Optimal. Leaf size=36 \[ \frac {2 a \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2} \]
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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} \[ \frac {2 a \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 a \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 29, normalized size = 0.81 \[ \frac {2 \sqrt {a+\frac {b}{x}} (2 a x-b)}{3 b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 27, normalized size = 0.75 \[ \frac {2 \, {\left (2 \, a x - b\right )} \sqrt {\frac {a x + b}{x}}}{3 \, b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 41, normalized size = 1.14 \[ \frac {2 \, {\left (3 \, a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{3 \, b^{2}} \]
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 \[ \frac {2 \left (a x +b \right ) \left (2 a x -b \right )}{3 \sqrt {\frac {a x +b}{x}}\, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 30, normalized size = 0.83 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b^{2}} + \frac {2 \, \sqrt {a + \frac {b}{x}} a}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 23, normalized size = 0.64 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (b-2\,a\,x\right )}{3\,b^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.45, size = 248, normalized size = 6.89 \[ \frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} + \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{4} b x^{\frac {5}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{3} b^{2} x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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