Integrand size = 23, antiderivative size = 82 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=-\frac {1}{2} g p x^2+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2525, 45, 2463, 2436, 2332, 2441, 2352} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )-\frac {1}{2} g p x^2 \]
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Rule 45
Rule 2332
Rule 2352
Rule 2436
Rule 2441
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (g \log \left (c (d+e x)^p\right )+\frac {f \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+\frac {1}{2} g \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e}-\frac {1}{2} (e f p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right ) \\ & = -\frac {1}{2} g p x^2+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f p \text {Li}_2\left (1+\frac {e x^2}{d}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {1}{2} g \left (-p x^2+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )+\frac {1}{2} f \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^2}{d}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(74)=148\).
Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91
method | result | size |
parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g \,x^{2}}{2}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (x \right )-p e \left (\frac {g \,x^{2}}{2 e}-\frac {g d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+2 f \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )\) | \(157\) |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g \,x^{2}}{2}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (x \right )-\frac {g p \,x^{2}}{2}+\frac {p g d \ln \left (e \,x^{2}+d \right )}{2 e}-p f \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g \,x^{2}}{2}+f \ln \left (x \right )\right )\) | \(278\) |
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\[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+f\right )}{x} \,d x \]
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