Integrand size = 23, antiderivative size = 219 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=-\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {15 i b \sqrt {e} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}} \]
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Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a+15 b \log \left (c x^n\right )-8 b n\right )}{8 d^{7/2}}-\frac {15 a+15 b \log \left (c x^n\right )-8 b n}{8 d^3 x}+\frac {5 a+5 b \log \left (c x^n\right )-b n}{8 d^2 x \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {15 i b \sqrt {e} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 b n}{8 d^3 x} \]
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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 \left (d+e x^2\right )^2}-\frac {\int \frac {-5 a+b n-5 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^2} \, dx}{4 d} \\ & = \frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}+\frac {\int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx}{8 d^2} \\ & = \frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}+\frac {\int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{x^2} \, dx}{8 d^3}-\frac {e \int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d^3} \\ & = -\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {(15 b e n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d^3} \\ & = -\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {\left (15 b \sqrt {e} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{7/2}} \\ & = -\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {\left (15 i b \sqrt {e} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{7/2}}-\frac {\left (15 i b \sqrt {e} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{7/2}} \\ & = -\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {15 i b \sqrt {e} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(552\) vs. \(2(219)=438\).
Time = 0.94 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.52 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {16 b n}{d^3 x}-\frac {16 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac {d \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {7 \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {7 \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {7 b \sqrt {e} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{7/2}}-\frac {7 b \sqrt {e} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{7/2}}+\frac {b d \sqrt {e} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}-\frac {15 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}+\frac {b \sqrt {e} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left ((-d)^{3/2}+d \sqrt {e} x\right )}{d}\right )}{(-d)^{5/2}}+\frac {15 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}+\frac {15 b \sqrt {e} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}-\frac {15 b \sqrt {e} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.00 (sec) , antiderivative size = 964, normalized size of antiderivative = 4.40
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x^{2}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (e\,x^2+d\right )}^3} \,d x \]
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