Optimal. Leaf size=24 \[ a \tan ^{-1}(x)+\frac {b x^2}{2}-\frac {1}{2} b \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1810, 635, 203, 260} \[ a \tan ^{-1}(x)+\frac {b x^2}{2}-\frac {1}{2} b \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1810
Rubi steps
\begin {align*} \int \frac {a+b x^3}{1+x^2} \, dx &=\int \left (b x+\frac {a-b x}{1+x^2}\right ) \, dx\\ &=\frac {b x^2}{2}+\int \frac {a-b x}{1+x^2} \, dx\\ &=\frac {b x^2}{2}+a \int \frac {1}{1+x^2} \, dx-b \int \frac {x}{1+x^2} \, dx\\ &=\frac {b x^2}{2}+a \tan ^{-1}(x)-\frac {1}{2} b \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.92 \[ a \tan ^{-1}(x)+\frac {1}{2} b \left (x^2-\log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 20, normalized size = 0.83 \[ \frac {1}{2} \, b x^{2} + a \arctan \relax (x) - \frac {1}{2} \, b \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 20, normalized size = 0.83 \[ \frac {1}{2} \, b x^{2} + a \arctan \relax (x) - \frac {1}{2} \, b \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.88 \[ \frac {b \,x^{2}}{2}+a \arctan \relax (x )-\frac {b \ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.25, size = 20, normalized size = 0.83 \[ \frac {1}{2} \, b x^{2} + a \arctan \relax (x) - \frac {1}{2} \, b \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 20, normalized size = 0.83 \[ \frac {b\,x^2}{2}-\frac {b\,\ln \left (x^2+1\right )}{2}+a\,\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.17, size = 34, normalized size = 1.42 \[ \frac {b x^{2}}{2} + \left (- \frac {i a}{2} - \frac {b}{2}\right ) \log {\left (x - i \right )} + \left (\frac {i a}{2} - \frac {b}{2}\right ) \log {\left (x + i \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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