Optimal. Leaf size=55 \[ \frac {B \sqrt {b x^2+c x^4}}{c x}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1145, 2008, 206} \[ \frac {B \sqrt {b x^2+c x^4}}{c x}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1145
Rule 2008
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\sqrt {b x^2+c x^4}} \, dx &=\frac {B \sqrt {b x^2+c x^4}}{c x}+A \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {B \sqrt {b x^2+c x^4}}{c x}-A \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {B \sqrt {b x^2+c x^4}}{c x}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.33 \[ \frac {x \left (\sqrt {b} B \left (b+c x^2\right )-A c \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{\sqrt {b} c \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 138, normalized size = 2.51 \[ \left [\frac {A \sqrt {b} c x \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} B b}{2 \, b c x}, \frac {A \sqrt {-b} c x \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} B b}{b c x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 60, normalized size = 1.09 \[ \frac {A \log \left ({\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2}\right )}{2 \, \sqrt {b}} - \frac {2 \, B \sqrt {b}}{{\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2} - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 1.31 \[ -\frac {\sqrt {c \,x^{2}+b}\, \left (A c \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {c \,x^{2}+b}\, B \sqrt {b}\right ) x}{\sqrt {c \,x^{4}+b \,x^{2}}\, \sqrt {b}\, c} \]
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 \[ \int \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {B\,x^2+A}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
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 \[ \int \frac {A + B x^{2}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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