Optimal. Leaf size=188 \[ \frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {f^2 p \text {Li}_2\left (-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{2 g^3}+\frac {d p x^2}{4 e g}+\frac {f p x^2}{2 g^2}-\frac {p x^4}{8 g} \]
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Rubi [A] time = 0.28, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2475, 43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ \frac {f^2 p \text {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3}+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {d p x^2}{4 e g}+\frac {f p x^2}{2 g^2}-\frac {p x^4}{8 g} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 \log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {f \log \left (c (d+e x)^p\right )}{g^2}+\frac {x \log \left (c (d+e x)^p\right )}{g}+\frac {f^2 \log \left (c (d+e x)^p\right )}{g^2 (f+g x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {f \operatorname {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g^2}+\frac {f^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 g^2}+\frac {\operatorname {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g}\\ &=\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}+\frac {f^2 \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 g^3}-\frac {f \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e g^2}-\frac {\left (e f^2 p\right ) \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 g^3}-\frac {(e p) \operatorname {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right )}{4 g}\\ &=\frac {f p x^2}{2 g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {f^2 \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 g^3}-\frac {\left (f^2 p\right ) \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 g^3}-\frac {(e p) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )}{4 g}\\ &=\frac {f p x^2}{2 g^2}+\frac {d p x^2}{4 e g}-\frac {p x^4}{8 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {f^2 \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 g^3}+\frac {f^2 p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 143, normalized size = 0.76 \[ \frac {e \log \left (c \left (d+e x^2\right )^p\right ) \left (4 e f^2 \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+2 g \left (-2 d f-2 e f x^2+e g x^4\right )\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+4 e^2 f^2 p \text {Li}_2\left (\frac {g \left (e x^2+d\right )}{d g-e f}\right )+e g p x^2 \left (2 d g+4 e f-e g x^2\right )}{8 e^2 g^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.12, size = 902, normalized size = 4.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 182, normalized size = 0.97 \[ \frac {{\left (\log \left (e x^{2} + d\right ) \log \left (\frac {e g x^{2} + d g}{e f - d g} + 1\right ) + {\rm Li}_2\left (-\frac {e g x^{2} + d g}{e f - d g}\right )\right )} f^{2} p}{2 \, g^{3}} + \frac {f^{2} \log \left (g x^{2} + f\right ) \log \relax (c)}{2 \, g^{3}} - \frac {{\left (e^{2} g p - 2 \, e^{2} g \log \relax (c)\right )} x^{4} - 2 \, {\left (2 \, e^{2} f p + d e g p - 2 \, e^{2} f \log \relax (c)\right )} x^{2} - 2 \, {\left (e^{2} g p x^{4} - 2 \, e^{2} f p x^{2} - 2 \, d e f p - d^{2} g p\right )} \log \left (e x^{2} + d\right )}{8 \, e^{2} g^{2}} \]
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 \[ \int \frac {x^5\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]
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 \[ \int \frac {x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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