Optimal. Leaf size=40 \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2}{b \sqrt{x}} \]
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Rubi [A] time = 0.0149488, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2}{b \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) x^{5/2}} \, dx &=\int \frac{1}{x^{3/2} (b+a x)} \, dx\\ &=-\frac{2}{b \sqrt{x}}-\frac{a \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{b}\\ &=-\frac{2}{b \sqrt{x}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=-\frac{2}{b \sqrt{x}}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0043369, size = 25, normalized size = 0.62 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{a x}{b}\right )}{b \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -2\,{\frac{a}{b\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{1}{b\sqrt{x}}} \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 = 1.74061, size = 207, normalized size = 5.18 \begin{align*} \left [\frac{x \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 2 \, \sqrt{x}}{b x}, \frac{2 \,{\left (x \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - \sqrt{x}\right )}}{b x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.53649, size = 102, normalized size = 2.55 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} + \frac{i \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{i \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09826, size = 42, normalized size = 1.05 \begin{align*} -\frac{2 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{2}{b \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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