Optimal. Leaf size=19 \[ \frac {x^4}{4 a \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \[ \frac {x^4}{4 a \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^3} \, dx &=\frac {x^4}{4 a \left (a+b x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.26 \[ -\frac {a+2 b x^2}{4 b^2 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 36, normalized size = 1.89 \[ -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 22, normalized size = 1.16 \[ -\frac {2 \, b x^{2} + a}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.63 \[ \frac {a}{4 \left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {1}{2 \left (b \,x^{2}+a \right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 36, normalized size = 1.89 \[ -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 37, normalized size = 1.95 \[ -\frac {\frac {a}{4\,b^2}+\frac {x^2}{2\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 36, normalized size = 1.89 \[ \frac {- a - 2 b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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