Integrand size = 24, antiderivative size = 541 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Time = 0.60 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2522, 281, 211, 2463, 266, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Rule 211
Rule 266
Rule 281
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2522
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{f+g x^4} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {\sqrt {g} x \log \left (c (d+e x)^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}-g x^2\right )}-\frac {\sqrt {g} x \log \left (c (d+e x)^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}+g x^2\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\sqrt {g} \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\sqrt {-f} \sqrt {g}-g x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}-\frac {\sqrt {g} \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\sqrt {-f} \sqrt {g}+g x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {-f}} \\ & = -\frac {\sqrt {g} \text {Subst}\left (\int \left (-\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x\right )}+\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}-\frac {\sqrt {g} \text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}-\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-f}} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}-\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}-\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}-\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}+\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}+\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}} \\ & = -\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}} \\ & = -\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{g} x}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{g} x}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{g} x}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{g} x}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}} \\ & = -\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {i \sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{i e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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\[\int \frac {\ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )}{g \,x^{2}+f}d x\]
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\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )}{g\,x^2+f} \,d x \]
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