Optimal. Leaf size=41 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.07, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1961, 12, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 1961
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a+b x}{c+d x}}}{a+b x} \, dx &=(2 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{(b c-a d) \left (b-d x^2\right )} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{b-d x^2} \, dx,x,\sqrt {\frac {a+b x}{c+d x}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {a+b x}{c+d x}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 97, normalized size = 2.37 \[ \frac {2 \sqrt {b c-a d} \sqrt {\frac {a+b x}{c+d x}} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b \sqrt {d} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 105, normalized size = 2.56 \[ \left [\frac {\sqrt {b d} \log \left (2 \, b d x + b c + a d + 2 \, \sqrt {b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}\right )}{b d}, -\frac {2 \, \sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (d x + c\right )} \sqrt {\frac {b x + a}{d x + c}}}{b d x + a d}\right )}{b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 74, normalized size = 1.80 \[ -\frac {\sqrt {b d} \log \left ({\left | -2 \, {\left (\sqrt {b d} x - \sqrt {b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right ) \mathrm {sgn}\left (d x + c\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 80, normalized size = 1.95 \[ \frac {\left (d x +c \right ) \sqrt {\frac {b x +a}{d x +c}}\, \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.73, size = 59, normalized size = 1.44 \[ -\frac {\log \left (\frac {d \sqrt {\frac {b x + a}{d x + c}} - \sqrt {b d}}{d \sqrt {\frac {b x + a}{d x + c}} + \sqrt {b d}}\right )}{\sqrt {b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 31, normalized size = 0.76 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {\frac {a+b\,x}{c+d\,x}}}{\sqrt {b}}\right )}{\sqrt {b}\,\sqrt {d}} \]
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 \[ \int \frac {\sqrt {\frac {a + b x}{c + d x}}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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