Optimal. Leaf size=18 \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.0045454, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac{1}{2 b \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [A] time = 0.0038647, size = 18, normalized size = 1. \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966129, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46865, size = 51, normalized size = 2.83 \begin{align*} -\frac{\sqrt{b x^{4} + a}}{2 \,{\left (b^{2} x^{4} + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.777046, size = 26, normalized size = 1.44 \begin{align*} \begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09243, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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