Integrand size = 22, antiderivative size = 113 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2360, 2351, 223, 212, 197} \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {2 b n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}-\frac {b n x}{3 d^2 \sqrt {d+e x^2}} \]
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Rule 197
Rule 212
Rule 223
Rule 2351
Rule 2360
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}-\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d} \\ & = -\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^2} \\ & = -\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^2} \\ & = -\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\sqrt {e} x \left (-b n \left (d+e x^2\right )+a \left (3 d+2 e x^2\right )\right )+b \sqrt {e} x \left (3 d+2 e x^2\right ) \log \left (c x^n\right )-2 b n \left (d+e x^2\right )^{3/2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{3 d^2 \sqrt {e} \left (d+e x^2\right )^{3/2}} \]
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\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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none
Time = 0.33 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.98 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left ({\left (b e^{2} n - 2 \, a e^{2}\right )} x^{3} + {\left (b d e n - 3 \, a d e\right )} x - {\left (2 \, b e^{2} x^{3} + 3 \, b d e x\right )} \log \left (c\right ) - {\left (2 \, b e^{2} n x^{3} + 3 \, b d e n x\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {2 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left ({\left (b e^{2} n - 2 \, a e^{2}\right )} x^{3} + {\left (b d e n - 3 \, a d e\right )} x - {\left (2 \, b e^{2} x^{3} + 3 \, b d e x\right )} \log \left (c\right ) - {\left (2 \, b e^{2} n x^{3} + 3 \, b d e n x\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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