Optimal. Leaf size=52 \[ \frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4} \]
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Rubi [A] time = 0.0825069, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ \frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx &=-\frac{\sqrt{b x^2+c x^4}}{3 b x^4}-\frac{(2 c) \int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx}{3 b}\\ &=-\frac{\sqrt{b x^2+c x^4}}{3 b x^4}+\frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0153207, size = 35, normalized size = 0.67 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (2 c x^2-b\right )}{3 b^2 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 37, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -2\,c{x}^{2}+b \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{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 = 1.30803, size = 66, normalized size = 1.27 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (2 \, c x^{2} - b\right )}}{3 \, b^{2} x^{4}} \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{1}{x^{3} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14449, size = 36, normalized size = 0.69 \begin{align*} -\frac{{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{c + \frac{b}{x^{2}}} c}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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