Optimal. Leaf size=176 \[ -\frac {g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}+\frac {g p \text {Li}_2\left (-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{2 f^2}-\frac {g p \text {Li}_2\left (\frac {e x^2}{d}+1\right )}{2 f^2}-\frac {e p \log \left (d+e x^2\right )}{2 d f}+\frac {e p \log (x)}{d f} \]
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Rubi [A] time = 0.27, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2475, 44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac {g p \text {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {g p \text {PolyLog}\left (2,\frac {e x^2}{d}+1\right )}{2 f^2}-\frac {g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac {e p \log \left (d+e x^2\right )}{2 d f}+\frac {e p \log (x)}{d f} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^3 \left (f+g x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2 (f+g x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{f x^2}-\frac {g \log \left (c (d+e x)^p\right )}{f^2 x}+\frac {g^2 \log \left (c (d+e x)^p\right )}{f^2 (f+g x)}\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^2\right )}{2 f}-\frac {g \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )}{2 f^2}+\frac {g^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 f^2}\\ &=-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {(e p) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^2\right )}{2 f}+\frac {(e g p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}-\frac {(e g p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}\\ &=-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {g p \text {Li}_2\left (1+\frac {e x^2}{d}\right )}{2 f^2}+\frac {(e p) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d f}-\frac {\left (e^2 p\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 d f}-\frac {(g p) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 f^2}\\ &=\frac {e p \log (x)}{d f}-\frac {e p \log \left (d+e x^2\right )}{2 d f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {g p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {g p \text {Li}_2\left (1+\frac {e x^2}{d}\right )}{2 f^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 147, normalized size = 0.84 \[ \frac {g \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^2}-g \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \text {Li}_2\left (\frac {e x^2}{d}+1\right )\right )+g p \text {Li}_2\left (\frac {g \left (e x^2+d\right )}{d g-e f}\right )+\frac {e f p \left (2 \log (x)-\log \left (d+e x^2\right )\right )}{d}}{2 f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{5} + f x^{3}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.68, size = 942, normalized size = 5.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 178, normalized size = 1.01 \[ \frac {1}{2} \, e p {\left (\frac {{\left (2 \, \log \left (\frac {e x^{2}}{d} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {e x^{2}}{d}\right )\right )} g}{e f^{2}} - \frac {{\left (\log \left (g x^{2} + f\right ) \log \left (-\frac {e g x^{2} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{2} + e f}{e f - d g}\right )\right )} g}{e f^{2}} - \frac {\log \left (e x^{2} + d\right )}{d f} + \frac {2 \, \log \relax (x)}{d f}\right )} + \frac {1}{2} \, {\left (\frac {g \log \left (g x^{2} + f\right )}{f^{2}} - \frac {g \log \left (x^{2}\right )}{f^{2}} - \frac {1}{f x^{2}}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
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 \[ \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x^3\,\left (g\,x^2+f\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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