}+(2 b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = \frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}-\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = \frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^3} \\ & = \frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}+\frac {\left (2 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}+\frac {6 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac {2 b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac {2 b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=\frac {-18 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b n \left (-2 b e n \left (2 e^2-3 d e \sqrt [3]{x}+6 d^2 x^{2/3}\right )+9 b d e n \left (e-2 d \sqrt [3]{x}\right ) \sqrt [3]{x}+36 a d^2 e x^{2/3}-36 b d^2 e n x^{2/3}+30 b d^3 n x \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+36 b d^2 \left (e+d \sqrt [3]{x}\right ) x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-18 d e^2 \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-36 d^3 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (e+d \sqrt [3]{x}\right )-36 d^3 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+18 b d^3 n x \log \left (e+d \sqrt [3]{x}\right ) \left (\log \left (e+d \sqrt [3]{x}\right )-2 \log \left (-\frac {d \sqrt [3]{x}}{e}\right )\right )-36 b d^3 n x \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right )-36 b d^3 n x \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )}{e^3}}{18 x} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}}{x^{2}}d x\]
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Time = 0.33 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {4 \, b^{2} e^{3} n^{2} - 12 \, a b e^{3} n + 18 \, a^{2} e^{3} - 18 \, {\left (b^{2} e^{3} x - b^{2} e^{3}\right )} \log \left (c\right )^{2} + 18 \, {\left (b^{2} d^{3} n^{2} x + b^{2} e^{3} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{2} - 2 \, {\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3} - {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x\right )} \log \left (c\right ) - 6 \, {\left (6 \, b^{2} d^{2} e n^{2} x^{\frac {2}{3}} - 3 \, b^{2} d e^{2} n^{2} x^{\frac {1}{3}} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + {\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x - 6 \, {\left (b^{2} d^{3} n x + b^{2} e^{3} n\right )} \log \left (c\right )\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right ) + 6 \, {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac {2}{3}} - 3 \, {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac {1}{3}}}{18 \, e^{3} x} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{2}}{x^{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {1}{3} \, a b e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{4}} - \frac {2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {2}{3}} - 3 \, d e x^{\frac {1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} - \frac {1}{18} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{4}} - \frac {2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {2}{3}} - 3 \, d e x^{\frac {1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 2 \, d^{3} x \log \left (x\right )^{2} - 22 \, d^{3} x \log \left (x\right ) - 66 \, d^{2} e x^{\frac {2}{3}} + 15 \, d e^{2} x^{\frac {1}{3}} - 4 \, e^{3} - 6 \, {\left (2 \, d^{3} x \log \left (x\right ) - 11 \, d^{3} x\right )} \log \left (d x^{\frac {1}{3}} + e\right )\right )} n^{2}}{e^{3} x}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2}}{x} - \frac {2 \, a b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x} - \frac {a^{2}}{x} \]
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Time = 0.34 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {18 \, {\left (\frac {3 \, {\left (d x^{\frac {1}{3}} + e\right )} b^{2} d^{2} n^{2}}{e^{2} x^{\frac {1}{3}}} - \frac {3 \, {\left (d x^{\frac {1}{3}} + e\right )}^{2} b^{2} d n^{2}}{e^{2} x^{\frac {2}{3}}} + \frac {{\left (d x^{\frac {1}{3}} + e\right )}^{3} b^{2} n^{2}}{e^{2} x}\right )} \log \left (\frac {d x^{\frac {1}{3}} + e}{x^{\frac {1}{3}}}\right )^{2} - 6 \, {\left (\frac {2 \, {\left (b^{2} n^{2} - 3 \, b^{2} n \log \left (c\right ) - 3 \, a b n\right )} {\left (d x^{\frac {1}{3}} + e\right )}^{3}}{e^{2} x} - \frac {9 \, {\left (b^{2} d n^{2} - 2 \, b^{2} d n \log \left (c\right ) - 2 \, a b d n\right )} {\left (d x^{\frac {1}{3}} + e\right )}^{2}}{e^{2} x^{\frac {2}{3}}} + \frac {18 \, {\left (b^{2} d^{2} n^{2} - b^{2} d^{2} n \log \left (c\right ) - a b d^{2} n\right )} {\left (d x^{\frac {1}{3}} + e\right )}}{e^{2} x^{\frac {1}{3}}}\right )} \log \left (\frac {d x^{\frac {1}{3}} + e}{x^{\frac {1}{3}}}\right ) + \frac {2 \, {\left (2 \, b^{2} n^{2} - 6 \, b^{2} n \log \left (c\right ) + 9 \, b^{2} \log \left (c\right )^{2} - 6 \, a b n + 18 \, a b \log \left (c\right ) + 9 \, a^{2}\right )} {\left (d x^{\frac {1}{3}} + e\right )}^{3}}{e^{2} x} - \frac {27 \, {\left (b^{2} d n^{2} - 2 \, b^{2} d n \log \left (c\right ) + 2 \, b^{2} d \log \left (c\right )^{2} - 2 \, a b d n + 4 \, a b d \log \left (c\right ) + 2 \, a^{2} d\right )} {\left (d x^{\frac {1}{3}} + e\right )}^{2}}{e^{2} x^{\frac {2}{3}}} + \frac {54 \, {\left (2 \, b^{2} d^{2} n^{2} - 2 \, b^{2} d^{2} n \log \left (c\right ) + b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} n + 2 \, a b d^{2} \log \left (c\right ) + a^{2} d^{2}\right )} {\left (d x^{\frac {1}{3}} + e\right )}}{e^{2} x^{\frac {1}{3}}}}{18 \, e} \]
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Time = 1.81 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^2} \, dx=\frac {\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}}{x^{2/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2\,\left (\frac {b^2}{x}+\frac {b^2\,d^3}{e^3}\right )-\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\,\left (\frac {2\,b\,\left (3\,a-b\,n\right )}{3\,x}-\frac {\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}}{x^{2/3}}+\frac {d\,\left (\frac {2\,b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {6\,a\,b\,d}{e}\right )}{e\,x^{1/3}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{e}\right )}{e}+\frac {2\,b^2\,d^2\,n^2}{e^2}}{x^{1/3}}-\frac {a^2-\frac {2\,a\,b\,n}{3}+\frac {2\,b^2\,n^2}{9}}{x}+\frac {\ln \left (d+\frac {e}{x^{1/3}}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{3\,e^3} \]
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