Optimal. Leaf size=41 \[ -\frac {\sqrt {a+b x}}{a x}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {44, 65, 214}
\begin {gather*} \frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x}}{a x}-\frac {b \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a}\\ &=-\frac {\sqrt {a+b x}}{a x}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a}\\ &=-\frac {\sqrt {a+b x}}{a x}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x}}{a x}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 40, normalized size = 0.98
method | result | size |
risch | \(\frac {b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {\sqrt {b x +a}}{a x}\) | \(34\) |
derivativedivides | \(2 b \left (-\frac {\sqrt {b x +a}}{2 a b x}+\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(40\) |
default | \(2 b \left (-\frac {\sqrt {b x +a}}{2 a b x}+\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 60, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {b x + a} b}{{\left (b x + a\right )} a - a^{2}} - \frac {b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 93, normalized size = 2.27 \begin {gather*} \left [\frac {\sqrt {a} b x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} a}{2 \, a^{2} x}, -\frac {\sqrt {-a} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \sqrt {b x + a} a}{a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.08, size = 44, normalized size = 1.07 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a \sqrt {x}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 47, normalized size = 1.15 \begin {gather*} -\frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {\sqrt {b x + a} b}{a x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 33, normalized size = 0.80 \begin {gather*} \frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b\,x}}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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