Optimal. Leaf size=94 \[ \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {p (e f-d g)^2 \log \left (d+e x^2\right )}{4 e^2 g}-\frac {p x^2 (e f-d g)}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g} \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2475, 2395, 43} \[ \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {p (e f-d g)^2 \log \left (d+e x^2\right )}{4 e^2 g}-\frac {p x^2 (e f-d g)}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 2475
Rubi steps
\begin {align*} \int x \left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (f+g x) \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {(e p) \operatorname {Subst}\left (\int \frac {(f+g x)^2}{d+e x} \, dx,x,x^2\right )}{4 g}\\ &=\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {(e p) \operatorname {Subst}\left (\int \left (\frac {g (e f-d g)}{e^2}+\frac {(e f-d g)^2}{e^2 (d+e x)}+\frac {g (f+g x)}{e}\right ) \, dx,x,x^2\right )}{4 g}\\ &=-\frac {(e f-d g) p x^2}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g}-\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 98, normalized size = 1.04 \[ \frac {1}{2} f \left (\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}-p x^2\right )+\frac {1}{4} g x^4 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 g p \log \left (d+e x^2\right )}{4 e^2}+\frac {d g p x^2}{4 e}-\frac {1}{8} g p x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 99, normalized size = 1.05 \[ -\frac {e^{2} g p x^{4} + 2 \, {\left (2 \, e^{2} f - d e g\right )} p x^{2} - 2 \, {\left (e^{2} g p x^{4} + 2 \, e^{2} f p x^{2} + {\left (2 \, d e f - d^{2} g\right )} p\right )} \log \left (e x^{2} + d\right ) - 2 \, {\left (e^{2} g x^{4} + 2 \, e^{2} f x^{2}\right )} \log \relax (c)}{8 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 148, normalized size = 1.57 \[ \frac {1}{8} \, {\left ({\left (2 \, {\left (x^{2} e + d\right )}^{2} \log \left (x^{2} e + d\right ) - 4 \, {\left (x^{2} e + d\right )} d \log \left (x^{2} e + d\right ) - {\left (x^{2} e + d\right )}^{2} + 4 \, {\left (x^{2} e + d\right )} d\right )} g p e^{\left (-1\right )} + 2 \, {\left ({\left (x^{2} e + d\right )}^{2} - 2 \, {\left (x^{2} e + d\right )} d\right )} g e^{\left (-1\right )} \log \relax (c) - 4 \, {\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} f p + 4 \, {\left (x^{2} e + d\right )} f \log \relax (c)\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 361, normalized size = 3.84 \[ -\frac {i \pi g \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{8}+\frac {i \pi g \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{8}+\frac {i \pi g \,x^{4} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{8}-\frac {i \pi g \,x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{8}-\frac {i \pi f \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{4}+\frac {i \pi f \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4}+\frac {i \pi f \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4}-\frac {i \pi f \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{4}-\frac {g p \,x^{4}}{8}+\frac {g \,x^{4} \ln \relax (c )}{4}+\frac {d g p \,x^{2}}{4 e}-\frac {f p \,x^{2}}{2}+\frac {f \,x^{2} \ln \relax (c )}{2}-\frac {d^{2} g p \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {d f p \ln \left (e \,x^{2}+d \right )}{2 e}+\left (\frac {1}{4} g \,x^{4}+\frac {1}{2} f \,x^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 99, normalized size = 1.05 \[ -\frac {e p {\left (\frac {e g^{2} x^{4} + 2 \, {\left (2 \, e f g - d g^{2}\right )} x^{2}}{e^{2}} + \frac {2 \, {\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{3}}\right )}}{8 \, g} + \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{4 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 78, normalized size = 0.83 \[ \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^4}{4}+\frac {f\,x^2}{2}\right )-x^2\,\left (\frac {f\,p}{2}-\frac {d\,g\,p}{4\,e}\right )-\frac {g\,p\,x^4}{8}-\frac {\ln \left (e\,x^2+d\right )\,\left (d^2\,g\,p-2\,d\,e\,f\,p\right )}{4\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 44.85, size = 139, normalized size = 1.48 \[ \begin {cases} - \frac {d^{2} g p \log {\left (d + e x^{2} \right )}}{4 e^{2}} + \frac {d f p \log {\left (d + e x^{2} \right )}}{2 e} + \frac {d g p x^{2}}{4 e} + \frac {f p x^{2} \log {\left (d + e x^{2} \right )}}{2} - \frac {f p x^{2}}{2} + \frac {f x^{2} \log {\relax (c )}}{2} + \frac {g p x^{4} \log {\left (d + e x^{2} \right )}}{4} - \frac {g p x^{4}}{8} + \frac {g x^{4} \log {\relax (c )}}{4} & \text {for}\: e \neq 0 \\\left (\frac {f x^{2}}{2} + \frac {g x^{4}}{4}\right ) \log {\left (c d^{p} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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