Optimal. Leaf size=34 \[ \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2013, 620, 206} \[ \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 2013
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {b \sqrt {x}+a x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )\\ &=\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 65, normalized size = 1.91 \[ \frac {4 \sqrt {b} \sqrt [4]{x} \sqrt {\frac {a \sqrt {x}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 37, normalized size = 1.09 \[ -\frac {2 \, \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 136, normalized size = 4.00 \[ -\frac {\sqrt {a x +b \sqrt {x}}\, \left (-b \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-b \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}-2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt {x}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {x}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x} \sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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