Optimal. Leaf size=44 \[ -\frac {2}{3} a^3 \tan ^{-1}\left (\frac {x}{a}\right )+\frac {1}{3} x^3 \log \left (a^2+x^2\right )+\frac {2 a^2 x}{3}-\frac {2 x^3}{9} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2455, 302, 203} \[ \frac {1}{3} x^3 \log \left (a^2+x^2\right )+\frac {2 a^2 x}{3}-\frac {2}{3} a^3 \tan ^{-1}\left (\frac {x}{a}\right )-\frac {2 x^3}{9} \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 2455
Rubi steps
\begin {align*} \int x^2 \log \left (a^2+x^2\right ) \, dx &=\frac {1}{3} x^3 \log \left (a^2+x^2\right )-\frac {2}{3} \int \frac {x^4}{a^2+x^2} \, dx\\ &=\frac {1}{3} x^3 \log \left (a^2+x^2\right )-\frac {2}{3} \int \left (-a^2+x^2+\frac {a^4}{a^2+x^2}\right ) \, dx\\ &=\frac {2 a^2 x}{3}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log \left (a^2+x^2\right )-\frac {1}{3} \left (2 a^4\right ) \int \frac {1}{a^2+x^2} \, dx\\ &=\frac {2 a^2 x}{3}-\frac {2 x^3}{9}-\frac {2}{3} a^3 \tan ^{-1}\left (\frac {x}{a}\right )+\frac {1}{3} x^3 \log \left (a^2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 44, normalized size = 1.00 \[ -\frac {2}{3} a^3 \tan ^{-1}\left (\frac {x}{a}\right )+\frac {1}{3} x^3 \log \left (a^2+x^2\right )+\frac {2 a^2 x}{3}-\frac {2 x^3}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 36, normalized size = 0.82 \[ -\frac {2}{3} \, a^{3} \arctan \left (\frac {x}{a}\right ) + \frac {1}{3} \, x^{3} \log \left (a^{2} + x^{2}\right ) + \frac {2}{3} \, a^{2} x - \frac {2}{9} \, x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 36, normalized size = 0.82 \[ -\frac {2}{3} \, a^{3} \arctan \left (\frac {x}{a}\right ) + \frac {1}{3} \, x^{3} \log \left (a^{2} + x^{2}\right ) + \frac {2}{3} \, a^{2} x - \frac {2}{9} \, x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.84 \[ -\frac {2 a^{3} \arctan \left (\frac {x}{a}\right )}{3}+\frac {x^{3} \ln \left (a^{2}+x^{2}\right )}{3}+\frac {2 a^{2} x}{3}-\frac {2 x^{3}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 36, normalized size = 0.82 \[ -\frac {2}{3} \, a^{3} \arctan \left (\frac {x}{a}\right ) + \frac {1}{3} \, x^{3} \log \left (a^{2} + x^{2}\right ) + \frac {2}{3} \, a^{2} x - \frac {2}{9} \, x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 65, normalized size = 1.48 \[ \frac {2\,a^2\,x}{3}-\frac {\ln \left (x-\sqrt {-a^2}\right )\,{\left (-a^2\right )}^{3/2}}{3}+\frac {x^3\,\ln \left (a^2+x^2\right )}{3}+\frac {\ln \left (x+\sqrt {-a^2}\right )\,{\left (-a^2\right )}^{3/2}}{3}-\frac {2\,x^3}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.19, size = 53, normalized size = 1.20 \[ - 2 a^{3} \left (- \frac {i \log {\left (- i a + x \right )}}{6} + \frac {i \log {\left (i a + x \right )}}{6}\right ) + \frac {2 a^{2} x}{3} + \frac {x^{3} \log {\left (a^{2} + x^{2} \right )}}{3} - \frac {2 x^{3}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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