Optimal. Leaf size=38 \[ \frac {2}{a \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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 = {51, 63, 208} \[ \frac {2}{a \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{3/2}} \, dx &=\frac {2}{a \sqrt {a+b x}}+\frac {\int \frac {1}{x \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2}{a \sqrt {a+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a b}\\ &=\frac {2}{a \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.79 \[ \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x}{a}+1\right )}{a \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 110, normalized size = 2.89 \[ \left [\frac {{\left (b x + a\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} a}{a^{2} b x + a^{3}}, \frac {2 \, {\left ({\left (b x + a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \sqrt {b x + a} a\right )}}{a^{2} b x + a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 37, normalized size = 0.97 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2}{\sqrt {b x + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 0.82 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{\sqrt {b x +a}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 45, normalized size = 1.18 \[ \frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {b x + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 0.79 \[ \frac {2}{a\,\sqrt {a+b\,x}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.87, size = 146, normalized size = 3.84 \[ \frac {2 a^{3} \sqrt {1 + \frac {b x}{a}}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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