Optimal. Leaf size=28 \[ -\frac {a \log \left (a-b x^2\right )}{2 b^2}-\frac {x^2}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {266, 43} \[ -\frac {a \log \left (a-b x^2\right )}{2 b^2}-\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{a-b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a-b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{b}-\frac {a}{b (-a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^2}{2 b}-\frac {a \log \left (a-b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ -\frac {a \log \left (a-b x^2\right )}{2 b^2}-\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 23, normalized size = 0.82 \[ -\frac {b x^{2} + a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 26, normalized size = 0.93 \[ -\frac {x^{2}}{2 \, b} - \frac {a \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.93 \[ -\frac {x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}-a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 25, normalized size = 0.89 \[ -\frac {x^{2}}{2 \, b} - \frac {a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 23, normalized size = 0.82 \[ -\frac {b\,x^2+a\,\ln \left (b\,x^2-a\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 22, normalized size = 0.79 \[ - \frac {a \log {\left (- a + b x^{2} \right )}}{2 b^{2}} - \frac {x^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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