Optimal. Leaf size=44 \[ \frac {4 b}{a^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {4 b}{a^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx &=\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}}-\frac {(2 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{a}\\ &=\frac {4 b}{a^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.64 \[ \frac {2 (a x+2 b)}{a^2 \sqrt {x} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 36, normalized size = 0.82 \[ \frac {2 \, {\left (a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a^{3} x + a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 38, normalized size = 0.86 \[ \frac {2 \, {\left (\frac {\sqrt {a x + b}}{a} + \frac {b}{\sqrt {a x + b} a}\right )}}{a} - \frac {4 \, \sqrt {b}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.73 \[ \frac {2 \left (a x +b \right ) \left (a x +2 b \right )}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 36, normalized size = 0.82 \[ \frac {2 \, \sqrt {a + \frac {b}{x}} \sqrt {x}}{a^{2}} + \frac {2 \, b}{\sqrt {a + \frac {b}{x}} a^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 37, normalized size = 0.84 \[ \frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^2}+\frac {4\,b\,\sqrt {x}}{a^3}\right )}{x+\frac {b}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.82, size = 39, normalized size = 0.89 \[ \frac {2 x}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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