Optimal. Leaf size=34 \[ -\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 0.74 \[ -\frac {2 (2 a x+b)}{b^2 x \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 31, normalized size = 0.91 \[ -\frac {2 \, {\left (2 \, a x + b\right )} \sqrt {\frac {a x + b}{x}}}{a b^{2} x + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 0.88 \[ -\frac {2 \, {\left (\frac {a}{\sqrt {\frac {a x + b}{x}}} + \sqrt {\frac {a x + b}{x}}\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.91 \[ -\frac {2 \left (a x +b \right ) \left (2 a x +b \right )}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 30, normalized size = 0.88 \[ -\frac {2 \, \sqrt {a + \frac {b}{x}}}{b^{2}} - \frac {2 \, a}{\sqrt {a + \frac {b}{x}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 36, normalized size = 1.06 \[ -\frac {x\,\sqrt {a+\frac {b}{x}}\,\left (\frac {2}{b}+\frac {4\,a\,x}{b^2}\right )}{a\,x^2+b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.37, size = 42, normalized size = 1.24 \[ \begin {cases} - \frac {4 a}{b^{2} \sqrt {a + \frac {b}{x}}} - \frac {2}{b x \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {3}{2}} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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