Optimal. Leaf size=51 \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]
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Rubi [A] time = 0.0202861, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 191} \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]
Antiderivative was successfully verified.
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Rule 453
Rule 191
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) x^2}{\sqrt{c+\frac{d}{x^2}}} \, dx &=\frac{a \sqrt{c+\frac{d}{x^2}} x^3}{3 c}+\frac{(3 b c-2 a d) \int \frac{1}{\sqrt{c+\frac{d}{x^2}}} \, dx}{3 c}\\ &=\frac{(3 b c-2 a d) \sqrt{c+\frac{d}{x^2}} x}{3 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^3}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0278078, size = 34, normalized size = 0.67 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (a c x^2-2 a d+3 b c\right )}{3 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.9 \begin{align*}{\frac{ \left ( a{x}^{2}c-2\,ad+3\,bc \right ) \left ( c{x}^{2}+d \right ) }{3\,x{c}^{2}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9383, size = 66, normalized size = 1.29 \begin{align*} \frac{b \sqrt{c + \frac{d}{x^{2}}} x}{c} + \frac{{\left ({\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 3 \, \sqrt{c + \frac{d}{x^{2}}} d x\right )} a}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28396, size = 82, normalized size = 1.61 \begin{align*} \frac{{\left (a c x^{3} +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.37479, size = 70, normalized size = 1.37 \begin{align*} \frac{a \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} - \frac{2 a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{2}} + \frac{b \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{2}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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