Optimal. Leaf size=25 \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
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Rubi [A] time = 0.0058113, antiderivative size = 25, 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 = {2000} \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 2000
Rubi steps
\begin{align*} \int \sqrt{b x^2+c x^4} \, dx &=\frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3}\\ \end{align*}
Mathematica [A] time = 0.0053323, size = 25, normalized size = 1. \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 c x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 29, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+b}{3\,cx}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984779, size = 19, normalized size = 0.76 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51192, size = 58, normalized size = 2.32 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{3 \, 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 \sqrt{b x^{2} + c x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30426, size = 36, normalized size = 1.44 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, c} - \frac{b^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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