Optimal. Leaf size=36 \[ \frac {2 \sqrt {b-\frac {a}{x}} x^{1+m}}{(1+2 m) \sqrt {a-b x}} \]
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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 = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {529, 23, 30}
\begin {gather*} \frac {2 x^{m+1} \sqrt {b-\frac {a}{x}}}{(2 m+1) \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^m}{\sqrt {a-b x}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m} \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^{-\frac {1}{2}+m} \, dx}{\sqrt {a-b x}}\\ &=\frac {2 \sqrt {b-\frac {a}{x}} x^{1+m}}{(1+2 m) \sqrt {a-b x}}\\ \end {align*}
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Mathematica [A]
time = 1.01, size = 35, normalized size = 0.97 \begin {gather*} \frac {\sqrt {b-\frac {a}{x}} x^{1+m}}{\left (\frac {1}{2}+m\right ) \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 36, normalized size = 1.00
method | result | size |
gosper | \(\frac {2 x^{1+m} \sqrt {-\frac {-b x +a}{x}}}{\left (1+2 m \right ) \sqrt {-b x +a}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 15, normalized size = 0.42 \begin {gather*} \frac {2 \, \sqrt {x} x^{m}}{2 i \, m + i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 44, normalized size = 1.22 \begin {gather*} \frac {2 \, \sqrt {-b x + a} x x^{m} \sqrt {\frac {b x - a}{x}}}{2 \, a m - {\left (2 \, b m + b\right )} x + a} \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^{m} \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.26, size = 32, normalized size = 0.89 \begin {gather*} \frac {2\,x^{m+1}\,\sqrt {b-\frac {a}{x}}}{\left (2\,m+1\right )\,\sqrt {a-b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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