Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]
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Rubi [A] time = 0.0056614, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^2} \, dx &=-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0115148, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 1.4 \begin{align*} -{\frac{2\,ax+2\,b}{7\,bx} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08417, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72129, size = 100, normalized size = 5.56 \begin{align*} -\frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{7 \, b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.11032, size = 80, normalized size = 4.44 \begin{align*} \begin{cases} - \frac{2 a^{3} \sqrt{a + \frac{b}{x}}}{7 b} - \frac{6 a^{2} \sqrt{a + \frac{b}{x}}}{7 x} - \frac{6 a b \sqrt{a + \frac{b}{x}}}{7 x^{2}} - \frac{2 b^{2} \sqrt{a + \frac{b}{x}}}{7 x^{3}} & \text{for}\: b \neq 0 \\- \frac{a^{\frac{5}{2}}}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1665, size = 279, normalized size = 15.5 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} \mathrm{sgn}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b \mathrm{sgn}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm{sgn}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4} \mathrm{sgn}\left (x\right ) + 7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5} \mathrm{sgn}\left (x\right ) + b^{6} \mathrm{sgn}\left (x\right )\right )}}{7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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