Optimal. Leaf size=48 \[ -\frac {2 \sqrt {x}}{b \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {49, 65, 223,
212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(a+b x)^{3/2}} \, dx &=-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{b}\\ &=-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b}\\ &=-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 50, normalized size = 1.04 \begin {gather*} -\frac {2 \sqrt {x}}{b \sqrt {a+b x}}-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.58, size = 38, normalized size = 0.79 \begin {gather*} \frac {2 \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{b^{\frac {3}{2}}}-\frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1+\frac {b x}{a}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 57, normalized size = 1.19 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{b^{\frac {3}{2}}} - \frac {2 \, \sqrt {x}}{\sqrt {b x + a} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 119, normalized size = 2.48 \begin {gather*} \left [\frac {{\left (b x + a\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, \sqrt {b x + a} b \sqrt {x}}{b^{3} x + a b^{2}}, -\frac {2 \, {\left ({\left (b x + a\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + a} b \sqrt {x}\right )}}{b^{3} x + a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.86, size = 46, normalized size = 0.96 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 65, normalized size = 1.35 \begin {gather*} 2 \left (-\frac {\frac {1}{2}\cdot 2 \sqrt {x} \sqrt {a+b x}}{b \left (a+b x\right )}-\frac {\ln \left |\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right |}{b \sqrt {b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x}}{{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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