Optimal. Leaf size=50 \[ \frac {\sqrt {b x^2+c x^4}}{x}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2021, 2008, 206} \[ \frac {\sqrt {b x^2+c x^4}}{x}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2021
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx &=\frac {\sqrt {b x^2+c x^4}}{x}+b \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {\sqrt {b x^2+c x^4}}{x}-b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {\sqrt {b x^2+c x^4}}{x}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 1.20 \[ \frac {x \left (-\sqrt {b} \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )+b+c x^2\right )}{\sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 117, normalized size = 2.34 \[ \left [\frac {\sqrt {b} x \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}}}{2 \, x}, \frac {\sqrt {-b} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 1.38 \[ \frac {b \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} + \sqrt {c x^{2} + b} \mathrm {sgn}\relax (x) - \frac {{\left (b \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b} \sqrt {b}\right )} \mathrm {sgn}\relax (x)}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 1.30 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {c \,x^{2}+b}\right )}{\sqrt {c \,x^{2}+b}\, x} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 68, normalized size = 1.36 \[ \frac {\sqrt {c\,x^4+b\,x^2}}{x}+\frac {\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,1{}\mathrm {i}}{\sqrt {c}\,x}\right )\,\sqrt {c\,x^4+b\,x^2}\,1{}\mathrm {i}}{\sqrt {c}\,x^2\,\sqrt {\frac {b}{c\,x^2}+1}} \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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