Integrand size = 25, antiderivative size = 153 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=-f g p x^2+\frac {d g^2 p x^2}{4 e}-\frac {1}{8} g^2 p x^4-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2525, 45, 2463, 2436, 2332, 2441, 2352, 2442} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{2} f^2 p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )+\frac {d g^2 p x^2}{4 e}-f g p x^2-\frac {1}{8} g^2 p x^4 \]
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Rule 45
Rule 2332
Rule 2352
Rule 2436
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (2 f g \log \left (c (d+e x)^p\right )+\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+g^2 x \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+(f g) \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^2 \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e}-\frac {1}{2} \left (e f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )-\frac {1}{4} \left (e g^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right ) \\ & = -f g p x^2+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right )-\frac {1}{4} \left (e g^2 p\right ) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -f g p x^2+\frac {d g^2 p x^2}{4 e}-\frac {1}{8} g^2 p x^4-\frac {d^2 g^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{4} g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {f g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} f^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {-e g p x^2 \left (8 e f-2 d g+e g x^2\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+2 e \left (g \left (4 d f+4 e f x^2+e g x^4\right )+2 e f^2 \log \left (-\frac {e x^2}{d}\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+4 e^2 f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{8 e^2} \]
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Time = 1.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.37
method | result | size |
parts | \(\frac {g^{2} x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f g \,x^{2}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (x \right )-\frac {p e \left (g \left (\frac {\frac {1}{2} e g \,x^{4}-d g \,x^{2}+4 f e \,x^{2}}{2 e^{2}}+\frac {d \left (d g -4 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}\right )+4 f^{2} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )}{2}\) | \(209\) |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{4}}{4}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f g \,x^{2}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (x \right )-\frac {g^{2} p \,x^{4}}{8}+\frac {d \,g^{2} p \,x^{2}}{4 e}-f g p \,x^{2}-\frac {d^{2} g^{2} p \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {p g d \ln \left (e \,x^{2}+d \right ) f}{e}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,f^{2} \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{4}}{4}+f g \,x^{2}+f^{2} \ln \left (x \right )\right )\) | \(359\) |
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x} \,d x \]
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