Optimal. Leaf size=25 \[ -\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]
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Rubi [A] time = 0.0459559, 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} \[ -\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]
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^9} \, dx &=-\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}\\ \end{align*}
Mathematica [A] time = 0.0141597, size = 25, normalized size = 1. \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 b x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 29, normalized size = 1.2 \begin{align*} -{\frac{c{x}^{2}+b}{5\,{x}^{8}b} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26809, size = 84, normalized size = 3.36 \begin{align*} -\frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \, b x^{6}} \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^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26731, size = 124, normalized size = 4.96 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{2} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + b^{4} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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