Optimal. Leaf size=48 \[ \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, 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} \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx &=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b}\\ &=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b}\\ &=\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 68, normalized size = 1.42 \begin {gather*} \frac {\sqrt {b} \sqrt {x} (a+b x)-a^{3/2} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 49, normalized size = 1.02 \begin {gather*} \frac {a \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{b^{3/2}}+\frac {\sqrt {x} \sqrt {a+b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 91, normalized size = 1.90 \begin {gather*} \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^{2}}, \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^{2}}\right ] \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 65, normalized size = 1.35 \begin {gather*} -\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}\, b^{\frac {3}{2}} \sqrt {x}}+\frac {\sqrt {b x +a}\, \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.91, size = 73, normalized size = 1.52 \begin {gather*} \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 \, b^{\frac {3}{2}}} - \frac {\sqrt {b x + a} a}{{\left (b^{2} - \frac {{\left (b x + a\right )} b}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 44, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x}\,\sqrt {a+b\,x}}{b}-\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.19, size = 44, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{b} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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