Optimal. Leaf size=25 \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.0475792, antiderivative size = 25, 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 = {2014} \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx &=\frac{\left (b x^2+c x^4\right )^{5/2}}{5 c x^5}\\ \end{align*}
Mathematica [A] time = 0.0090662, size = 25, normalized size = 1. \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 c x^5} \]
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}{5\,c{x}^{3}} \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 = 1.02987, size = 43, normalized size = 1.72 \begin{align*} \frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{2} + b}}{5 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3784, size = 80, normalized size = 3.2 \begin{align*} \frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \, 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{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13411, size = 78, normalized size = 3.12 \begin{align*} -\frac{b^{\frac{5}{2}} \mathrm{sgn}\left (x\right )}{5 \, c} + \frac{5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) +{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} \mathrm{sgn}\left (x\right )}{15 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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