Optimal. Leaf size=44 \[ \sqrt {x} \sqrt {a+b x}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \[ \sqrt {x} \sqrt {a+b x}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {a+b x}+\frac {1}{2} a \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\sqrt {x} \sqrt {a+b x}+a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {a+b x}+a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\sqrt {x} \sqrt {a+b x}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 62, normalized size = 1.41 \[ \frac {\frac {a^{3/2} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b}}+\sqrt {x} (a+b x)}{\sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 93, normalized size = 2.11 \[ \left [\frac {a \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} b \sqrt {x}}{2 \, b}, -\frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} b \sqrt {x}}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.00, size = 62, normalized size = 1.41 \[ \frac {\sqrt {\left (b x +a \right ) x}\, a \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {b}\, \sqrt {x}}+\sqrt {b x +a}\, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.99, size = 70, normalized size = 1.59 \[ -\frac {a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{2 \, \sqrt {b}} - \frac {\sqrt {b x + a} a}{{\left (b - \frac {b x + a}{x}\right )} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 41, normalized size = 0.93 \[ \sqrt {x}\,\sqrt {a+b\,x}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.92, size = 42, normalized size = 0.95 \[ \sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}} + \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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