Integrand size = 25, antiderivative size = 135 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=-\frac {1}{2} g^2 p x^2+\frac {e f^2 p \log (x)}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f g p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2525, 45, 2463, 2436, 2332, 2442, 36, 29, 31, 2441, 2352} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}+\frac {e f^2 p \log (x)}{d}+f g p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )-\frac {1}{2} g^2 p x^2 \]
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Rule 29
Rule 31
Rule 36
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^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (g^2 \log \left (c (d+e x)^p\right )+\frac {f^2 \log \left (c (d+e x)^p\right )}{x^2}+\frac {2 f g \log \left (c (d+e x)^p\right )}{x}\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^2} \, dx,x,x^2\right )+(f g) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+\frac {1}{2} g^2 \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g^2 \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e}+\frac {1}{2} \left (e f^2 p\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^2\right )-(e f g 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^2 p x^2-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f g p \text {Li}_2\left (1+\frac {e x^2}{d}\right )+\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (e^2 f^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {1}{2} g^2 p x^2+\frac {e f^2 p \log (x)}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f g p \text {Li}_2\left (1+\frac {e x^2}{d}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\frac {e f^2 p \log (x)}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}-\frac {1}{2} g^2 \left (p x^2-\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )+f g \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^2}{d}\right )\right ) \]
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Time = 1.72 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.56
method | result | size |
parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{2}}{2}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 x^{2}}+2 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f g \ln \left (x \right )-p e \left (\frac {x^{2} g^{2}}{2 e}-\frac {\left (g^{2} d^{2}-e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 d \,e^{2}}-\frac {f^{2} \ln \left (x \right )}{d}+4 f g \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )\) | \(211\) |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{2} g^{2}}{2}+2 \ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f g \ln \left (x \right )-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{2 x^{2}}-\frac {g^{2} p \,x^{2}}{2}+\frac {p d \ln \left (e \,x^{2}+d \right ) g^{2}}{2 e}-\frac {e \,f^{2} p \ln \left (e \,x^{2}+d \right )}{2 d}+\frac {e \,f^{2} p \ln \left (x \right )}{d}-2 p f g \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p f g \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p f g \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p f g \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{2}}{2}-\frac {f^{2}}{2 x^{2}}+2 f g \ln \left (x \right )\right )\) | \(349\) |
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x^3} \,d x \]
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