Optimal. Leaf size=63 \[ \frac {1}{2} x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \sqrt {a+\frac {b}{x^2}}+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 50, 63, 208} \[ \frac {1}{2} x^2 \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {3}{2} b \sqrt {a+\frac {b}{x^2}}+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^{3/2} x \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{3/2} x^2-\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {3}{2} b \sqrt {a+\frac {b}{x^2}}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{3/2} x^2-\frac {1}{4} (3 a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {3}{2} b \sqrt {a+\frac {b}{x^2}}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{3/2} x^2-\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=-\frac {3}{2} b \sqrt {a+\frac {b}{x^2}}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{3/2} x^2+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.75 \[ -\frac {b \sqrt {a+\frac {b}{x^2}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {a x^2}{b}\right )}{\sqrt {\frac {a x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 129, normalized size = 2.05 \[ \left [\frac {3}{4} \, \sqrt {a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + \frac {1}{2} \, {\left (a x^{2} - 2 \, b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}, -\frac {3}{2} \, \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + \frac {1}{2} \, {\left (a x^{2} - 2 \, b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 79, normalized size = 1.25 \[ \frac {1}{2} \, \sqrt {a x^{2} + b} a x \mathrm {sgn}\relax (x) - \frac {3}{4} \, \sqrt {a} b \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, \sqrt {a} b^{2} \mathrm {sgn}\relax (x)}{{\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 107, normalized size = 1.70 \[ -\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} \left (-3 a \,b^{2} x \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )-3 \sqrt {a \,x^{2}+b}\, a^{\frac {3}{2}} b \,x^{2}-2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{\frac {3}{2}} x^{2}+2 \left (a \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {a}\right ) x^{2}}{2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 66, normalized size = 1.05 \[ \frac {1}{2} \, \sqrt {a + \frac {b}{x^{2}}} a x^{2} - \frac {3}{4} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \sqrt {a + \frac {b}{x^{2}}} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 48, normalized size = 0.76 \[ \frac {a\,x^2\,\sqrt {a+\frac {b}{x^2}}}{2}-b\,\sqrt {a+\frac {b}{x^2}}+\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.72, size = 88, normalized size = 1.40 \[ \frac {3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{2} + \frac {a^{2} x^{3}}{2 \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {a \sqrt {b} x}{2 \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {b^{\frac {3}{2}}}{x \sqrt {\frac {a x^{2}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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