Optimal. Leaf size=29 \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {515, 23, 30} \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 515
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x}} x}{\sqrt {a-b x}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {\sqrt {x} \sqrt {-a+b x}}{\sqrt {a-b x}} \, dx}{\sqrt {-a+b x}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \sqrt {x} \, dx}{\sqrt {a-b x}}\\ &=\frac {2 \sqrt {b-\frac {a}{x}} x^2}{3 \sqrt {a-b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 35, normalized size = 1.21 \[ -\frac {2 \, \sqrt {-b x + a} x^{2} \sqrt {\frac {b x - a}{x}}}{3 \, {\left (b x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 56, normalized size = 1.93 \[ \frac {2 \, \sqrt {-a b} a {\left | b \right |} \mathrm {sgn}\relax (x)}{3 \, b^{3}} - \frac {2 \, {\left (\sqrt {-a b} a + \frac {{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac {3}{2}}}{b}\right )} {\left | b \right |} \mathrm {sgn}\relax (x)}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.93 \[ \frac {2 \sqrt {-\frac {-b x +a}{x}}\, x^{2}}{3 \sqrt {-b x +a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.73, size = 5, normalized size = 0.17 \[ -\frac {2}{3} i \, x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.06, size = 23, normalized size = 0.79 \[ \frac {2\,x^2\,\sqrt {b-\frac {a}{x}}}{3\,\sqrt {a-b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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