Integrand size = 25, antiderivative size = 176 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {8 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {277, 197, 2392, 12, 1279, 462, 223, 212} \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}-\frac {8 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3} \]
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Rule 12
Rule 197
Rule 212
Rule 223
Rule 277
Rule 462
Rule 1279
Rule 2392
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-(b n) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt {d+e x^2}} \, dx \\ & = -\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt {d+e x^2}} \, dx}{3 d^3} \\ & = -\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {-14 d^2 e-24 d e^2 x^2}{x^2 \sqrt {d+e x^2}} \, dx}{9 d^4} \\ & = -\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (8 b e^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^3} \\ & = -\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (8 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^3} \\ & = -\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-3 a d^2-b d^2 n+12 a d e x^2+13 b d e n x^2+24 a e^2 x^4+14 b e^2 n x^4-3 b \left (d^2-4 d e x^2-8 e^2 x^4\right ) \log \left (c x^n\right )-24 b e^{3/2} n x^3 \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{9 d^3 x^3 \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Time = 0.38 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.10 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {12 \, {\left (b e^{2} n x^{5} + b d e n x^{3}\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + {\left (2 \, {\left (7 \, b e^{2} n + 12 \, a e^{2}\right )} x^{4} - b d^{2} n - 3 \, a d^{2} + {\left (13 \, b d e n + 12 \, a d e\right )} x^{2} + 3 \, {\left (8 \, b e^{2} x^{4} + 4 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (8 \, b e^{2} n x^{4} + 4 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{3} e x^{5} + d^{4} x^{3}\right )}}, \frac {24 \, {\left (b e^{2} n x^{5} + b d e n x^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, {\left (7 \, b e^{2} n + 12 \, a e^{2}\right )} x^{4} - b d^{2} n - 3 \, a d^{2} + {\left (13 \, b d e n + 12 \, a d e\right )} x^{2} + 3 \, {\left (8 \, b e^{2} x^{4} + 4 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (8 \, b e^{2} n x^{4} + 4 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{9 \, {\left (d^{3} e x^{5} + d^{4} x^{3}\right )}}\right ] \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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