Optimal. Leaf size=25 \[ -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{b \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 25, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 19, normalized size = 0.76 \begin {gather*} -\frac {4 \, \sqrt {a x + b \sqrt {x}}}{b \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 25, normalized size = 1.00 \begin {gather*} \frac {4}{\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 159, normalized size = 6.36 \begin {gather*} \frac {\sqrt {a x +b \sqrt {x}}\, \left (-a b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+a b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} x +2 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} x -4 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b \sqrt {x}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \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 {1}{x \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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