Optimal. Leaf size=155 \[ -\frac {e f p \log \left (d+e x^2\right )}{2 g^2 (e f-d g)}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {e f p \log \left (f+g x^2\right )}{2 g^2 (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2525, 45,
2463, 2442, 36, 31, 2441, 2440, 2438} \begin {gather*} \frac {p \text {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\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^2}-\frac {e f p \log \left (d+e x^2\right )}{2 g^2 (e f-d g)}+\frac {e f p \log \left (f+g x^2\right )}{2 g^2 (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 45
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {f \log \left (c (d+e x)^p\right )}{g (f+g x)^2}+\frac {\log \left (c (d+e x)^p\right )}{g (f+g x)}\right ) \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 g}-\frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right )}{2 g}\\ &=\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\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^2}-\frac {(e p) \text {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^2}-\frac {(e f p) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^2\right )}{2 g^2}\\ &=\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\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^2}-\frac {p \text {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^2}-\frac {\left (e^2 f p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 g^2 (e f-d g)}+\frac {(e f p) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^2\right )}{2 g (e f-d g)}\\ &=-\frac {e f p \log \left (d+e x^2\right )}{2 g^2 (e f-d g)}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 g^2 \left (f+g x^2\right )}+\frac {e f p \log \left (f+g x^2\right )}{2 g^2 (e f-d g)}+\frac {\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^2}+\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 131, normalized size = 0.85 \begin {gather*} \frac {\frac {e f p \log \left (d+e x^2\right )}{-e f+d g}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}+\frac {e f p \log \left (f+g x^2\right )}{e f-d 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 )+p \text {Li}_2\left (\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{2 g^2} \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 = 0.46, size = 732, normalized size = 4.72
method | result | size |
risch | \(\text {Expression too large to display}\) | \(732\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 193, normalized size = 1.25 \begin {gather*} \frac {{\left (d g \log \left (c\right ) - {\left (f p + f \log \left (c\right )\right )} e\right )} \log \left (g x^{2} + f\right )}{2 \, {\left (d g^{3} - f g^{2} e\right )}} + \frac {d f g \log \left (c\right ) - f^{2} e \log \left (c\right ) + {\left (f g p x^{2} e + d f g p\right )} \log \left (x^{2} e + d\right )}{2 \, {\left (d f g^{3} - f^{2} g^{2} e + {\left (d g^{4} - f g^{3} e\right )} x^{2}\right )}} + \frac {{\left (\log \left (x^{2} e + d\right ) \log \left (-\frac {g x^{2} e + d g}{d g - f e} + 1\right ) + {\rm Li}_2\left (\frac {g x^{2} e + d g}{d g - f e}\right )\right )} p}{2 \, g^{2}} \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {x^3\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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