Optimal. Leaf size=35 \[ \frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0199261, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0137037, size = 31, normalized size = 0.89 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (2 a x^2-b\right )}{3 b^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 39, normalized size = 1.1 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}-b \right ) }{3\,{b}^{2}{x}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991083, size = 39, normalized size = 1.11 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{\sqrt{a + \frac{b}{x^{2}}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52537, size = 69, normalized size = 1.97 \begin{align*} \frac{{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.12436, size = 231, normalized size = 6.6 \begin{align*} \frac{2 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} + \frac{a^{\frac{5}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{4} b x^{5}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{3} b^{2} x^{3}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} \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{1}{\sqrt{a + \frac{b}{x^{2}}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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