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 = 0.01, 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 = {679, 223, 209}
\begin {gather*} \frac {a \text {ArcTan}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}+\frac {\sqrt {a^2-b^2 x^2}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 679
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 \text {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 = 0.11, size = 63, normalized size = 1.37 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2}}{b}-\frac {a \log \left (-\sqrt {-b^2} x+\sqrt {a^2-b^2 x^2}\right )}{\sqrt {-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 78, normalized size = 1.70
method | result | size |
risch | \(\frac {\sqrt {-b^{2} x^{2}+a^{2}}}{b}+\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{\sqrt {b^{2}}}\) | \(49\) |
default | \(\frac {\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}+\frac {a b \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{\sqrt {b^{2}}}}{b}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 31, normalized size = 0.67 \begin {gather*} \frac {a \arcsin \left (\frac {b x}{a}\right )}{b} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.60, size = 52, normalized size = 1.13 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 36, normalized size = 0.78 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\sqrt {a^2-b^2\,x^2}}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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