Optimal. Leaf size=172 \[ -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {e^2 f^2 p \log \left (d+e x^2\right )}{4 d^2}-\frac {e^2 f^2 p \log (x)}{2 d^2}-\frac {e f^2 p}{4 d x^2}-\frac {e f g p \log \left (d+e x^2\right )}{d}+\frac {2 e f g p \log (x)}{d}+\frac {1}{2} g^2 p \text {Li}_2\left (\frac {e x^2}{d}+1\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2475, 43, 2416, 2395, 44, 36, 29, 31, 2394, 2315} \[ \frac {1}{2} g^2 p \text {PolyLog}\left (2,\frac {e x^2}{d}+1\right )-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {e^2 f^2 p \log \left (d+e x^2\right )}{4 d^2}-\frac {e^2 f^2 p \log (x)}{2 d^2}-\frac {e f^2 p}{4 d x^2}-\frac {e f g p \log \left (d+e x^2\right )}{d}+\frac {2 e f g p \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 44
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {f^2 \log \left (c (d+e x)^p\right )}{x^3}+\frac {2 f g \log \left (c (d+e x)^p\right )}{x^2}+\frac {g^2 \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} f^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^2\right )+(f g) \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{2} g^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} \left (e f^2 p\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,x^2\right )+(e f g p) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^2\right )-\frac {1}{2} \left (e g^2 p\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right )+\frac {1}{4} \left (e f^2 p\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )+\frac {(e f g p) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{d}-\frac {\left (e^2 f g p\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{d}\\ &=-\frac {e f^2 p}{4 d x^2}-\frac {e^2 f^2 p \log (x)}{2 d^2}+\frac {2 e f g p \log (x)}{d}+\frac {e^2 f^2 p \log \left (d+e x^2\right )}{4 d^2}-\frac {e f g p \log \left (d+e x^2\right )}{d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g^2 p \text {Li}_2\left (1+\frac {e x^2}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 148, normalized size = 0.86 \[ \frac {1}{4} \left (-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^4}-\frac {4 f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+2 g^2 \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \text {Li}_2\left (\frac {e x^2}{d}+1\right )\right )-\frac {e f^2 p \left (-e x^2 \log \left (d+e x^2\right )+d+2 e x^2 \log (x)\right )}{d^2 x^2}+\frac {4 e f g p \left (2 \log (x)-\log \left (d+e x^2\right )\right )}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g^{2} x^{4} + 2 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 663, normalized size = 3.85 \[ -g^{2} p \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-g^{2} p \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-\frac {f g \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{x^{2}}+g^{2} \ln \relax (c ) \ln \relax (x )-\frac {f^{2} \ln \relax (c )}{4 x^{4}}+g^{2} \ln \relax (x ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {f^{2} \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}+\frac {i \pi f 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 )}{2 x^{2}}-g^{2} p \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-g^{2} p \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-\frac {f g \ln \relax (c )}{x^{2}}+\frac {i \pi \,f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{8 x^{4}}-\frac {i \pi \,g^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3} \ln \relax (x )}{2}+\frac {i \pi \,g^{2} \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^{2} \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 f g \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{2 x^{2}}-\frac {i \pi \,f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{8 x^{4}}-\frac {e^{2} f^{2} p \ln \relax (x )}{2 d^{2}}+\frac {e^{2} f^{2} p \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {i \pi \,f^{2} \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}}{8 x^{4}}+\frac {2 e f g p \ln \relax (x )}{d}-\frac {e f g p \ln \left (e \,x^{2}+d \right )}{d}-\frac {e \,f^{2} p}{4 d \,x^{2}}-\frac {i \pi \,g^{2} \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 f g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{2 x^{2}}-\frac {i \pi f 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}}{2 x^{2}}+\frac {i \pi \,f^{2} \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 )}{8 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 166, normalized size = 0.97 \[ \frac {1}{2} \, {\left (\log \left (e x^{2} + d\right ) \log \left (-\frac {e x^{2} + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x^{2} + d}{d}\right )\right )} g^{2} p - \frac {{\left (e^{2} f^{2} p - 4 \, d e f g p - 2 \, d^{2} g^{2} \log \relax (c)\right )} \log \relax (x)}{2 \, d^{2}} - \frac {d^{2} f^{2} \log \relax (c) + {\left (d e f^{2} p + 4 \, d^{2} f g \log \relax (c)\right )} x^{2} + {\left (4 \, d^{2} f g p x^{2} + d^{2} f^{2} p - {\left (e^{2} f^{2} p - 4 \, d e f g p\right )} x^{4}\right )} \log \left (e x^{2} + d\right )}{4 \, d^{2} x^{4}} \]
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 \[ \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x^5} \,d x \]
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 \[ \int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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