Integrand size = 25, antiderivative size = 196 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {2 b e^3 n \sqrt {d+e x^2}}{35 d^2 x}+\frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {2 b e^{7/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{35 d^2}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5} \]
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Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {277, 270, 2392, 12, 462, 283, 223, 212} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}-\frac {2 b e^{7/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{35 d^2}+\frac {2 b e^3 n \sqrt {d+e x^2}}{35 d^2 x}+\frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5} \]
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Rule 12
Rule 212
Rule 223
Rule 270
Rule 277
Rule 283
Rule 462
Rule 2392
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8} \, dx \\ & = -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8} \, dx}{35 d^2} \\ & = -\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {(2 b e n) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{35 d^2} \\ & = \frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (2 b e^2 n\right ) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{35 d^2} \\ & = \frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (2 b e^3 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{35 d^2} \\ & = \frac {2 b e^3 n \sqrt {d+e x^2}}{35 d^2 x}+\frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (2 b e^4 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{35 d^2} \\ & = \frac {2 b e^3 n \sqrt {d+e x^2}}{35 d^2 x}+\frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{35 d^2} \\ & = \frac {2 b e^3 n \sqrt {d+e x^2}}{35 d^2 x}+\frac {2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac {2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac {2 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{35 d^2}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {\sqrt {d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b n \left (75 d^3+183 d^2 e x^2+71 d e^2 x^4-247 e^3 x^6\right )\right )+105 b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^{5/2} \log \left (c x^n\right )+210 b e^{7/2} n x^7 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{3675 d^2 x^7} \]
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\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{8}}d x\]
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Time = 0.39 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.16 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\left [\frac {105 \, b e^{\frac {7}{2}} n x^{7} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + {\left ({\left (247 \, b e^{3} n + 210 \, a e^{3}\right )} x^{6} - 75 \, b d^{3} n - {\left (71 \, b d e^{2} n + 105 \, a d e^{2}\right )} x^{4} - 525 \, a d^{3} - 3 \, {\left (61 \, b d^{2} e n + 280 \, a d^{2} e\right )} x^{2} + 105 \, {\left (2 \, b e^{3} x^{6} - b d e^{2} x^{4} - 8 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (2 \, b e^{3} n x^{6} - b d e^{2} n x^{4} - 8 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3675 \, d^{2} x^{7}}, \frac {210 \, b \sqrt {-e} e^{3} n x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left ({\left (247 \, b e^{3} n + 210 \, a e^{3}\right )} x^{6} - 75 \, b d^{3} n - {\left (71 \, b d e^{2} n + 105 \, a d e^{2}\right )} x^{4} - 525 \, a d^{3} - 3 \, {\left (61 \, b d^{2} e n + 280 \, a d^{2} e\right )} x^{2} + 105 \, {\left (2 \, b e^{3} x^{6} - b d e^{2} x^{4} - 8 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (2 \, b e^{3} n x^{6} - b d e^{2} n x^{4} - 8 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3675 \, d^{2} x^{7}}\right ] \]
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\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{8}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]
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