Optimal. Leaf size=27 \[ \frac {x^2}{2 a}-\frac {b \log \left (a x^2+b\right )}{2 a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {x^2}{2 a}-\frac {b \log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{b+a x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{b+a x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a}-\frac {b \log \left (b+a x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {x^2}{2 a}-\frac {b \log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 22, normalized size = 0.81 \[ \frac {a x^{2} - b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 24, normalized size = 0.89 \[ \frac {x^{2}}{2 \, a} - \frac {b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ \frac {x^{2}}{2 a}-\frac {b \ln \left (a \,x^{2}+b \right )}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 23, normalized size = 0.85 \[ \frac {x^{2}}{2 \, a} - \frac {b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 22, normalized size = 0.81 \[ -\frac {b\,\ln \left (a\,x^2+b\right )-a\,x^2}{2\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.74 \[ \frac {x^{2}}{2 a} - \frac {b \log {\left (a x^{2} + b \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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