Optimal. Leaf size=40 \[ \frac {2 B \sqrt {a+b x}}{b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 63, 208} \begin {gather*} \frac {2 B \sqrt {a+b x}}{b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x \sqrt {a+b x}} \, dx &=\frac {2 B \sqrt {a+b x}}{b}+A \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=\frac {2 B \sqrt {a+b x}}{b}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 B \sqrt {a+b x}}{b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {a+b x}}{b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {a+b x}}{b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 94, normalized size = 2.35 \begin {gather*} \left [\frac {A \sqrt {a} b \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} B a}{a b}, \frac {2 \, {\left (A \sqrt {-a} b \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \sqrt {b x + a} B a\right )}}{a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 36, normalized size = 0.90 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, \sqrt {b x + a} B}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.88 \begin {gather*} \frac {-\frac {2 A b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}+2 \sqrt {b x +a}\, B}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 47, normalized size = 1.18 \begin {gather*} \frac {A \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {b x + a} B}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 32, normalized size = 0.80 \begin {gather*} \frac {2\,B\,\sqrt {a+b\,x}}{b}-\frac {2\,A\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.77, size = 56, normalized size = 1.40 \begin {gather*} \frac {2 A \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{a}} \sqrt {a + b x}} \right )}}{a \sqrt {- \frac {1}{a}}} - B \left (\begin {cases} - \frac {x}{\sqrt {a}} & \text {for}\: b = 0 \\- \frac {2 \sqrt {a + b x}}{b} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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