Optimal. Leaf size=48 \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}{5 a}-\frac {4 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}{15 a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 48, 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} \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}{5 a}-\frac {4 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}{15 a^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x}} x^{3/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}}{5 a}-\frac {(2 b) \int \sqrt {a+\frac {b}{x}} \sqrt {x} \, dx}{5 a}\\ &=-\frac {4 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}{15 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 0.75 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (a x+b) (3 a x-2 b)}{15 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 37, normalized size = 0.77 \[ \frac {2 \, {\left (3 \, a^{2} x^{2} + a b x - 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{15 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 37, normalized size = 0.77 \[ \frac {2}{15} \, {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.69 \[ \frac {2 \left (a x +b \right ) \left (3 a x -2 b \right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{15 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 35, normalized size = 0.73 \[ \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}} - 5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b x^{\frac {3}{2}}\right )}}{15 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 36, normalized size = 0.75 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{5/2}}{5}+\frac {2\,b\,x^{3/2}}{15\,a}-\frac {4\,b^2\,\sqrt {x}}{15\,a^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.24, size = 65, normalized size = 1.35 \[ \frac {2 \sqrt {b} x^{2} \sqrt {\frac {a x}{b} + 1}}{5} + \frac {2 b^{\frac {3}{2}} x \sqrt {\frac {a x}{b} + 1}}{15 a} - \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{15 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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