Optimal. Leaf size=32 \[ 2 a^2 \sqrt {x}-\frac {4 a b}{\sqrt {x}}-\frac {2 b^2}{3 x^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 32, 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 = {263, 43} \[ 2 a^2 \sqrt {x}-\frac {4 a b}{\sqrt {x}}-\frac {2 b^2}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx &=\int \frac {(b+a x)^2}{x^{5/2}} \, dx\\ &=\int \left (\frac {b^2}{x^{5/2}}+\frac {2 a b}{x^{3/2}}+\frac {a^2}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 b^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 a^2 \sqrt {x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.81 \[ -\frac {2 \left (-3 a^2 x^2+6 a b x+b^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 24, normalized size = 0.75 \[ \frac {2 \, {\left (3 \, a^{2} x^{2} - 6 \, a b x - b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 0.72 \[ 2 \, a^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.78 \[ \frac {2 a^{2} x^{2}-4 a b x -\frac {2}{3} b^{2}}{x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 24, normalized size = 0.75 \[ 2 \, a^{2} \sqrt {x} - \frac {4 \, a b}{\sqrt {x}} - \frac {2 \, b^{2}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 24, normalized size = 0.75 \[ -\frac {-6\,a^2\,x^2+12\,a\,b\,x+2\,b^2}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 31, normalized size = 0.97 \[ 2 a^{2} \sqrt {x} - \frac {4 a b}{\sqrt {x}} - \frac {2 b^{2}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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