Optimal. Leaf size=82 \[ -\frac {1}{2} g p x^2+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f p \text {Li}_2\left (1+\frac {e x^2}{d}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2525, 45, 2463,
2436, 2332, 2441, 2352} \begin {gather*} \frac {1}{2} f p \text {PolyLog}\left (2,\frac {e x^2}{d}+1\right )+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}-\frac {1}{2} g p x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2352
Rule 2436
Rule 2441
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (g \log \left (c (d+e x)^p\right )+\frac {f \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )+\frac {1}{2} g \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e}-\frac {1}{2} (e f 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 p x^2+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f p \text {Li}_2\left (1+\frac {e x^2}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 80, normalized size = 0.98 \begin {gather*} \frac {1}{2} g \left (-p x^2+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )+\frac {1}{2} f \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 {d+e x^2}{d}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.16, size = 419, normalized size = 5.11
method | result | size |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g \,x^{2}}{2}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (x \right )-\frac {g p \,x^{2}}{2}+\frac {p g d \ln \left (e \,x^{2}+d \right )}{2 e}-p f \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )-p f \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )-p f \dilog \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )-p f \dilog \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )+\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f \ln \left (x \right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} f \ln \left (x \right )}{2}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3} f \ln \left (x \right )}{2}-\frac {i \pi g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{4}+\frac {i \pi g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4}+\frac {i \pi g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{4}-\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f \ln \left (x \right )}{2}-\frac {i \pi g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{4}+\frac {\ln \left (c \right ) g \,x^{2}}{2}+\ln \left (c \right ) f \ln \left (x \right )\) | \(419\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 95, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, {\left (\log \left (x^{2} e + d\right ) \log \left (-\frac {x^{2} e + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {x^{2} e + d}{d}\right )\right )} f p + f \log \left (c\right ) \log \left (x\right ) - \frac {1}{2} \, {\left ({\left (g p - g \log \left (c\right )\right )} x^{2} e - {\left (g p x^{2} e + d g p\right )} \log \left (x^{2} e + d\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+f\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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