Optimal. Leaf size=43 \[ \frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {B \sqrt {a+b x^2}}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {641, 217, 206} \[ \frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {B \sqrt {a+b x^2}}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {a+b x^2}} \, dx &=\frac {B \sqrt {a+b x^2}}{b}+A \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {B \sqrt {a+b x^2}}{b}+A \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {B \sqrt {a+b x^2}}{b}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 46, normalized size = 1.07 \[ \frac {A \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )}{\sqrt {b}}+\frac {B \sqrt {a+b x^2}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 92, normalized size = 2.14 \[ \left [\frac {A \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, \sqrt {b x^{2} + a} B}{2 \, b}, -\frac {A \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - \sqrt {b x^{2} + a} B}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 39, normalized size = 0.91 \[ -\frac {A \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{\sqrt {b}} + \frac {\sqrt {b x^{2} + a} B}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.86 \[ \frac {A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {\sqrt {b \,x^{2}+a}\, B}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 29, normalized size = 0.67 \[ \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {\sqrt {b x^{2} + a} B}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 36, normalized size = 0.84 \[ \frac {B\,\sqrt {b\,x^2+a}}{b}+\frac {A\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.64, size = 102, normalized size = 2.37 \[ A \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) + B \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: b = 0 \\\frac {\sqrt {a + b x^{2}}}{b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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