Optimal. Leaf size=108 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {2 e^{3/2} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {2 e f p}{3 d x}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}} \]
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Rubi [A] time = 0.10, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2476, 2455, 325, 205} \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {2 e^{3/2} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}-\frac {2 e f p}{3 d x}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 2455
Rule 2476
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx &=\int \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^4}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right ) \, dx\\ &=f \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx+g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx\\ &=-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {1}{3} (2 e f p) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx+(2 e g p) \int \frac {1}{d+e x^2} \, dx\\ &=-\frac {2 e f p}{3 d x}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {\left (2 e^2 f p\right ) \int \frac {1}{d+e x^2} \, dx}{3 d}\\ &=-\frac {2 e f p}{3 d x}-\frac {2 e^{3/2} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 96, normalized size = 0.89 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {2 e f p \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {e x^2}{d}\right )}{3 d x}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 191, normalized size = 1.77 \[ \left [-\frac {{\left (e f - 3 \, d g\right )} p x^{3} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} + 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \relax (c)}{3 \, d x^{3}}, -\frac {2 \, {\left (e f - 3 \, d g\right )} p x^{3} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \relax (c)}{3 \, d x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 92, normalized size = 0.85 \[ \frac {2 \, {\left (3 \, d g p e - f p e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{3 \, d^{\frac {3}{2}}} - \frac {3 \, d g p x^{2} \log \left (x^{2} e + d\right ) + 2 \, f p x^{2} e + 3 \, d g x^{2} \log \relax (c) + d f p \log \left (x^{2} e + d\right ) + d f \log \relax (c)}{3 \, d x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.66, size = 430, normalized size = 3.98 \[ -\frac {\left (3 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}+\frac {3 i \pi d g \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )-3 i \pi d g \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-3 i \pi d g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+3 i \pi d g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+i \pi d f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )-i \pi d f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-i \pi d f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+i \pi d f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-6 d g \,x^{2} \ln \relax (c )+2 d \,x^{3} \RootOf \left (9 d^{2} e \,g^{2} p^{2}-6 d \,e^{2} f g \,p^{2}+e^{3} f^{2} p^{2}+d^{3} \textit {\_Z}^{2}\right ) \ln \left (\left (-3 d^{3} g p +d^{2} e f p \right ) \RootOf \left (9 d^{2} e \,g^{2} p^{2}-6 d \,e^{2} f g \,p^{2}+e^{3} f^{2} p^{2}+d^{3} \textit {\_Z}^{2}\right )+\left (18 d^{2} e \,g^{2} p^{2}-12 d \,e^{2} f g \,p^{2}+2 e^{3} f^{2} p^{2}+3 \RootOf \left (9 d^{2} e \,g^{2} p^{2}-6 d \,e^{2} f g \,p^{2}+e^{3} f^{2} p^{2}+d^{3} \textit {\_Z}^{2}\right )^{2} d^{3}\right ) x \right )-4 e f p \,x^{2}-2 d f \ln \relax (c )}{6 d \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 65, normalized size = 0.60 \[ -\frac {2}{3} \, e p {\left (\frac {{\left (e f - 3 \, d g\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d} + \frac {f}{d x}\right )} - \frac {{\left (3 \, g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 65, normalized size = 0.60 \[ \frac {2\,\sqrt {e}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,d\,g-e\,f\right )}{3\,d^{3/2}}-\frac {2\,e\,f\,p}{3\,d\,x}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+\frac {f}{3}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 105.07, size = 1454, normalized size = 13.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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