Optimal. Leaf size=46 \[ -\frac {4 b}{3 a^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 46, 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} \[ -\frac {4 b}{3 a^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}} \]
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 )^{5/2} x^{3/2}} \, dx &=-\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}+\frac {(2 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}} \, dx}{a}\\ &=-\frac {4 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 0.83 \[ -\frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (3 a x+2 b)}{3 a^2 (a x+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 48, normalized size = 1.04 \[ -\frac {2 \, {\left (3 \, a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 29, normalized size = 0.63 \[ -\frac {2 \, {\left (3 \, a x + 2 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2}} + \frac {4}{3 \, a^{2} \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.72 \[ -\frac {2 \left (a x +b \right ) \left (3 a x +2 b \right )}{3 \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} a^{2} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 31, normalized size = 0.67 \[ -\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x - b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 49, normalized size = 1.07 \[ -\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^3}+\frac {4\,b\,\sqrt {x}}{3\,a^4}\right )}{x^2+\frac {b^2}{a^2}+\frac {2\,b\,x}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.29, size = 94, normalized size = 2.04 \[ - \frac {6 a x}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {4 b}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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