Integrand size = 24, antiderivative size = 503 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\frac {b e n \log (d+e x)}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {b e n \log (d+e x)}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}} \]
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Time = 0.30 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2456, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {b e n \log (d+e x)}{4 f \sqrt {g} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log (d+e x)}{4 \sqrt {g} \left (d f \sqrt {g}+e (-f)^{3/2}\right )}-\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f \sqrt {g} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \sqrt {g} \left (d f \sqrt {g}+e (-f)^{3/2}\right )} \]
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Rule 31
Rule 36
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2456
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx \\ & = -\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{-f g-g^2 x^2} \, dx}{2 f} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {g \int \left (-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{2 f}+\frac {(b e n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}-g x\right )} \, dx}{4 f}-\frac {(b e n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}+g x\right )} \, dx}{4 f} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {1}{\sqrt {-f} \sqrt {g}+g x} \, dx}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {1}{\sqrt {-f} \sqrt {g}-g x} \, dx}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right )} \\ & = \frac {b e n \log (d+e x)}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {b e n \log (d+e x)}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{4 (-f)^{3/2} \sqrt {g}}-\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{4 (-f)^{3/2} \sqrt {g}} \\ & = \frac {b e n \log (d+e x)}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {b e n \log (d+e x)}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 (-f)^{3/2} \sqrt {g}} \\ & = \frac {b e n \log (d+e x)}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {b e n \log (d+e x)}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\frac {1}{4} \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f \left (\sqrt {-f} \sqrt {g}+g x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{(-f)^{3/2} \sqrt {g}+f g x}+\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f} f \sqrt {g}+d f g}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2} \sqrt {g}}+\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{e (-f)^{3/2} \sqrt {g}+d f g}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2} \sqrt {g}}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2} \sqrt {g}}+\frac {b f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2} \sqrt {g}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.31 (sec) , antiderivative size = 1406, normalized size of antiderivative = 2.80
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
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