Optimal. Leaf size=40 \[ \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 65, 214}
\begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{-a+b x} \, dx &=\frac {2 \sqrt {x}}{b}+\frac {a \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{b}\\ &=\frac {2 \sqrt {x}}{b}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.86, size = 98, normalized size = 2.45 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\sqrt {x}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2 x^{\frac {3}{2}}}{3 a},b\text {==}0\right \},\left \{\frac {2 \sqrt {x}}{b},a\text {==}0\right \}\right \},-\frac {a \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{b^2 \sqrt {\frac {a}{b}}}+\frac {a \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{b^2 \sqrt {\frac {a}{b}}}+\frac {2 \sqrt {x}}{b}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 32, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b}-\frac {2 a \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(32\) |
default | \(\frac {2 \sqrt {x}}{b}-\frac {2 a \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(32\) |
risch | \(\frac {2 \sqrt {x}}{b}-\frac {2 a \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 47, normalized size = 1.18 \begin {gather*} \frac {a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2 \, \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 83, normalized size = 2.08 \begin {gather*} \left [\frac {\sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, \sqrt {x}}{b}, \frac {2 \, {\left (\sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + \sqrt {x}\right )}}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 83, normalized size = 2.08 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{b^{2} \sqrt {\frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{b^{2} \sqrt {\frac {a}{b}}} + \frac {2 \sqrt {x}}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 47, normalized size = 1.18 \begin {gather*} -2 \left (-\frac {\sqrt {x}}{b}-\frac {2 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{b\cdot 2 \sqrt {-a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 28, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {x}}{b}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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