Optimal. Leaf size=43 \[ 4 \sqrt {a+b \sqrt {x}}-4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {266, 50, 63, 208} \[ 4 \sqrt {a+b \sqrt {x}}-4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {x}}}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sqrt {x}\right )\\ &=4 \sqrt {a+b \sqrt {x}}+(2 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=4 \sqrt {a+b \sqrt {x}}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {x}}\right )}{b}\\ &=4 \sqrt {a+b \sqrt {x}}-4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ 4 \sqrt {a+b \sqrt {x}}-4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 88, normalized size = 2.05 \[ \left [2 \, \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) + 4 \, \sqrt {b \sqrt {x} + a}, 4 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \sqrt {x} + a} \sqrt {-a}}{a}\right ) + 4 \, \sqrt {b \sqrt {x} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 0.95 \[ \frac {4 \, {\left (\frac {a b \arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {b \sqrt {x} + a} b\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.74 \[ -4 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \sqrt {x}+a}}{\sqrt {a}}\right )+4 \sqrt {b \sqrt {x}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 49, normalized size = 1.14 \[ 2 \, \sqrt {a} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right ) + 4 \, \sqrt {b \sqrt {x} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.89, size = 31, normalized size = 0.72 \[ 4\,\sqrt {a+b\,\sqrt {x}}-4\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,\sqrt {x}}}{\sqrt {a}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.57, size = 75, normalized size = 1.74 \[ - 4 \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )} + \frac {4 a}{\sqrt {b} \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {4 \sqrt {b} \sqrt [4]{x}}{\sqrt {\frac {a}{b \sqrt {x}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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