Optimal. Leaf size=29 \[ \frac {2 x^3 \sqrt {b-\frac {a}{x}}}{5 \sqrt {a-b x}} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {515, 23, 30} \[ \frac {2 x^3 \sqrt {b-\frac {a}{x}}}{5 \sqrt {a-b x}} \]
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^2}{\sqrt {a-b x}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {x^{3/2} \sqrt {-a+b x}}{\sqrt {a-b x}} \, dx}{\sqrt {-a+b x}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int x^{3/2} \, dx}{\sqrt {a-b x}}\\ &=\frac {2 \sqrt {b-\frac {a}{x}} x^3}{5 \sqrt {a-b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {2 x^3 \sqrt {b-\frac {a}{x}}}{5 \sqrt {a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 35, normalized size = 1.21 \[ -\frac {2 \, \sqrt {-b x + a} x^{3} \sqrt {\frac {b x - a}{x}}}{5 \, {\left (b x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 76, normalized size = 2.62 \[ \frac {2 \, \sqrt {-a b} a^{2} {\left | b \right |} \mathrm {sgn}\relax (x)}{5 \, b^{4}} - \frac {2 \, {\left (\sqrt {-a b} a^{2} - \frac {{\left ({\left (b x - a\right )} b + a b\right )}^{2} \sqrt {-{\left (b x - a\right )} b - a b}}{b^{2}}\right )} {\left | b \right |} \mathrm {sgn}\relax (x)}{5 \, b^{4}} \]
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 \[ \frac {2 \sqrt {-\frac {-b x +a}{x}}\, x^{3}}{5 \sqrt {-b x +a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.81, size = 5, normalized size = 0.17 \[ -\frac {2}{5} i \, x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 23, normalized size = 0.79 \[ \frac {2\,x^3\,\sqrt {b-\frac {a}{x}}}{5\,\sqrt {a-b\,x}} \]
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 \[ \int \frac {x^{2} \sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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