Integrand size = 23, antiderivative size = 72 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=-2 g p x+\frac {2 (e f+d g) p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right ) \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2526, 2498, 327, 211, 2505} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\frac {2 \sqrt {e} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {2 \sqrt {d} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right )-2 g p x \]
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Rule 211
Rule 327
Rule 2498
Rule 2505
Rule 2526
Rubi steps \begin{align*} \text {integral}& = \int \left (g \log \left (c \left (d+e x^2\right )^p\right )+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right ) \, dx \\ & = f \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx+g \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right )+(2 e f p) \int \frac {1}{d+e x^2} \, dx-(2 e g p) \int \frac {x^2}{d+e x^2} \, dx \\ & = -2 g p x+\frac {2 \sqrt {e} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right )+(2 d g p) \int \frac {1}{d+e x^2} \, dx \\ & = -2 g p x+\frac {2 \sqrt {e} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {2 \sqrt {d} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x}+g x \log \left (c \left (d+e x^2\right )^p\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=-2 g p x+\frac {2 (e f+d g) p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+\left (-\frac {f}{x}+g x\right ) \log \left (c \left (d+e x^2\right )^p\right ) \]
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Time = 0.67 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01
method | result | size |
parts | \(g x \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x}-2 p e \left (\frac {g x}{e}+\frac {\left (-d g -e f \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e \sqrt {d e}}\right )\) | \(73\) |
risch | \(-\frac {\left (-g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{x}+\frac {i \pi g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} d e -i \pi g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right ) d e -i \pi g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3} d e +i \pi g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right ) d e -i \pi d e f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+i \pi d e f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+i \pi d e f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-i \pi d e f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \sqrt {-d e}\, p \ln \left (-\sqrt {-d e}\, x +d \right ) g d x +2 \sqrt {-d e}\, p \ln \left (-\sqrt {-d e}\, x +d \right ) f e x -2 \sqrt {-d e}\, p \ln \left (-\sqrt {-d e}\, x -d \right ) g d x -2 \sqrt {-d e}\, p \ln \left (-\sqrt {-d e}\, x -d \right ) f e x +2 \ln \left (c \right ) g \,x^{2} d e -4 d g p \,x^{2} e -2 \ln \left (c \right ) d e f}{2 d e x}\) | \(427\) |
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Time = 0.35 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.76 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\left [-\frac {2 \, d e g p x^{2} + \sqrt {-d e} {\left (e f + d g\right )} p x \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - {\left (d e g p x^{2} - d e f p\right )} \log \left (e x^{2} + d\right ) - {\left (d e g x^{2} - d e f\right )} \log \left (c\right )}{d e x}, -\frac {2 \, d e g p x^{2} - 2 \, \sqrt {d e} {\left (e f + d g\right )} p x \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (d e g p x^{2} - d e f p\right )} \log \left (e x^{2} + d\right ) - {\left (d e g x^{2} - d e f\right )} \log \left (c\right )}{d e x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (71) = 142\).
Time = 15.37 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.83 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\begin {cases} \left (- \frac {f}{x} + g x\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\\left (- \frac {f}{x} + g x\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 f p}{x} - \frac {f \log {\left (c \left (e x^{2}\right )^{p} \right )}}{x} - 2 g p x + g x \log {\left (c \left (e x^{2}\right )^{p} \right )} & \text {for}\: d = 0 \\\frac {2 d g p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{e \sqrt {- \frac {d}{e}}} - \frac {d g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{e \sqrt {- \frac {d}{e}}} + \frac {2 f p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x} - 2 g p x + g x \log {\left (c \left (d + e x^{2}\right )^{p} \right )} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=-{\left (2 \, g p - g \log \left (c\right )\right )} x + {\left (g p x - \frac {f p}{x}\right )} \log \left (e x^{2} + d\right ) + \frac {2 \, {\left (e f p + d g p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} - \frac {f \log \left (c\right )}{x} \]
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Time = 1.61 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (2\,g\,x-\frac {g\,x^2+f}{x}\right )-2\,g\,p\,x+\frac {2\,p\,\mathrm {atan}\left (\frac {2\,\sqrt {e}\,p\,x\,\left (d\,g+e\,f\right )}{\sqrt {d}\,\left (2\,d\,g\,p+2\,e\,f\,p\right )}\right )\,\left (d\,g+e\,f\right )}{\sqrt {d}\,\sqrt {e}} \]
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