Integrand size = 27, antiderivative size = 430 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )} \]
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Time = 0.39 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2460, 2465, 2437, 2338, 2441, 2440, 2438} \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=-\frac {b e n \left (d \sqrt {-f} \sqrt {g}+e f\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}-\frac {b e n \left (e f-d \sqrt {-f} \sqrt {g}\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (d^2 g+e^2 f\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b^2 e n^2 \left (d \sqrt {g}+e \sqrt {-f}\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g \left (d^2 g+e^2 f\right )}-\frac {b^2 e n^2 \left (d \sqrt {-f} \sqrt {g}+e f\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 f g \left (d^2 g+e^2 f\right )} \]
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Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2460
Rule 2465
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}+\frac {(b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f+g x^2\right )} \, dx}{g} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}+\frac {(b e n) \int \left (\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) (d+e x)}-\frac {g (-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}\right ) \, dx}{g} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {(b e n) \int \frac {(-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{e^2 f+d^2 g}+\frac {\left (b e^3 n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{g \left (e^2 f+d^2 g\right )} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {(b e n) \int \left (\frac {\left (-d \sqrt {-f}-\frac {e f}{\sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\left (-d \sqrt {-f}+\frac {e f}{\sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{e^2 f+d^2 g}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{g \left (e^2 f+d^2 g\right )} \\ & = \frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {\left (b e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \left (e^2 f+d^2 g\right )}+\frac {\left (b e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \left (e^2 f+d^2 g\right )} \\ & = \frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}-\frac {b e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e^2 \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e^2 \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {g} \left (e^2 f+d^2 g\right )} \\ & = \frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}-\frac {b e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n^2\right ) \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n^2\right ) \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )} \\ & = \frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}-\frac {b e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n \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 )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {g} \left (e^2 f+d^2 g\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {-\frac {2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}+\frac {2 b n \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (2 \sqrt {f} g \left (d^2-e^2 x^2\right ) \log (d+e x)+e \left (f+g x^2\right ) \left (\left (e \sqrt {f}+i d \sqrt {g}\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )+\left (e \sqrt {f}-i d \sqrt {g}\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )\right )}{\sqrt {f} \left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}+\frac {i b^2 n^2 \left (\frac {-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+\frac {\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}\right )}{\sqrt {f}}}{4 g} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.54 (sec) , antiderivative size = 1231, normalized size of antiderivative = 2.86
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \]
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