Optimal. Leaf size=66 \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}+\frac {3}{4} b x \sqrt {a+\frac {b}{x}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 47, 63, 208} \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}+\frac {3}{4} b x \sqrt {a+\frac {b}{x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{8} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )\\ &=\frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 1.15 \[ \frac {x \sqrt {a+\frac {b}{x}} \left (2 a^2 x^2+3 b^2 \sqrt {\frac {b}{a x}+1} \tanh ^{-1}\left (\sqrt {\frac {b}{a x}+1}\right )+7 a b x+5 b^2\right )}{4 (a x+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 130, normalized size = 1.97 \[ \left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 79, normalized size = 1.20 \[ -\frac {3 \, b^{2} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right ) \mathrm {sgn}\relax (x)}{8 \, \sqrt {a}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{8 \, \sqrt {a}} + \frac {1}{4} \, \sqrt {a x^{2} + b x} {\left (2 \, a x \mathrm {sgn}\relax (x) + 5 \, b \mathrm {sgn}\relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.45 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a \,b^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+4 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x +10 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b \right ) x}{8 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.23, size = 98, normalized size = 1.48 \[ -\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} - 2 \, {\left (a + \frac {b}{x}\right )} a + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 52, normalized size = 0.79 \[ \frac {5\,x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4}+\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,\sqrt {a}}-\frac {3\,a\,x^2\,\sqrt {a+\frac {b}{x}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.11, size = 75, normalized size = 1.14 \[ \frac {a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{2} + \frac {5 b^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{4} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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