Optimal. Leaf size=29 \[ \frac {2 \sqrt {b-\frac {a}{x}} x^2}{3 \sqrt {a-b x}} \]
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Rubi [A]
time = 0.01, 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 = {529, 23, 30}
\begin {gather*} \frac {2 x^2 \sqrt {b-\frac {a}{x}}}{3 \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 529
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 = 5.94, size = 27, normalized size = 0.93 \begin {gather*} -\frac {2 x \sqrt {a-b x}}{3 \sqrt {b-\frac {a}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 27, normalized size = 0.93
method | result | size |
gosper | \(\frac {2 x^{2} \sqrt {-\frac {-b x +a}{x}}}{3 \sqrt {-b x +a}}\) | \(27\) |
default | \(\frac {2 x^{2} \sqrt {-\frac {-b x +a}{x}}}{3 \sqrt {-b x +a}}\) | \(27\) |
risch | \(-\frac {2 \sqrt {-\frac {-b x +a}{x}}\, \sqrt {-\left (-b x +a \right ) x}\, \sqrt {\frac {x \left (-b x +a \right )}{b x -a}}\, \left (b x -a \right ) x^{2}}{3 \left (-b x +a \right )^{\frac {3}{2}} \sqrt {\left (b x -a \right ) x}\, \sqrt {-x}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.30, size = 5, normalized size = 0.17 \begin {gather*} -\frac {2}{3} i \, x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 35, normalized size = 1.21 \begin {gather*} -\frac {2 \, \sqrt {-b x + a} x^{2} \sqrt {\frac {b x - a}{x}}}{3 \, {\left (b x - a\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x \sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (23) = 46\).
time = 4.68, size = 56, normalized size = 1.93 \begin {gather*} \frac {2 \, \sqrt {-a b} a {\left | b \right |} \mathrm {sgn}\left (x\right )}{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}\left (x\right )}{3 \, b^{3}} \end {gather*}
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 \begin {gather*} \frac {2\,x^2\,\sqrt {b-\frac {a}{x}}}{3\,\sqrt {a-b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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