Integrand size = 29, antiderivative size = 694 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=-\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {2 b^2 e g n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}} \]
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Time = 0.76 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {2463, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46, 2444, 2441, 2352, 2456, 2443, 2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\frac {2 b e^3 n \log \left (1-\frac {d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {2 b^2 e g n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{(-f)^{5/2}} \]
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Rule 31
Rule 46
Rule 2351
Rule 2352
Rule 2356
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2441
Rule 2443
Rule 2444
Rule 2445
Rule 2456
Rule 2458
Rule 2463
Rule 2481
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^4}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x^2}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx}{f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx}{f^2}+\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^2 \int \left (\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^2}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx}{3 f}-\frac {(2 b e g n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f^2} \\ & = -\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 (-f)^{5/2}}-\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 (-f)^{5/2}}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )}{3 f}+\frac {\left (2 b^2 e^2 g n^2\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d f^2} \\ & = -\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {2 b^2 e g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )}{3 d f}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d f}-\frac {\left (b e g^{3/2} n\right ) \int \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 )}{d+e x} \, dx}{(-f)^{5/2}}+\frac {\left (b e g^{3/2} n\right ) \int \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 )}{d+e x} \, dx}{(-f)^{5/2}} \\ & = -\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {2 b^2 e g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d^2 f}+\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{3 d^2 f}-\frac {\left (b g^{3/2} n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}+d \sqrt {g}}{e}-\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}+\frac {\left (b g^{3/2} n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}-d \sqrt {g}}{e}+\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d f} \\ & = -\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {2 b^2 e g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+e x\right )}{3 d f}-\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{3 d^3 f}-\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e x\right )}{3 d^3 f}+\frac {\left (b^2 g^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}}-\frac {\left (b^2 g^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{5/2}} \\ & = -\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {2 b^2 e g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\frac {-2 d^3 f^{3/2} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 d^3 \sqrt {f} g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 d^3 g^{3/2} x^3 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 i b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (6 i d^2 \sqrt {f} g x^2 (e x \log (x)-(d+e x) \log (d+e x))+i f^{3/2} \left (d e x (d-2 e x)-2 e^3 x^3 \log (x)+2 \left (d^3+e^3 x^3\right ) \log (d+e x)\right )-3 d^3 g^{3/2} x^3 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+3 d^3 g^{3/2} x^3 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )\right )+i b^2 n^2 \left (6 i d^2 \sqrt {f} g x^2 \left (2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-(d+e x) \log ^2(d+e x)+2 e x \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )+2 i f^{3/2} \left (d e^2 x^2+3 e^3 x^3 \log (x)+d^2 e x \log (d+e x)-2 d e^2 x^2 \log (d+e x)-3 e^3 x^3 \log (d+e x)-2 e^3 x^3 \log \left (-\frac {e x}{d}\right ) \log (d+e x)+d^3 \log ^2(d+e x)+e^3 x^3 \log ^2(d+e x)-2 e^3 x^3 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )+3 d^3 g^{3/2} x^3 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )-3 d^3 g^{3/2} x^3 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{6 d^3 f^{5/2} x^3} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{4} \left (g \,x^{2}+f \right )}d x\]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^4\,\left (g\,x^2+f\right )} \,d x \]
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