Optimal. Leaf size=56 \[ \frac {2 \sqrt {a x+b \sqrt {x}}}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 56, 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 = {2010, 2013, 620, 206} \[ \frac {2 \sqrt {a x+b \sqrt {x}}}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 2010
Rule 2013
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \sqrt {x}+a x}} \, dx &=\frac {2 \sqrt {b \sqrt {x}+a x}}{a}-\frac {b \int \frac {1}{\sqrt {x} \sqrt {b \sqrt {x}+a x}} \, dx}{2 a}\\ &=\frac {2 \sqrt {b \sqrt {x}+a x}}{a}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {2 \sqrt {b \sqrt {x}+a x}}{a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{a}\\ &=\frac {2 \sqrt {b \sqrt {x}+a x}}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 1.57 \[ \frac {2 \sqrt {a} \sqrt {x} \left (a \sqrt {x}+b\right )-2 b^{3/2} \sqrt [4]{x} \sqrt {\frac {a \sqrt {x}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )}{a^{3/2} \sqrt {a x+b \sqrt {x}}} \]
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 = 0.28, size = 54, normalized size = 0.96 \[ \frac {b \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {a x + b \sqrt {x}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.48 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-b \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 72, normalized size = 1.29 \[ \frac {4\,x\,\left (\frac {3\,\sqrt {b}\,\sqrt {b+a\,\sqrt {x}}}{2\,a\,\sqrt {x}}+\frac {b^{3/2}\,\mathrm {asin}\left (\frac {\sqrt {a}\,x^{1/4}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,3{}\mathrm {i}}{2\,a^{3/2}\,x^{3/4}}\right )\,\sqrt {\frac {a\,\sqrt {x}}{b}+1}}{3\,\sqrt {a\,x+b\,\sqrt {x}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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