Optimal. Leaf size=31 \[ \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {369, 266, 63, 208} \begin {gather*} \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}} x} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (\frac {4 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}\right )}{b \sqrt {c}},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 31, normalized size = 1.00 \begin {gather*} \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} \frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 81, normalized size = 2.61 \begin {gather*} \left [\frac {2 \, \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right )}{\sqrt {a}}, -\frac {4 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \sqrt {\frac {c}{x}} + a} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 200, normalized size = 6.45 \begin {gather*} \frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (\sqrt {\frac {c}{x}}\, b \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+\sqrt {\frac {c}{x}}\, b \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}-2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}\right )}{\sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {\frac {c}{x}}\, \sqrt {a}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 45, normalized size = 1.45 \begin {gather*} -\frac {2 \, \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x\,\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x \sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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