Optimal. Leaf size=67 \[ -\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x} \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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, 47, 63, 208} \[ -\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 368
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx &=\frac {\sqrt {c x^2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\sqrt {c x^2}\right )}{x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}+\frac {\left (b \sqrt {c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{2 x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}+\frac {\sqrt {c x^2} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 87, normalized size = 1.30 \[ -\frac {b \sqrt {c x^2} \sqrt {\frac {b \sqrt {c x^2}}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b \sqrt {c x^2}}{a}+1}\right )+a+b \sqrt {c x^2}}{x \sqrt {a+b \sqrt {c x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 180, normalized size = 2.69 \[ \left [\frac {b x \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b + a}}{2 \, x}, -\frac {b x \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b + a}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 54, normalized size = 0.81 \[ \frac {\frac {b^{2} c \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b \sqrt {c} x + a} b \sqrt {c}}{x}}{b \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.81 \[ -\frac {\sqrt {c \,x^{2}}\, b \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{2}}\, b}}{\sqrt {a}}\right )+\sqrt {a +\sqrt {c \,x^{2}}\, b}\, \sqrt {a}}{\sqrt {a}\, x} \]
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 \[ \int \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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