Integrand size = 25, antiderivative size = 256 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}+\frac {8 b e^{9/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5} \]
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Time = 0.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {277, 270, 2392, 12, 1279, 462, 283, 223, 212} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {8 b e^{9/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9} \]
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Rule 12
Rule 212
Rule 223
Rule 270
Rule 277
Rule 283
Rule 462
Rule 1279
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 )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{315 d^3 x^{10}} \, dx \\ & = -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{x^{10}} \, dx}{315 d^3} \\ & = -\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-250 d^2 e+72 d e^2 x^2\right )}{x^8} \, dx}{2835 d^4} \\ & = -\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^2 n\right ) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{315 d^3} \\ & = -\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^3 n\right ) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{315 d^3} \\ & = -\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^4 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{315 d^3} \\ & = -\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^5 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{315 d^3} \\ & = -\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^5 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{315 d^3} \\ & = -\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}+\frac {8 b e^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {\sqrt {d+e x^2} \left (315 a \left (d+e x^2\right )^2 \left (35 d^2-20 d e x^2+8 e^2 x^4\right )+b n \left (1225 d^4+2425 d^3 e x^2+429 d^2 e^2 x^4-677 d e^3 x^6+2614 e^4 x^8\right )\right )+315 b \left (d+e x^2\right )^{5/2} \left (35 d^2-20 d e x^2+8 e^2 x^4\right ) \log \left (c x^n\right )-2520 b e^{9/2} n x^9 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{99225 d^3 x^9} \]
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\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{10}}d x\]
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Time = 0.41 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.05 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\left [\frac {1260 \, b e^{\frac {9}{2}} n x^{9} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, {\left (1307 \, b e^{4} n + 1260 \, a e^{4}\right )} x^{8} - {\left (677 \, b d e^{3} n + 1260 \, a d e^{3}\right )} x^{6} + 1225 \, b d^{4} n + 11025 \, a d^{4} + 3 \, {\left (143 \, b d^{2} e^{2} n + 315 \, a d^{2} e^{2}\right )} x^{4} + 25 \, {\left (97 \, b d^{3} e n + 630 \, a d^{3} e\right )} x^{2} + 315 \, {\left (8 \, b e^{4} x^{8} - 4 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} + 50 \, b d^{3} e x^{2} + 35 \, b d^{4}\right )} \log \left (c\right ) + 315 \, {\left (8 \, b e^{4} n x^{8} - 4 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} + 50 \, b d^{3} e n x^{2} + 35 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{99225 \, d^{3} x^{9}}, -\frac {2520 \, b \sqrt {-e} e^{4} n x^{9} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, {\left (1307 \, b e^{4} n + 1260 \, a e^{4}\right )} x^{8} - {\left (677 \, b d e^{3} n + 1260 \, a d e^{3}\right )} x^{6} + 1225 \, b d^{4} n + 11025 \, a d^{4} + 3 \, {\left (143 \, b d^{2} e^{2} n + 315 \, a d^{2} e^{2}\right )} x^{4} + 25 \, {\left (97 \, b d^{3} e n + 630 \, a d^{3} e\right )} x^{2} + 315 \, {\left (8 \, b e^{4} x^{8} - 4 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} + 50 \, b d^{3} e x^{2} + 35 \, b d^{4}\right )} \log \left (c\right ) + 315 \, {\left (8 \, b e^{4} n x^{8} - 4 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} + 50 \, b d^{3} e n x^{2} + 35 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{99225 \, d^{3} x^{9}}\right ] \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{10}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{10}} \,d x \]
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