Optimal. Leaf size=46 \[ \frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
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Rubi [A] time = 0.0161951, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {665, 217, 203} \[ \frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 665
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2-b^2 x^2}}{a+b x} \, dx &=\frac{\sqrt{a^2-b^2 x^2}}{b}+a \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{\sqrt{a^2-b^2 x^2}}{b}+a \operatorname{Subst}\left (\int \frac{1}{1+b^2 x^2} \, dx,x,\frac{x}{\sqrt{a^2-b^2 x^2}}\right )\\ &=\frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0358361, size = 43, normalized size = 0.93 \[ \frac{\sqrt{a^2-b^2 x^2}+a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 77, normalized size = 1.7 \begin{align*}{\frac{1}{b}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}+{a\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77752, size = 101, normalized size = 2.2 \begin{align*} -\frac{2 \, a \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) - \sqrt{-b^{2} x^{2} + a^{2}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28768, size = 49, normalized size = 1.07 \begin{align*} \frac{a \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} + \frac{\sqrt{-b^{2} x^{2} + a^{2}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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