Integrand size = 24, antiderivative size = 276 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2} \]
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Time = 0.20 (sec) , antiderivative size = 276, 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} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3} \]
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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 (\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{x^{2/3}}\right )\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{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}{x^{2/3}}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{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}{x^{2/3}}\right ) \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\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)}{6 e^3 x} \, dx,x,d+\frac {e}{x^{2/3}}\right ) \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\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}{x^{2/3}}\right )}{6 e^3} \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\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}{x^{2/3}}\right )}{6 e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-18 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+b n \left (9 b d n x^{2/3} \left (e \left (e-2 d x^{2/3}\right )+2 d^2 x^{4/3} \log \left (d+\frac {e}{x^{2/3}}\right )\right )-2 b n \left (e \left (2 e^2-3 d e x^{2/3}+6 d^2 x^{4/3}\right )-6 d^3 x^2 \log \left (d+\frac {e}{x^{2/3}}\right )\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-18 d e^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+36 d^2 x^{4/3} \left (e (a-b n)+b \left (e+d x^{2/3}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+b n \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{36 e^3 x^2} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{2}}{x^{3}}d x\]
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Time = 0.35 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {4 \, b^{2} e^{3} n^{2} + 18 \, b^{2} e^{3} \log \left (c\right )^{2} - 12 \, a b e^{3} n + 18 \, a^{2} e^{3} + 18 \, {\left (b^{2} d^{3} n^{2} x^{2} + b^{2} e^{3} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{2} - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} \log \left (c\right ) - 6 \, {\left (6 \, b^{2} d^{2} e n^{2} x^{\frac {4}{3}} - 3 \, b^{2} d e^{2} n^{2} x^{\frac {2}{3}} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + {\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x^{2} - 6 \, {\left (b^{2} d^{3} n x^{2} + b^{2} e^{3} n\right )} \log \left (c\right )\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 3 \, {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac {2}{3}} - 6 \, {\left (6 \, b^{2} d^{2} e n x \log \left (c\right ) - {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, a b d^{2} e n\right )} x\right )} x^{\frac {1}{3}}}{36 \, e^{3} x^{2}} \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {1}{6} \, a b e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{4}} - \frac {6 \, d^{3} \log \left (x^{\frac {2}{3}}\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {4}{3}} - 3 \, d e x^{\frac {2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} - \frac {1}{36} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{4}} - \frac {6 \, d^{3} \log \left (x^{\frac {2}{3}}\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {4}{3}} - 3 \, d e x^{\frac {2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} e x^{\frac {4}{3}} + 15 \, d e^{2} x^{\frac {2}{3}} - 4 \, e^{3} - 6 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right )\right )} n^{2}}{e^{3} x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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Time = 1.65 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{4\,e}}{x^{4/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2\,\left (\frac {b^2}{2\,x^2}+\frac {b^2\,d^3}{2\,e^3}\right )-\frac {\frac {a^2}{2}-\frac {a\,b\,n}{3}+\frac {b^2\,n^2}{9}}{x^2}-\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\,\left (\frac {b\,\left (3\,a-b\,n\right )}{3\,x^2}-\frac {\frac {b\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {3\,a\,b\,d}{2\,e}}{x^{4/3}}+\frac {d\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )}{e\,x^{2/3}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{e^2}}{x^{2/3}}+\frac {\ln \left (d+\frac {e}{x^{2/3}}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{6\,e^3} \]
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