Optimal. Leaf size=51 \[ 2 \sqrt {a+b \sqrt {c x^2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 51, 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 = {368, 50, 63, 208} \begin {gather*} 2 \sqrt {a+b \sqrt {c x^2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 368
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sqrt {c x^2}\right )\\ &=2 \sqrt {a+b \sqrt {c x^2}}+a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )\\ &=2 \sqrt {a+b \sqrt {c x^2}}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{b}\\ &=2 \sqrt {a+b \sqrt {c x^2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 1.00 \begin {gather*} 2 \sqrt {a+b \sqrt {c x^2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 57, normalized size = 1.12 \begin {gather*} 2 \sqrt {a+b \sqrt {c} \sqrt {x^2}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c} \sqrt {x^2}}}{\sqrt {a}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.83, size = 114, normalized size = 2.24 \begin {gather*} \left [\sqrt {a} \log \left (\frac {b c x^{2} - 2 \, \sqrt {c x^{2}} \sqrt {\sqrt {c x^{2}} b + a} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, \sqrt {\sqrt {c x^{2}} b + a}, 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} \sqrt {-a}}{a}\right ) + 2 \, \sqrt {\sqrt {c x^{2}} b + a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 38, normalized size = 0.75 \begin {gather*} \frac {2 \, a \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b \sqrt {c} x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.78 \begin {gather*} -2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{2}}\, b}}{\sqrt {a}}\right )+2 \sqrt {a +\sqrt {c \,x^{2}}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 60, normalized size = 1.18 \begin {gather*} \sqrt {a} \log \left (\frac {\sqrt {\sqrt {c x^{2}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{2}} b + a} + \sqrt {a}}\right ) + 2 \, \sqrt {\sqrt {c x^{2}} b + 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 = -0.02 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{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 {\sqrt {a + b \sqrt {c x^{2}}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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