Integrand size = 22, antiderivative size = 82 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=\frac {b n}{3 x}-\frac {b d n}{2 e x^{2/3}}+\frac {b d^2 n}{e^2 \sqrt [3]{x}}-\frac {b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \]
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Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 45} \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x}-\frac {b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}+\frac {b d^2 n}{e^2 \sqrt [3]{x}}-\frac {b d n}{2 e x^{2/3}}+\frac {b n}{3 x} \]
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Rule 45
Rule 2442
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 ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x}+(b e n) \text {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x}+(b e n) \text {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = \frac {b n}{3 x}-\frac {b d n}{2 e x^{2/3}}+\frac {b d^2 n}{e^2 \sqrt [3]{x}}-\frac {b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=-\frac {a}{x}+\frac {b n}{3 x}-\frac {b d n}{2 e x^{2/3}}+\frac {b d^2 n}{e^2 \sqrt [3]{x}}-\frac {b d^3 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}-\frac {b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \]
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\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x^{2}}d x\]
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Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=\frac {6 \, b d^{2} e n x^{\frac {2}{3}} - 3 \, b d e^{2} n x^{\frac {1}{3}} + 2 \, b e^{3} n - 6 \, a e^{3} - 2 \, {\left (b e^{3} n - 3 \, a e^{3}\right )} x + 6 \, {\left (b e^{3} x - b e^{3}\right )} \log \left (c\right ) - 6 \, {\left (b d^{3} n x + b e^{3} n\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )}{6 \, e^{3} x} \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=-\frac {1}{6} \, b e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{4}} - \frac {2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {2}{3}} - 3 \, d e x^{\frac {1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x} - \frac {a}{x} \]
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Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=-\frac {1}{6} \, {\left (e {\left (\frac {6 \, d^{3} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{e^{4}} - \frac {2 \, d^{3} \log \left ({\left | x \right |}\right )}{e^{4}} - \frac {6 \, d^{2} e x^{\frac {2}{3}} - 3 \, d e^{2} x^{\frac {1}{3}} + 2 \, e^{3}}{e^{4} x}\right )} + \frac {6 \, \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right )}{x}\right )} b n - \frac {b \log \left (c\right )}{x} - \frac {a}{x} \]
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Time = 1.59 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx=\frac {b\,n}{3\,x}-\frac {a}{x}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x}-\frac {b\,d\,n}{2\,e\,x^{2/3}}-\frac {b\,d^3\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{e^3}+\frac {b\,d^2\,n}{e^2\,x^{1/3}} \]
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