Optimal. Leaf size=221 \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )-\frac {4 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {2 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}-\frac {2 d^2 g^2 p x}{5 e^2}+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {4 d f g p x}{3 e}+\frac {2 d g^2 p x^3}{15 e}-2 f^2 p x-\frac {4}{9} f g p x^3-\frac {2}{25} g^2 p x^5 \]
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Rubi [A] time = 0.16, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2471, 2448, 321, 205, 2455, 302} \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )-\frac {4 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {2 d^2 g^2 p x}{5 e^2}+\frac {2 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {4 d f g p x}{3 e}+\frac {2 d g^2 p x^3}{15 e}-2 f^2 p x-\frac {4}{9} f g p x^3-\frac {2}{25} g^2 p x^5 \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rule 321
Rule 2448
Rule 2455
Rule 2471
Rubi steps
\begin {align*} \int \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^2 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )-\left (2 e f^2 p\right ) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{3} (4 e f g p) \int \frac {x^4}{d+e x^2} \, dx-\frac {1}{5} \left (2 e g^2 p\right ) \int \frac {x^6}{d+e x^2} \, dx\\ &=-2 f^2 p x+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^2 p\right ) \int \frac {1}{d+e x^2} \, dx-\frac {1}{3} (4 e f g p) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac {1}{5} \left (2 e g^2 p\right ) \int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f^2 p x+\frac {4 d f g p x}{3 e}-\frac {2 d^2 g^2 p x}{5 e^2}-\frac {4}{9} f g p x^3+\frac {2 d g^2 p x^3}{15 e}-\frac {2}{25} g^2 p x^5+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )-\frac {\left (4 d^2 f g p\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}+\frac {\left (2 d^3 g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{5 e^2}\\ &=-2 f^2 p x+\frac {4 d f g p x}{3 e}-\frac {2 d^2 g^2 p x}{5 e^2}-\frac {4}{9} f g p x^3+\frac {2 d g^2 p x^3}{15 e}-\frac {2}{25} g^2 p x^5+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {4 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {2 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 151, normalized size = 0.68 \[ \frac {\sqrt {e} x \left (15 e^2 \left (15 f^2+10 f g x^2+3 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-2 p \left (45 d^2 g^2-15 d e g \left (10 f+g x^2\right )+e^2 \left (225 f^2+50 f g x^2+9 g^2 x^4\right )\right )\right )+30 \sqrt {d} p \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{225 e^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 404, normalized size = 1.83 \[ \left [-\frac {18 \, e^{2} g^{2} p x^{5} + 10 \, {\left (10 \, e^{2} f g - 3 \, d e g^{2}\right )} p x^{3} - 15 \, {\left (15 \, e^{2} f^{2} - 10 \, d e f g + 3 \, d^{2} g^{2}\right )} p \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 30 \, {\left (15 \, e^{2} f^{2} - 10 \, d e f g + 3 \, d^{2} g^{2}\right )} p x - 15 \, {\left (3 \, e^{2} g^{2} p x^{5} + 10 \, e^{2} f g p x^{3} + 15 \, e^{2} f^{2} p x\right )} \log \left (e x^{2} + d\right ) - 15 \, {\left (3 \, e^{2} g^{2} x^{5} + 10 \, e^{2} f g x^{3} + 15 \, e^{2} f^{2} x\right )} \log \relax (c)}{225 \, e^{2}}, -\frac {18 \, e^{2} g^{2} p x^{5} + 10 \, {\left (10 \, e^{2} f g - 3 \, d e g^{2}\right )} p x^{3} - 30 \, {\left (15 \, e^{2} f^{2} - 10 \, d e f g + 3 \, d^{2} g^{2}\right )} p \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 30 \, {\left (15 \, e^{2} f^{2} - 10 \, d e f g + 3 \, d^{2} g^{2}\right )} p x - 15 \, {\left (3 \, e^{2} g^{2} p x^{5} + 10 \, e^{2} f g p x^{3} + 15 \, e^{2} f^{2} p x\right )} \log \left (e x^{2} + d\right ) - 15 \, {\left (3 \, e^{2} g^{2} x^{5} + 10 \, e^{2} f g x^{3} + 15 \, e^{2} f^{2} x\right )} \log \relax (c)}{225 \, e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 201, normalized size = 0.91 \[ \frac {2 \, {\left (3 \, d^{3} g^{2} p - 10 \, d^{2} f g p e + 15 \, d f^{2} p e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )}}{15 \, \sqrt {d}} + \frac {1}{225} \, {\left (45 \, g^{2} p x^{5} e^{2} \log \left (x^{2} e + d\right ) - 18 \, g^{2} p x^{5} e^{2} + 45 \, g^{2} x^{5} e^{2} \log \relax (c) + 30 \, d g^{2} p x^{3} e + 150 \, f g p x^{3} e^{2} \log \left (x^{2} e + d\right ) - 100 \, f g p x^{3} e^{2} + 150 \, f g x^{3} e^{2} \log \relax (c) - 90 \, d^{2} g^{2} p x + 300 \, d f g p x e + 225 \, f^{2} p x e^{2} \log \left (x^{2} e + d\right ) - 450 \, f^{2} p x e^{2} + 225 \, f^{2} x e^{2} \log \relax (c)\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 686, normalized size = 3.10 \[ -\frac {i \pi \,g^{2} x^{5} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{10}-\frac {\sqrt {-d e}\, f^{2} p \ln \left (d +\sqrt {-d e}\, x \right )}{e}+\frac {\sqrt {-d e}\, f^{2} p \ln \left (d -\sqrt {-d e}\, x \right )}{e}-2 f^{2} p x -\frac {2 g^{2} p \,x^{5}}{25}+\frac {g^{2} x^{5} \ln \relax (c )}{5}+f^{2} x \ln \relax (c )+\frac {\sqrt {-d e}\, d^{2} g^{2} p \ln \left (d -\sqrt {-d e}\, x \right )}{5 e^{3}}+\left (\frac {1}{5} g^{2} x^{5}+\frac {2}{3} f g \,x^{3}+f^{2} x \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {\sqrt {-d e}\, d^{2} g^{2} p \ln \left (d +\sqrt {-d e}\, x \right )}{5 e^{3}}-\frac {i \pi f g \,x^{3} \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}+\frac {2 f g \,x^{3} \ln \relax (c )}{3}+\frac {i \pi \,f^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{2}+\frac {i \pi \,f^{2} x \,\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}}{2}-\frac {4 f g p \,x^{3}}{9}-\frac {2 d^{2} g^{2} p x}{5 e^{2}}+\frac {2 d \,g^{2} p \,x^{3}}{15 e}-\frac {i \pi \,g^{2} x^{5} \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 )}{10}+\frac {i \pi f g \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{3}+\frac {i \pi f g \,x^{3} \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}-\frac {i \pi \,f^{2} x \,\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 )}{2}-\frac {i \pi \,f^{2} x \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{2}+\frac {i \pi \,g^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{10}+\frac {i \pi \,g^{2} x^{5} \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}}{10}-\frac {i \pi f g \,x^{3} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{3}+\frac {4 d f g p x}{3 e}+\frac {2 \sqrt {-d e}\, d f g p \ln \left (d +\sqrt {-d e}\, x \right )}{3 e^{2}}-\frac {2 \sqrt {-d e}\, d f g p \ln \left (d -\sqrt {-d e}\, x \right )}{3 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 150, normalized size = 0.68 \[ \frac {2}{225} \, e p {\left (\frac {15 \, {\left (15 \, d e^{2} f^{2} - 10 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{3}} - \frac {9 \, e^{2} g^{2} x^{5} + 5 \, {\left (10 \, e^{2} f g - 3 \, d e g^{2}\right )} x^{3} + 15 \, {\left (15 \, e^{2} f^{2} - 10 \, d e f g + 3 \, d^{2} g^{2}\right )} x}{e^{3}}\right )} + \frac {1}{15} \, {\left (3 \, g^{2} x^{5} + 10 \, f g x^{3} + 15 \, f^{2} x\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 193, normalized size = 0.87 \[ \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (f^2\,x+\frac {2\,f\,g\,x^3}{3}+\frac {g^2\,x^5}{5}\right )-x\,\left (2\,f^2\,p-\frac {d\,\left (\frac {4\,f\,g\,p}{3}-\frac {2\,d\,g^2\,p}{5\,e}\right )}{e}\right )-x^3\,\left (\frac {4\,f\,g\,p}{9}-\frac {2\,d\,g^2\,p}{15\,e}\right )-\frac {2\,g^2\,p\,x^5}{25}+\frac {2\,\sqrt {d}\,p\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e}\,p\,x\,\left (3\,d^2\,g^2-10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{3\,p\,d^3\,g^2-10\,p\,d^2\,e\,f\,g+15\,p\,d\,e^2\,f^2}\right )\,\left (3\,d^2\,g^2-10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{15\,e^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 95.79, size = 415, normalized size = 1.88 \[ \begin {cases} \frac {i d^{\frac {5}{2}} g^{2} p \log {\left (d + e x^{2} \right )}}{5 e^{3} \sqrt {\frac {1}{e}}} - \frac {2 i d^{\frac {5}{2}} g^{2} p \log {\left (- i \sqrt {d} \sqrt {\frac {1}{e}} + x \right )}}{5 e^{3} \sqrt {\frac {1}{e}}} - \frac {2 i d^{\frac {3}{2}} f g p \log {\left (d + e x^{2} \right )}}{3 e^{2} \sqrt {\frac {1}{e}}} + \frac {4 i d^{\frac {3}{2}} f g p \log {\left (- i \sqrt {d} \sqrt {\frac {1}{e}} + x \right )}}{3 e^{2} \sqrt {\frac {1}{e}}} + \frac {i \sqrt {d} f^{2} p \log {\left (d + e x^{2} \right )}}{e \sqrt {\frac {1}{e}}} - \frac {2 i \sqrt {d} f^{2} p \log {\left (- i \sqrt {d} \sqrt {\frac {1}{e}} + x \right )}}{e \sqrt {\frac {1}{e}}} - \frac {2 d^{2} g^{2} p x}{5 e^{2}} + \frac {4 d f g p x}{3 e} + \frac {2 d g^{2} p x^{3}}{15 e} + f^{2} p x \log {\left (d + e x^{2} \right )} - 2 f^{2} p x + f^{2} x \log {\relax (c )} + \frac {2 f g p x^{3} \log {\left (d + e x^{2} \right )}}{3} - \frac {4 f g p x^{3}}{9} + \frac {2 f g x^{3} \log {\relax (c )}}{3} + \frac {g^{2} p x^{5} \log {\left (d + e x^{2} \right )}}{5} - \frac {2 g^{2} p x^{5}}{25} + \frac {g^{2} x^{5} \log {\relax (c )}}{5} & \text {for}\: e \neq 0 \\\left (f^{2} x + \frac {2 f g x^{3}}{3} + \frac {g^{2} x^{5}}{5}\right ) \log {\left (c d^{p} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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