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 = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 203
Rule 217
Rule 665
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.90, size = 52, normalized size = 1.13 \[ -\frac {2 \, a \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - \sqrt {-b^{2} x^{2} + a^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 36, normalized size = 0.78 \[ \frac {a \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b)}{{\left | b \right |}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.67 \[ \frac {a \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {2 \left (x +\frac {a}{b}\right ) a b -\left (x +\frac {a}{b}\right )^{2} b^{2}}}\right )}{\sqrt {b^{2}}}+\frac {\sqrt {2 \left (x +\frac {a}{b}\right ) a b -\left (x +\frac {a}{b}\right )^{2} b^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.89, size = 31, normalized size = 0.67 \[ \frac {a \arcsin \left (\frac {b x}{a}\right )}{b} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} \]
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 \[ \int \frac {\sqrt {a^2-b^2\,x^2}}{a+b\,x} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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