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 = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 51
Rule 63
Rule 205
Rule 263
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 1.02, size = 93, normalized size = 2.32 \[ \left [\frac {x \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, \sqrt {x}}{b x}, \frac {2 \, {\left (x \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a \sqrt {x}}\right ) - \sqrt {x}\right )}}{b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 31, normalized size = 0.78 \[ -\frac {2 \, a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {2}{b \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.80 \[ -\frac {2 a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {2}{b \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.48, size = 31, normalized size = 0.78 \[ \frac {2 \, a \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b} - \frac {2}{b \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 28, normalized size = 0.70 \[ -\frac {2}{b\,\sqrt {x}}-\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.59, size = 102, normalized size = 2.55 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: a = 0 \\- \frac {2}{b \sqrt {x}} + \frac {i \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {i \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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