Optimal. Leaf size=47 \[ -\frac {A \sqrt {a+c x^2}}{a x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {807, 266, 63, 208} \[ -\frac {A \sqrt {a+c x^2}}{a x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \sqrt {a+c x^2}} \, dx &=-\frac {A \sqrt {a+c x^2}}{a x}+B \int \frac {1}{x \sqrt {a+c x^2}} \, dx\\ &=-\frac {A \sqrt {a+c x^2}}{a x}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {a+c x^2}}{a x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c}\\ &=-\frac {A \sqrt {a+c x^2}}{a x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \[ -\frac {A \sqrt {a+c x^2}}{a x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 101, normalized size = 2.15 \[ \left [\frac {B \sqrt {a} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, \sqrt {c x^{2} + a} A}{2 \, a x}, \frac {B \sqrt {-a} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - \sqrt {c x^{2} + a} A}{a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 65, normalized size = 1.38 \[ \frac {2 \, B \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, A \sqrt {c}}{{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 49, normalized size = 1.04 \[ -\frac {B \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{\sqrt {a}}-\frac {\sqrt {c \,x^{2}+a}\, A}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 37, normalized size = 0.79 \[ -\frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{\sqrt {a}} - \frac {\sqrt {c x^{2} + a} A}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 39, normalized size = 0.83 \[ -\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {A\,\sqrt {c\,x^2+a}}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.64, size = 41, normalized size = 0.87 \[ - \frac {A \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{a} - \frac {B \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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