Optimal. Leaf size=37 \[ \frac {\log (a+b x)}{b}-\frac {2 \tan ^{-1}\left (\frac {a-2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1868, 31, 617, 204} \[ \frac {\log (a+b x)}{b}-\frac {2 \tan ^{-1}\left (\frac {a-2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 617
Rule 1868
Rubi steps
\begin {align*} \int \frac {2 a^2+b^2 x^2}{a^3+b^3 x^3} \, dx &=\frac {a \int \frac {1}{\frac {a^2}{b^2}-\frac {a x}{b}+x^2} \, dx}{b^2}+\frac {\int \frac {1}{\frac {a}{b}+x} \, dx}{b}\\ &=\frac {\log (a+b x)}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 b x}{a}\right )}{b}\\ &=-\frac {2 \tan ^{-1}\left (\frac {a-2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b}+\frac {\log (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 1.95 \[ \frac {\log \left (a^3+b^3 x^3\right )-\log \left (a^2-a b x+b^2 x^2\right )+2 \log (a+b x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 b x-a}{\sqrt {3} a}\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 36, normalized size = 0.97 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b x - a\right )}}{3 \, a}\right ) + 3 \, \log \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 37, normalized size = 1.00 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b x - a\right )}}{3 \, a}\right )}{3 \, b} + \frac {\log \left ({\left | b x + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 43, normalized size = 1.16 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x -a b \right ) \sqrt {3}}{3 a b}\right )}{3 b}+\frac {\ln \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 42, normalized size = 1.14 \[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b^{2} x - a b\right )}}{3 \, a b}\right )}{3 \, b} + \frac {\log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 84, normalized size = 2.27 \[ \frac {\ln \left (a+b\,x\right )}{b}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {4\,\sqrt {3}\,a^3\,b^4}{4\,a^3\,b^4+4\,x\,a^2\,b^5}-\frac {4\,\sqrt {3}\,a^2\,b^5\,x}{4\,a^3\,b^4+4\,x\,a^2\,b^5}\right )}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.50, size = 60, normalized size = 1.62 \[ \frac {- \frac {\sqrt {3} i \log {\left (x + \frac {- a - \sqrt {3} i a}{2 b} \right )}}{3} + \frac {\sqrt {3} i \log {\left (x + \frac {- a + \sqrt {3} i a}{2 b} \right )}}{3} + \log {\left (\frac {a}{b} + x \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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