Optimal. Leaf size=93 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac {e f p \log \left (d+e x^2\right )}{2 d}+\frac {e f p \log (x)}{d}+\frac {1}{2} g p \text {Li}_2\left (\frac {e x^2}{d}+1\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2475, 43, 2416, 2395, 36, 29, 31, 2394, 2315} \[ \frac {1}{2} g p \text {PolyLog}\left (2,\frac {e x^2}{d}+1\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac {e f p \log \left (d+e x^2\right )}{2 d}+\frac {e f p \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {f \log \left (c (d+e x)^p\right )}{x^2}+\frac {g \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} f \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{2} g \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} (e f p) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^2\right )-\frac {1}{2} (e g p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )\\ &=-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g p \text {Li}_2\left (1+\frac {e x^2}{d}\right )+\frac {(e f p) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (e^2 f p\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 d}\\ &=\frac {e f p \log (x)}{d}-\frac {e f p \log \left (d+e x^2\right )}{2 d}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g p \text {Li}_2\left (1+\frac {e x^2}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 92, normalized size = 0.99 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \text {Li}_2\left (\frac {e x^2+d}{d}\right )\right )-\frac {e f p \log \left (d+e x^2\right )}{2 d}+\frac {e f p \log (x)}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 421, normalized size = 4.53 \[ -\frac {i \pi g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \ln \relax (x )}{2}+\frac {i \pi g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi g \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi g \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3} \ln \relax (x )}{2}-g p \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-g p \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\frac {e f p \ln \relax (x )}{d}-\frac {e f p \ln \left (e \,x^{2}+d \right )}{2 d}-g p \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-g p \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+g \ln \relax (c ) \ln \relax (x )+g \ln \relax (x ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )+\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{2}}-\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4 x^{2}}-\frac {i \pi f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4 x^{2}}+\frac {i \pi f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{4 x^{2}}-\frac {f \ln \relax (c )}{2 x^{2}}-\frac {f \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 93, normalized size = 1.00 \[ \frac {1}{2} \, {\left (\log \left (e x^{2} + d\right ) \log \left (-\frac {e x^{2} + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x^{2} + d}{d}\right )\right )} g p + \frac {{\left (e f p + d g \log \relax (c)\right )} \log \relax (x)}{d} - \frac {d f \log \relax (c) + {\left (e f p x^{2} + d f p\right )} \log \left (e x^{2} + d\right )}{2 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+f\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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