Optimal. Leaf size=57 \[ \frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {c}}-\frac {A \sqrt {b x^2+c x^4}}{b x^2} \]
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Rubi [A] time = 0.15, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2034, 792, 620, 206} \[ \frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {c}}-\frac {A \sqrt {b x^2+c x^4}}{b x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \sqrt {b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x \sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {b x^2+c x^4}}{b x^2}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {b x^2+c x^4}}{b x^2}+B \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )\\ &=-\frac {A \sqrt {b x^2+c x^4}}{b x^2}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 1.30 \[ \frac {b B x \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b+c x^2}}\right )-A \sqrt {c} \left (b+c x^2\right )}{b \sqrt {c} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 136, normalized size = 2.39 \[ \left [\frac {B b \sqrt {c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} A c}{2 \, b c x^{2}}, -\frac {B b \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} A c}{b c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 66, normalized size = 1.16 \[ -\frac {B \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} + b \right |}\right )}{2 \, \sqrt {c}} + \frac {A}{\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 67, normalized size = 1.18 \[ -\frac {\sqrt {c \,x^{2}+b}\, \left (-B b x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+\sqrt {c \,x^{2}+b}\, A \sqrt {c}\right )}{\sqrt {c \,x^{4}+b \,x^{2}}\, b \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 56, normalized size = 0.98 \[ \frac {B \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{2 \, \sqrt {c}} - \frac {\sqrt {c x^{4} + b x^{2}} A}{b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 57, normalized size = 1.00 \[ \frac {B\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{2\,\sqrt {c}}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{b\,x^2} \]
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 \[ \int \frac {A + B x^{2}}{x \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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