Integrand size = 25, antiderivative size = 230 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^4}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {277, 198, 197, 2392, 12, 1819, 1279, 462, 223, 212} \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}-\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^4}-\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3} \]
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Rule 12
Rule 197
Rule 198
Rule 212
Rule 223
Rule 277
Rule 462
Rule 1279
Rule 1819
Rule 2392
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}-(b n) \int \frac {-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{3 d^4 x^4 \left (d+e x^2\right )^{3/2}} \, dx \\ & = -\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{x^4 \left (d+e x^2\right )^{3/2}} \, dx}{3 d^4} \\ & = -\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {d^3-7 d^2 e x^2-16 d e^2 x^4}{x^4 \sqrt {d+e x^2}} \, dx}{3 d^5} \\ & = -\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {23 d^3 e+48 d^2 e^2 x^2}{x^2 \sqrt {d+e x^2}} \, dx}{9 d^6} \\ & = -\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {\left (16 b e^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^4} \\ & = -\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {\left (16 b e^2 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^4} \\ & = -\frac {b e^2 n x}{3 d^4 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{9 d^3 x^3}+\frac {23 b e n \sqrt {d+e x^2}}{9 d^4 x}-\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^4}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 \sqrt {d+e x^2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\frac {-3 a d^3-b d^3 n+18 a d^2 e x^2+21 b d^2 e n x^2+72 a d e^2 x^4+42 b d e^2 n x^4+48 a e^3 x^6+20 b e^3 n x^6+3 b \left (-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6\right ) \log \left (c x^n\right )-48 b e^{3/2} n x^3 \left (d+e x^2\right )^{3/2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{9 d^4 x^3 \left (d+e x^2\right )^{3/2}} \]
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\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Time = 0.38 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.26 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {24 \, {\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + {\left (4 \, {\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \, {\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \, {\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \, {\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}, \frac {48 \, {\left (b e^{3} n x^{7} + 2 \, b d e^{2} n x^{5} + b d^{2} e n x^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (4 \, {\left (5 \, b e^{3} n + 12 \, a e^{3}\right )} x^{6} - b d^{3} n + 6 \, {\left (7 \, b d e^{2} n + 12 \, a d e^{2}\right )} x^{4} - 3 \, a d^{3} + 3 \, {\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x^{2} + 3 \, {\left (16 \, b e^{3} x^{6} + 24 \, b d e^{2} x^{4} + 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (16 \, b e^{3} n x^{6} + 24 \, b d e^{2} n x^{4} + 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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