Optimal. Leaf size=20 \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]
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Rubi [A] time = 0.004762, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {264} \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{a x}\\ \end{align*}
Mathematica [A] time = 0.0042253, size = 20, normalized size = 1. \[ -\frac{\sqrt [4]{a-b x^4}}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt [4]{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00684, size = 24, normalized size = 1.2 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7578, size = 36, normalized size = 1.8 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.990024, size = 83, normalized size = 4.15 \begin{align*} \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{1}{4}\right )}{4 a \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\- \frac{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{- \frac{3 i \pi }{4}} \Gamma \left (- \frac{1}{4}\right )}{4 a \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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