Integrand size = 31, antiderivative size = 148 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \]
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Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2387, 2376, 272, 52, 65, 214} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 2376
Rule 2387
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{x} \, dx}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {e^2 x}{d^2}}} \, dx,x,x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {d^2 x^2}{e^2}} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b d n \log (x)}{e^2}-\frac {b n \sqrt {d-e x} \sqrt {d+e x} \log (x)}{e^2}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a-b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{e^2}-\frac {b d n \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{e^2} \]
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\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}}d x\]
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Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b d n \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )} \sqrt {e x + d} \sqrt {-e x + d}}{e^{2}} \]
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d - e x} \sqrt {d + e x}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \sqrt {-e^{2} x^{2} + d^{2}}\right )} b n}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b \log \left (c x^{n}\right )}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{e^{2}} \]
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]
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