Optimal. Leaf size=278 \[ \frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 d^{3/2} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {4 d^{5/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {4 d^2 f g p x}{5 e^2}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 d f^2 p x}{3 e}+\frac {4 d f g p x^3}{15 e}+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{9} f^2 p x^3-\frac {4}{25} f g p x^5-\frac {2}{49} g^2 p x^7 \]
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Rubi [A] time = 0.24, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2476, 2455, 302, 205} \[ \frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 d^{3/2} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {4 d^2 f g p x}{5 e^2}+\frac {4 d^{5/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 d f^2 p x}{3 e}+\frac {4 d f g p x^3}{15 e}+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{9} f^2 p x^3-\frac {4}{25} f g p x^5-\frac {2}{49} g^2 p x^7 \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rule 2455
Rule 2476
Rubi steps
\begin {align*} \int x^2 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=\frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{3} \left (2 e f^2 p\right ) \int \frac {x^4}{d+e x^2} \, dx-\frac {1}{5} (4 e f g p) \int \frac {x^6}{d+e x^2} \, dx-\frac {1}{7} \left (2 e g^2 p\right ) \int \frac {x^8}{d+e x^2} \, dx\\ &=\frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{3} \left (2 e f^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-\frac {1}{5} (4 e f g p) \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-\frac {1}{7} \left (2 e g^2 p\right ) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {2 d f^2 p x}{3 e}-\frac {4 d^2 f g p x}{5 e^2}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2}{9} f^2 p x^3+\frac {4 d f g p x^3}{15 e}-\frac {2 d^2 g^2 p x^3}{21 e^2}-\frac {4}{25} f g p x^5+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {\left (2 d^2 f^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}+\frac {\left (4 d^3 f g p\right ) \int \frac {1}{d+e x^2} \, dx}{5 e^2}-\frac {\left (2 d^4 g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}\\ &=\frac {2 d f^2 p x}{3 e}-\frac {4 d^2 f g p x}{5 e^2}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2}{9} f^2 p x^3+\frac {4 d f g p x^3}{15 e}-\frac {2 d^2 g^2 p x^3}{21 e^2}-\frac {4}{25} f g p x^5+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7-\frac {2 d^{3/2} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {4 d^{5/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 188, normalized size = 0.68 \[ \frac {\sqrt {e} x \left (105 e^3 x^2 \left (35 f^2+42 f g x^2+15 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+2 p \left (1575 d^3 g^2-105 d^2 e g \left (42 f+5 g x^2\right )+105 d e^2 \left (35 f^2+14 f g x^2+3 g^2 x^4\right )-e^3 x^2 \left (1225 f^2+882 f g x^2+225 g^2 x^4\right )\right )\right )-210 d^{3/2} p \left (15 d^2 g^2-42 d e f g+35 e^2 f^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{11025 e^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 492, normalized size = 1.77 \[ \left [-\frac {450 \, e^{3} g^{2} p x^{7} + 126 \, {\left (14 \, e^{3} f g - 5 \, d e^{2} g^{2}\right )} p x^{5} + 70 \, {\left (35 \, e^{3} f^{2} - 42 \, d e^{2} f g + 15 \, d^{2} e g^{2}\right )} p x^{3} - 105 \, {\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) - 210 \, {\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p x - 105 \, {\left (15 \, e^{3} g^{2} p x^{7} + 42 \, e^{3} f g p x^{5} + 35 \, e^{3} f^{2} p x^{3}\right )} \log \left (e x^{2} + d\right ) - 105 \, {\left (15 \, e^{3} g^{2} x^{7} + 42 \, e^{3} f g x^{5} + 35 \, e^{3} f^{2} x^{3}\right )} \log \relax (c)}{11025 \, e^{3}}, -\frac {450 \, e^{3} g^{2} p x^{7} + 126 \, {\left (14 \, e^{3} f g - 5 \, d e^{2} g^{2}\right )} p x^{5} + 70 \, {\left (35 \, e^{3} f^{2} - 42 \, d e^{2} f g + 15 \, d^{2} e g^{2}\right )} p x^{3} + 210 \, {\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) - 210 \, {\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p x - 105 \, {\left (15 \, e^{3} g^{2} p x^{7} + 42 \, e^{3} f g p x^{5} + 35 \, e^{3} f^{2} p x^{3}\right )} \log \left (e x^{2} + d\right ) - 105 \, {\left (15 \, e^{3} g^{2} x^{7} + 42 \, e^{3} f g x^{5} + 35 \, e^{3} f^{2} x^{3}\right )} \log \relax (c)}{11025 \, e^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 246, normalized size = 0.88 \[ -\frac {2 \, {\left (15 \, d^{4} g^{2} p - 42 \, d^{3} f g p e + 35 \, d^{2} f^{2} p e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {7}{2}\right )}}{105 \, \sqrt {d}} + \frac {1}{11025} \, {\left (1575 \, g^{2} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 450 \, g^{2} p x^{7} e^{3} + 1575 \, g^{2} x^{7} e^{3} \log \relax (c) + 630 \, d g^{2} p x^{5} e^{2} + 4410 \, f g p x^{5} e^{3} \log \left (x^{2} e + d\right ) - 1764 \, f g p x^{5} e^{3} - 1050 \, d^{2} g^{2} p x^{3} e + 4410 \, f g x^{5} e^{3} \log \relax (c) + 2940 \, d f g p x^{3} e^{2} + 3675 \, f^{2} p x^{3} e^{3} \log \left (x^{2} e + d\right ) + 3150 \, d^{3} g^{2} p x - 2450 \, f^{2} p x^{3} e^{3} - 8820 \, d^{2} f g p x e + 3675 \, f^{2} x^{3} e^{3} \log \relax (c) + 7350 \, d f^{2} p x e^{2}\right )} e^{\left (-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 761, normalized size = 2.74 \[ \frac {g^{2} x^{7} \ln \relax (c )}{7}+\frac {f^{2} x^{3} \ln \relax (c )}{3}+\frac {2 f g \,x^{5} \ln \relax (c )}{5}-\frac {2 g^{2} p \,x^{7}}{49}-\frac {2 f^{2} p \,x^{3}}{9}-\frac {\sqrt {-d e}\, d \,f^{2} p \ln \left (-d +\sqrt {-d e}\, x \right )}{3 e^{2}}+\frac {\sqrt {-d e}\, d^{3} g^{2} p \ln \left (-d -\sqrt {-d e}\, x \right )}{7 e^{4}}-\frac {\sqrt {-d e}\, d^{3} g^{2} p \ln \left (-d +\sqrt {-d e}\, x \right )}{7 e^{4}}+\frac {\sqrt {-d e}\, d \,f^{2} p \ln \left (-d -\sqrt {-d e}\, x \right )}{3 e^{2}}+\frac {i \pi f g \,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}}{5}-\frac {i \pi \,f^{2} 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 )}{6}-\frac {4 f g p \,x^{5}}{25}+\frac {2 d^{3} g^{2} p x}{7 e^{3}}+\left (\frac {1}{7} g^{2} x^{7}+\frac {2}{5} f g \,x^{5}+\frac {1}{3} f^{2} x^{3}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {2 d^{2} g^{2} p \,x^{3}}{21 e^{2}}+\frac {2 d \,g^{2} p \,x^{5}}{35 e}+\frac {2 \sqrt {-d e}\, d^{2} f g p \ln \left (-d +\sqrt {-d e}\, x \right )}{5 e^{3}}-\frac {2 \sqrt {-d e}\, d^{2} f g p \ln \left (-d -\sqrt {-d e}\, x \right )}{5 e^{3}}-\frac {i \pi f g \,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 )}{5}+\frac {i \pi \,g^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{14}+\frac {i \pi \,g^{2} x^{7} \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}}{14}-\frac {i \pi f g \,x^{5} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{5}+\frac {i \pi \,f^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{6}+\frac {i \pi \,f^{2} 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}}{6}+\frac {2 d \,f^{2} p x}{3 e}-\frac {i \pi \,f^{2} x^{3} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{6}-\frac {i \pi \,g^{2} x^{7} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{14}-\frac {i \pi \,g^{2} x^{7} \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 )}{14}+\frac {i \pi f g \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{5}-\frac {4 d^{2} f g p x}{5 e^{2}}+\frac {4 d f g p \,x^{3}}{15 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 189, normalized size = 0.68 \[ -\frac {2}{11025} \, e p {\left (\frac {105 \, {\left (35 \, d^{2} e^{2} f^{2} - 42 \, d^{3} e f g + 15 \, d^{4} g^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {225 \, e^{3} g^{2} x^{7} + 63 \, {\left (14 \, e^{3} f g - 5 \, d e^{2} g^{2}\right )} x^{5} + 35 \, {\left (35 \, e^{3} f^{2} - 42 \, d e^{2} f g + 15 \, d^{2} e g^{2}\right )} x^{3} - 105 \, {\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} x}{e^{4}}\right )} + \frac {1}{105} \, {\left (15 \, g^{2} x^{7} + 42 \, f g x^{5} + 35 \, f^{2} x^{3}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 235, normalized size = 0.85 \[ \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^3}{3}+\frac {2\,f\,g\,x^5}{5}+\frac {g^2\,x^7}{7}\right )-x^3\,\left (\frac {2\,f^2\,p}{9}-\frac {d\,\left (\frac {4\,f\,g\,p}{5}-\frac {2\,d\,g^2\,p}{7\,e}\right )}{3\,e}\right )-x^5\,\left (\frac {4\,f\,g\,p}{25}-\frac {2\,d\,g^2\,p}{35\,e}\right )-\frac {2\,g^2\,p\,x^7}{49}+\frac {d\,x\,\left (\frac {2\,f^2\,p}{3}-\frac {d\,\left (\frac {4\,f\,g\,p}{5}-\frac {2\,d\,g^2\,p}{7\,e}\right )}{e}\right )}{e}-\frac {2\,d^{3/2}\,p\,\mathrm {atan}\left (\frac {d^{3/2}\,\sqrt {e}\,p\,x\,\left (15\,d^2\,g^2-42\,d\,e\,f\,g+35\,e^2\,f^2\right )}{15\,p\,d^4\,g^2-42\,p\,d^3\,e\,f\,g+35\,p\,d^2\,e^2\,f^2}\right )\,\left (15\,d^2\,g^2-42\,d\,e\,f\,g+35\,e^2\,f^2\right )}{105\,e^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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