Integrand size = 33, antiderivative size = 251 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]
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Time = 0.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2387, 272, 45, 2392, 12, 457, 81, 52, 65, 214} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rule 12
Rule 45
Rule 52
Rule 65
Rule 81
Rule 214
Rule 272
Rule 457
Rule 2387
Rule 2392
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x^3 \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 {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {d^2 \left (-2 d^2-e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{3 e^4 x} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \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 {\left (-2 d^2-e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{x} \, dx}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \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 {\left (-2 d^2-e^2 x\right ) \sqrt {1-\frac {e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{6 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \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 {\sqrt {1-\frac {e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \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}{x \sqrt {1-\frac {e^2 x}{d^2}}} \, dx,x,x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (2 b d^6 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 )}{3 e^6 \sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {-6 b d^3 n \log (x)+3 b n \sqrt {d-e x} \sqrt {d+e x} \left (2 d^2+e^2 x^2\right ) \log (x)+\sqrt {d-e x} \sqrt {d+e x} \left (e^2 x^2 \left (3 a-b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+d^2 \left (6 a-5 b n-6 b n \log (x)+6 b \log \left (c x^n\right )\right )\right )+6 b d^3 n \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{9 e^4} \]
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\[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}}d x\]
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Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.50 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {6 \, b d^{3} n \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) + {\left (5 \, b d^{2} n - 6 \, a d^{2} + {\left (b e^{2} n - 3 \, a e^{2}\right )} x^{2} - 3 \, {\left (b e^{2} x^{2} + 2 \, b d^{2}\right )} \log \left (c\right ) - 3 \, {\left (b e^{2} n x^{2} + 2 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x + d} \sqrt {-e x + d}}{9 \, e^{4}} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d - e x} \sqrt {d + e x}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {1}{9} \, b n {\left (\frac {3 \, d^{3} \log \left (d + \sqrt {-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac {3 \, d^{3} \log \left (-d + \sqrt {-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} - {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{e^{4}}\right )} - \frac {1}{3} \, b {\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{4}}\right )} \log \left (c x^{n}\right ) - \frac {1}{3} \, a {\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{4}}\right )} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]
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