Optimal. Leaf size=178 \[ -4 f g p x+\frac {2 d g^2 p x}{3 e}-\frac {2}{9} g^2 p x^3+\frac {2 \sqrt {e} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{3/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2526, 2498,
327, 211, 2505, 308} \begin {gather*} -\frac {2 d^{3/2} g^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {2 \sqrt {e} f^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d g^2 p x}{3 e}-4 f g p x-\frac {2}{9} g^2 p x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 308
Rule 327
Rule 2498
Rule 2505
Rule 2526
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx &=\int \left (2 f g \log \left (c \left (d+e x^2\right )^p\right )+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+g^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx+(2 f g) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 e f^2 p\right ) \int \frac {1}{d+e x^2} \, dx-(4 e f g p) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{3} \left (2 e g^2 p\right ) \int \frac {x^4}{d+e x^2} \, dx\\ &=-4 f g p x+\frac {2 \sqrt {e} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+(4 d f g p) \int \frac {1}{d+e x^2} \, dx-\frac {1}{3} \left (2 e g^2 p\right ) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-4 f g p x+\frac {2 d g^2 p x}{3 e}-\frac {2}{9} g^2 p x^3+\frac {2 \sqrt {e} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {\left (2 d^2 g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}\\ &=-4 f g p x+\frac {2 d g^2 p x}{3 e}-\frac {2}{9} g^2 p x^3+\frac {2 \sqrt {e} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{3/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 112, normalized size = 0.63 \begin {gather*} \frac {1}{9} \left (-\frac {2 g p x \left (18 e f-3 d g+e g x^2\right )}{e}+\frac {6 \left (3 e^2 f^2+6 d e f g-d^2 g^2\right ) p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}+\left (-\frac {9 f^2}{x}+18 f g x+3 g^2 x^3\right ) \log \left (c \left (d+e x^2\right )^p\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.72, size = 702, normalized size = 3.94
method | result | size |
risch | \(-\frac {\left (-g^{2} x^{4}-6 f g \,x^{2}+3 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x}+\frac {3 i \pi e \,g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-3 i \pi e \,g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+3 i \pi e \,g^{2} 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}-3 i \pi e \,g^{2} 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 ) \mathrm {csgn}\left (i c \right )-18 i \pi e f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+18 i \pi e f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-9 i \pi e \,f^{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}+9 i \pi e \,f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-18 i \pi e f 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 ) \mathrm {csgn}\left (i c \right )-9 i \pi e \,f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+18 i \pi e f 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}+9 i \pi e \,f^{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 ) \mathrm {csgn}\left (i c \right )+6 \ln \left (c \right ) e \,g^{2} x^{4}-4 e \,g^{2} p \,x^{4}+36 \ln \left (c \right ) e f g \,x^{2}+12 d \,g^{2} p \,x^{2}-72 e f g p \,x^{2}-18 \ln \left (c \right ) e \,f^{2}+6 \left (\munderset {\textit {\_R} =\RootOf \left (g^{4} p^{2} d^{4}-12 e f \,p^{2} g^{3} d^{3}+30 d^{2} e^{2} f^{2} g^{2} p^{2}+36 d \,e^{3} f^{3} g \,p^{2}+9 e^{4} f^{4} p^{2}+d \,\textit {\_Z}^{2} e \right )}{\sum }\textit {\_R} \ln \left (\left (2 g^{4} p^{2} d^{4}-24 e f \,p^{2} g^{3} d^{3}+60 d^{2} e^{2} f^{2} g^{2} p^{2}+72 d \,e^{3} f^{3} g \,p^{2}+18 e^{4} f^{4} p^{2}+3 \textit {\_R}^{2} d e \right ) x +\left (d^{3} g^{2} p -6 d^{2} f g p e -3 d \,f^{2} e^{2} p \right ) \textit {\_R} \right )\right ) x}{18 e x}\) | \(702\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 108, normalized size = 0.61 \begin {gather*} -\frac {2}{9} \, {\left (\frac {3 \, {\left (d^{2} g^{2} - 6 \, d f g e - 3 \, f^{2} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )}}{\sqrt {d}} + {\left (g^{2} x^{3} e - 3 \, {\left (d g^{2} - 6 \, f g e\right )} x\right )} e^{\left (-2\right )}\right )} p e + \frac {1}{3} \, {\left (g^{2} x^{3} + 6 \, f g x - \frac {3 \, f^{2}}{x}\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 345, normalized size = 1.94 \begin {gather*} \left [\frac {{\left (6 \, d^{2} g^{2} p x^{2} e + 3 \, {\left (d g^{2} p x^{4} + 6 \, d f g p x^{2} - 3 \, d f^{2} p\right )} e^{2} \log \left (x^{2} e + d\right ) + 3 \, {\left (d g^{2} x^{4} + 6 \, d f g x^{2} - 3 \, d f^{2}\right )} e^{2} \log \left (c\right ) + 3 \, {\left (d^{2} g^{2} p x - 6 \, d f g p x e - 3 \, f^{2} p x e^{2}\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) - 2 \, {\left (d g^{2} p x^{4} + 18 \, d f g p x^{2}\right )} e^{2}\right )} e^{\left (-2\right )}}{9 \, d x}, \frac {{\left (6 \, d^{2} g^{2} p x^{2} e - 6 \, {\left (d^{2} g^{2} p x - 6 \, d f g p x e - 3 \, f^{2} p x e^{2}\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} + 3 \, {\left (d g^{2} p x^{4} + 6 \, d f g p x^{2} - 3 \, d f^{2} p\right )} e^{2} \log \left (x^{2} e + d\right ) + 3 \, {\left (d g^{2} x^{4} + 6 \, d f g x^{2} - 3 \, d f^{2}\right )} e^{2} \log \left (c\right ) - 2 \, {\left (d g^{2} p x^{4} + 18 \, d f g p x^{2}\right )} e^{2}\right )} e^{\left (-2\right )}}{9 \, d x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs.
\(2 (182) = 364\).
time = 65.89, size = 400, normalized size = 2.25 \begin {gather*} \begin {cases} \left (- \frac {f^{2}}{x} + 2 f g x + \frac {g^{2} x^{3}}{3}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\- \frac {2 f^{2} p}{x} - \frac {f^{2} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{x} - 4 f g p x + 2 f g x \log {\left (c \left (e x^{2}\right )^{p} \right )} - \frac {2 g^{2} p x^{3}}{9} + \frac {g^{2} x^{3} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{3} & \text {for}\: d = 0 \\\left (- \frac {f^{2}}{x} + 2 f g x + \frac {g^{2} x^{3}}{3}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 d^{2} g^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{3 e^{2} \sqrt {- \frac {d}{e}}} + \frac {d^{2} g^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 e^{2} \sqrt {- \frac {d}{e}}} + \frac {4 d f g p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{e \sqrt {- \frac {d}{e}}} - \frac {2 d f g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{e \sqrt {- \frac {d}{e}}} + \frac {2 d g^{2} p x}{3 e} + \frac {2 f^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x} - 4 f g p x + 2 f g x \log {\left (c \left (d + e x^{2}\right )^{p} \right )} - \frac {2 g^{2} p x^{3}}{9} + \frac {g^{2} x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.25, size = 168, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (d^{2} g^{2} p - 6 \, d f g p e - 3 \, f^{2} p e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {3}{2}\right )}}{3 \, \sqrt {d}} + \frac {{\left (3 \, g^{2} p x^{4} e \log \left (x^{2} e + d\right ) - 2 \, g^{2} p x^{4} e + 3 \, g^{2} x^{4} e \log \left (c\right ) + 18 \, f g p x^{2} e \log \left (x^{2} e + d\right ) + 6 \, d g^{2} p x^{2} - 36 \, f g p x^{2} e + 18 \, f g x^{2} e \log \left (c\right ) - 9 \, f^{2} p e \log \left (x^{2} e + d\right ) - 9 \, f^{2} e \log \left (c\right )\right )} e^{\left (-1\right )}}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 180, normalized size = 1.01 \begin {gather*} \frac {2\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,p\,x\,\left (-d^2\,g^2+6\,d\,e\,f\,g+3\,e^2\,f^2\right )}{\sqrt {d}\,\left (-p\,d^2\,g^2+6\,p\,d\,e\,f\,g+3\,p\,e^2\,f^2\right )}\right )\,\left (-d^2\,g^2+6\,d\,e\,f\,g+3\,e^2\,f^2\right )}{3\,\sqrt {d}\,e^{3/2}}-x\,\left (4\,f\,g\,p-\frac {2\,d\,g^2\,p}{3\,e}\right )-\frac {2\,g^2\,p\,x^3}{9}-\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2+2\,f\,g\,x^2+g^2\,x^4}{x}-\frac {\frac {4\,g^2\,x^4}{3}+4\,f\,g\,x^2}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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