Optimal. Leaf size=36 \[ \frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0160528, antiderivative size = 36, 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} \[ \frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0171368, size = 29, normalized size = 0.81 \[ \frac{2 \sqrt{a+\frac{b}{x}} (2 a x-b)}{3 b^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-b \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01151, size = 41, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{2 \, \sqrt{a + \frac{b}{x}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43524, size = 58, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (2 \, a x - b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.1837, size = 248, normalized size = 6.89 \begin{align*} \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{4} b x^{\frac{5}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{3} b^{2} x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14855, size = 55, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (3 \, a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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