Optimal. Leaf size=54 \[ x \left (a+\frac {b}{x}\right )^{3/2}-3 b \sqrt {a+\frac {b}{x}}+3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {242, 47, 50, 63, 208} \[ x \left (a+\frac {b}{x}\right )^{3/2}-3 b \sqrt {a+\frac {b}{x}}+3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 242
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{3/2} \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {b}{x}\right )^{3/2} x-\frac {1}{2} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-3 b \sqrt {a+\frac {b}{x}}+\left (a+\frac {b}{x}\right )^{3/2} x-\frac {1}{2} (3 a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-3 b \sqrt {a+\frac {b}{x}}+\left (a+\frac {b}{x}\right )^{3/2} x-(3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )\\ &=-3 b \sqrt {a+\frac {b}{x}}+\left (a+\frac {b}{x}\right )^{3/2} x+3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.85 \[ \sqrt {a+\frac {b}{x}} (a x-2 b)+3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 100, normalized size = 1.85 \[ \left [\frac {3}{2} \, \sqrt {a} b \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (a x - 2 \, b\right )} \sqrt {\frac {a x + b}{x}}, -3 \, \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a x - 2 \, b\right )} \sqrt {\frac {a x + b}{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 100, normalized size = 1.85 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-3 a b \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} x^{2}+4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\right )}{2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 63, normalized size = 1.17 \[ \sqrt {a + \frac {b}{x}} a x - \frac {3}{2} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 2 \, \sqrt {a + \frac {b}{x}} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 34, normalized size = 0.63 \[ -\frac {2\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,x}{b}\right )}{{\left (\frac {a\,x}{b}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.75, size = 92, normalized size = 1.70 \[ 3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} + \frac {a^{2} x^{\frac {3}{2}}}{\sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {a \sqrt {b} \sqrt {x}}{\sqrt {\frac {a x}{b} + 1}} - \frac {2 b^{\frac {3}{2}}}{\sqrt {x} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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