Optimal. Leaf size=21 \[ -\frac{x}{c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.0194372, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1588} \[ -\frac{x}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \frac{x^4}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{x}{c \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0047655, size = 21, normalized size = 1. \[ -\frac{x}{c \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 29, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3} \left ( c{x}^{2}+b \right ) }{c} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00746, size = 19, normalized size = 0.9 \begin{align*} -\frac{1}{\sqrt{c x^{2} + b} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28105, size = 54, normalized size = 2.57 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}}{c^{2} x^{3} + b c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16458, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{\sqrt{c + \frac{b}{x^{2}}} c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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