Optimal. Leaf size=746 \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{4 \sqrt{f} g^{3/2}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{4 \sqrt{f} g^{3/2}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} g^{3/2}}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{\sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{g (e f-d g)}+\frac{p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} g^{3/2}} \]
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Rubi [A] time = 1.51254, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 14, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.56, Rules used = {2476, 2471, 2463, 801, 635, 205, 260, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{4 \sqrt{f} g^{3/2}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{4 \sqrt{f} g^{3/2}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} g^{3/2}}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{\sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{g (e f-d g)}+\frac{p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} g^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2476
Rule 2471
Rule 2463
Rule 801
Rule 635
Rule 205
Rule 260
Rule 2470
Rule 12
Rule 4928
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )^2}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{g}-\frac{f \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx}{g}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} g^{3/2}}-\frac{f \int \left (-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt{-f} \sqrt{g}-g x\right )^2}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt{-f} \sqrt{g}+g x\right )^2}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{g}-\frac{(2 e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (d+e x^2\right )} \, dx}{g}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} g^{3/2}}+\frac{1}{4} \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt{-f} \sqrt{g}-g x\right )^2} \, dx+\frac{1}{4} \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt{-f} \sqrt{g}+g x\right )^2} \, dx+\frac{1}{2} \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{-f g-g^2 x^2} \, dx-\frac{(2 e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^2} \, dx}{\sqrt{f} g^{3/2}}\\ &=\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}+(e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} g^{3/2} \left (d+e x^2\right )} \, dx-\frac{(2 e p) \int \left (-\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{\sqrt{f} g^{3/2}}-\frac{(e p) \int \frac{x}{\left (\sqrt{-f} \sqrt{g}-g x\right ) \left (d+e x^2\right )} \, dx}{2 g}+\frac{(e p) \int \frac{x}{\left (\sqrt{-f} \sqrt{g}+g x\right ) \left (d+e x^2\right )} \, dx}{2 g}\\ &=\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}+\frac{\left (\sqrt{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{\sqrt{f} g^{3/2}}-\frac{\left (\sqrt{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{\sqrt{f} g^{3/2}}+\frac{(e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^2} \, dx}{\sqrt{f} g^{3/2}}+\frac{(e p) \int \left (\frac{\sqrt{-f}}{(e f-d g) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{-d \sqrt{g}-e \sqrt{-f} x}{\sqrt{g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 g}-\frac{(e p) \int \left (\frac{\sqrt{-f}}{(e f-d g) \left (-\sqrt{-f}+\sqrt{g} x\right )}-\frac{d \sqrt{g}-e \sqrt{-f} x}{\sqrt{g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 g}\\ &=-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{2 p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} g^{3/2}}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}+\frac{(e p) \int \left (-\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{\sqrt{f} g^{3/2}}-2 \frac{p \int \frac{\log \left (\frac{2}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{1+\frac{g x^2}{f}} \, dx}{f g}+\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{f} \left (-i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{f g}+\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{f} \left (i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{f g}+\frac{(e p) \int \frac{-d \sqrt{g}-e \sqrt{-f} x}{d+e x^2} \, dx}{2 g^{3/2} (e f-d g)}-\frac{(e p) \int \frac{d \sqrt{g}-e \sqrt{-f} x}{d+e x^2} \, dx}{2 g^{3/2} (e f-d g)}\\ &=-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{2 p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} g^{3/2}}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}-2 \frac{(i p) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{\sqrt{f} g^{3/2}}-\frac{\left (\sqrt{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{f} g^{3/2}}+\frac{\left (\sqrt{e} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{f} g^{3/2}}-2 \frac{(d e p) \int \frac{1}{d+e x^2} \, dx}{2 g (e f-d g)}\\ &=-\frac{\sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{g (e f-d g)}-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}-\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+2 \frac{p \int \frac{\log \left (\frac{2}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{1+\frac{g x^2}{f}} \, dx}{2 f g}-\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{f} \left (-i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{2 f g}-\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{f} \left (i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{2 f g}\\ &=-\frac{\sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{g (e f-d g)}-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}-\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 \sqrt{f} g^{3/2}}+2 \frac{(i p) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{2 \sqrt{f} g^{3/2}}\\ &=-\frac{\sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{g (e f-d g)}-\frac{e \sqrt{-f} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{f} g^{3/2}}+\frac{e \sqrt{-f} p \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 g^{3/2} (e f-d g)}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt{f} g^{3/2}}-\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 \sqrt{f} g^{3/2}}+\frac{i p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 \sqrt{f} g^{3/2}}\\ \end{align*}
Mathematica [A] time = 3.30286, size = 1231, normalized size = 1.65 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )}{2 \sqrt{f} g^{3/2}}+\frac{p x \log \left (e x^2+d\right )-x \log \left (c \left (e x^2+d\right )^p\right )}{2 g^2 x^2+2 f g}+\frac{1}{4} p \left (-\frac{i \left (\frac{\log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )}{i \sqrt{g} x+\sqrt{f}}+\frac{\sqrt{e} \left (\log \left (i \sqrt{f}-\sqrt{g} x\right )-\log \left (i \sqrt{d}-\sqrt{e} x\right )\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}\right )}{g^{3/2}}-\frac{i \left (\frac{\log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )}{i \sqrt{g} x+\sqrt{f}}+\frac{\sqrt{e} \left (\log \left (i \sqrt{f}-\sqrt{g} x\right )-\log \left (\sqrt{e} x+i \sqrt{d}\right )\right )}{\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}}\right )}{g^{3/2}}+\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right ) \left (\log \left (i \sqrt{d}-\sqrt{e} x\right )-\log \left (\sqrt{g} x+i \sqrt{f}\right )\right )-\left (\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}\right ) \log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )}{\left (\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}\right ) g^{3/2} \left (\sqrt{g} x+i \sqrt{f}\right )}+\frac{-\frac{\log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )}{\sqrt{g} x+i \sqrt{f}}-\frac{i \sqrt{e} \left (\log \left (\sqrt{e} x+i \sqrt{d}\right )-\log \left (\sqrt{g} x+i \sqrt{f}\right )\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}}{g^{3/2}}+4 \left (\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{f} g^{3/2}}-\frac{x}{2 g \left (g x^2+f\right )}\right ) \left (-\log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right )-\log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\log \left (e x^2+d\right )\right )+\frac{i \left (\log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}\right )+\text{PolyLog}\left (2,-\frac{\sqrt{g} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}\right )\right )}{\sqrt{f} g^{3/2}}-\frac{i \left (\log \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} \left (i \sqrt{g} x+\sqrt{f}\right )}{\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{g} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}}\right )\right )}{\sqrt{f} g^{3/2}}-\frac{i \left (\log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} \left (i \sqrt{g} x+\sqrt{f}\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}\right )+\text{PolyLog}\left (2,-\frac{\sqrt{g} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{e} \sqrt{f}-\sqrt{d} \sqrt{g}}\right )\right )}{\sqrt{f} g^{3/2}}+\frac{i \left (\log \left (x-\frac{i \sqrt{d}}{\sqrt{e}}\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{g} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{e} \sqrt{f}+\sqrt{d} \sqrt{g}}\right )\right )}{\sqrt{f} g^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 1.414, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) }{ \left ( g{x}^{2}+f \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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