Optimal. Leaf size=461 \[ -\frac {b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}-\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}+\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (\frac {e x}{d}+1\right )}{d f} \]
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Rubi [A] time = 0.63, antiderivative size = 461, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2416, 2397, 2394, 2315, 2409, 2396, 2433, 2374, 6589} \[ -\frac {b \sqrt {g} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {b^2 \sqrt {g} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d f}+\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}-\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}+\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2374
Rule 2394
Rule 2396
Rule 2397
Rule 2409
Rule 2416
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 \left (f+g x^2\right )} \, dx &=\int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f \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^2} \, dx}{f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}-\frac {g \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}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 (-f)^{3/2}}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 (-f)^{3/2}}-\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d f}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \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)^{3/2}}-\frac {\sqrt {g} \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)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}-\frac {\left (b e \sqrt {g} 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)^{3/2}}+\frac {\left (b e \sqrt {g} 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)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \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)^{3/2}}-\frac {\sqrt {g} \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)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}-\frac {\left (b \sqrt {g} n\right ) \operatorname {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)^{3/2}}+\frac {\left (b \sqrt {g} n\right ) \operatorname {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)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \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)^{3/2}}-\frac {\sqrt {g} \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)^{3/2}}-\frac {b \sqrt {g} 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)^{3/2}}+\frac {b \sqrt {g} 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)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}+\frac {\left (b^2 \sqrt {g} n^2\right ) \operatorname {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)^{3/2}}-\frac {\left (b^2 \sqrt {g} n^2\right ) \operatorname {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)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \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)^{3/2}}-\frac {\sqrt {g} \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)^{3/2}}-\frac {b \sqrt {g} 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)^{3/2}}+\frac {b \sqrt {g} 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)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}+\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 668, normalized size = 1.45 \[ \frac {2 b n \left (i d \sqrt {g} x \left (\text {Li}_2\left (-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )-i d \sqrt {g} x \left (\text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{i \sqrt {g} d+e \sqrt {f}}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )\right )+2 \sqrt {f} (e x \log (x)-(d+e x) \log (d+e x))\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-2 d \sqrt {g} x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-2 d \sqrt {f} \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 n^2 \left (-i d \sqrt {g} x \left (-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right )+i d \sqrt {g} x \left (-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+i e \sqrt {f}}\right )\right )+2 \sqrt {f} \left (2 e x \text {Li}_2\left (\frac {e x}{d}+1\right )-\left ((d+e x) \log ^2(d+e x)\right )+2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)\right )\right )}{2 d f^{3/2} x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g x^{4} + f x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 62.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g \,x^{2}+f \right ) x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a^{2} {\left (\frac {g \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} f} + \frac {1}{f x}\right )} + \int \frac {b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x^{4} + f x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2\,\left (g\,x^2+f\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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