Integrand size = 23, antiderivative size = 260 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 i b e^{3/2} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}} \]
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Time = 0.30 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2385, 2380, 2341, 211, 2361, 12, 4940, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^{9/2}}+\frac {e \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^4 x}-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{24 d^3 x^3}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {35 i b e^{3/2} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 b e n}{8 d^4 x}-\frac {35 b n}{72 d^3 x^3} \]
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Rule 12
Rule 211
Rule 2341
Rule 2361
Rule 2380
Rule 2385
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-7 a+b n-7 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx}{4 d} \\ & = \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )} \, dx}{8 d^2} \\ & = \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4} \, dx}{8 d^3}-\frac {e \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx}{8 d^3} \\ & = -\frac {35 b n}{72 d^3 x^3}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}-\frac {e \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^2} \, dx}{8 d^4}+\frac {e^2 \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d^4} \\ & = -\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^2 n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d^4} \\ & = -\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{9/2}} \\ & = -\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}}+\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}} \\ & = -\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 i b e^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(584\) vs. \(2(260)=520\).
Time = 1.03 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.25 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\frac {1}{144} \left (-\frac {16 b n}{d^3 x^3}+\frac {432 b e n}{d^4 x}-\frac {48 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^3}+\frac {432 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}-\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}+\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left ((-d)^{3/2}+d \sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}+\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}+\frac {315 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}-\frac {315 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.39 (sec) , antiderivative size = 1029, normalized size of antiderivative = 3.96
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^3} \,d x \]
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