Optimal. Leaf size=19 \[ \frac{x^4}{4 a \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0038325, antiderivative size = 19, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0076782, 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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Maple [A] time = 0.007, size = 31, normalized size = 1.6 \begin{align*}{\frac{a}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{1}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.15848, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, b x^{2} + a}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.24151, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, b x^{2} + a}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.41778, size = 36, normalized size = 1.89 \begin{align*} - \frac{a + 2 b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36925, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, b x^{2} + a}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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