Optimal. Leaf size=36 \[ \frac {2}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 a}{3 b^2 \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, 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} \[ \frac {2}{b^2 \sqrt {a+\frac {b}{x}}}-\frac {2 a}{3 b^2 \left (a+\frac {b}{x}\right )^{3/2}} \]
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 )^{5/2} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{(a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{5/2}}+\frac {1}{b (a+b x)^{3/2}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a}{3 b^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.92 \[ \frac {4 a x+6 b}{3 b^2 \sqrt {a+\frac {b}{x}} (a x+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 47, normalized size = 1.31 \[ \frac {2 \, {\left (2 \, a x^{2} + 3 \, b x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 1.00 \[ -\frac {2 \, {\left (a - \frac {3 \, {\left (a x + b\right )}}{x}\right )} x}{3 \, {\left (a x + b\right )} b^{2} \sqrt {\frac {a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.92 \[ \frac {2 \left (a x +b \right ) \left (2 a x +3 b \right )}{3 \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 30, normalized size = 0.83 \[ \frac {2}{\sqrt {a + \frac {b}{x}} b^{2}} - \frac {2 \, a}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 25, normalized size = 0.69 \[ \frac {6\,b+4\,a\,x}{3\,b^2\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.04, size = 82, normalized size = 2.28 \[ \begin {cases} \frac {4 a x}{3 a b^{2} x \sqrt {a + \frac {b}{x}} + 3 b^{3} \sqrt {a + \frac {b}{x}}} + \frac {6 b}{3 a b^{2} x \sqrt {a + \frac {b}{x}} + 3 b^{3} \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {5}{2}} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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