Optimal. Leaf size=69 \[ -\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {3 b}{2 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2 a \sqrt {a+\frac {b}{x^2}}} \]
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Rubi [A] time = 0.03, antiderivative size = 69, 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, 51, 63, 208} \[ \frac {3 x^2 \sqrt {a+\frac {b}{x^2}}}{2 a^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {x^2}{a \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {x^2}{a \sqrt {a+\frac {b}{x^2}}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {x^2}{a \sqrt {a+\frac {b}{x^2}}}+\frac {3 \sqrt {a+\frac {b}{x^2}} x^2}{2 a^2}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{4 a^2}\\ &=-\frac {x^2}{a \sqrt {a+\frac {b}{x^2}}}+\frac {3 \sqrt {a+\frac {b}{x^2}} x^2}{2 a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{2 a^2}\\ &=-\frac {x^2}{a \sqrt {a+\frac {b}{x^2}}}+\frac {3 \sqrt {a+\frac {b}{x^2}} x^2}{2 a^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 1.07 \[ \frac {\sqrt {a} x \left (a x^2+3 b\right )-3 b^{3/2} \sqrt {\frac {a x^2}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{5/2} x \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 192, normalized size = 2.78 \[ \left [\frac {3 \, {\left (a b x^{2} + b^{2}\right )} \sqrt {a} \log \left (-2 \, a x^{2} + 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{4 \, {\left (a^{4} x^{2} + a^{3} b\right )}}, \frac {3 \, {\left (a b x^{2} + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{2 \, {\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \]
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 [A] time = 0.01, size = 73, normalized size = 1.06 \[ \frac {\left (a \,x^{2}+b \right ) \left (a^{\frac {5}{2}} x^{3}+3 a^{\frac {3}{2}} b x -3 \sqrt {a \,x^{2}+b}\, a b \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )\right )}{2 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.82, size = 86, normalized size = 1.25 \[ \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )} b - 2 \, a b}{2 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x^{2}}} a^{3}\right )}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 53, normalized size = 0.77 \[ \frac {3\,b}{2\,a^2\,\sqrt {a+\frac {b}{x^2}}}+\frac {x^2}{2\,a\,\sqrt {a+\frac {b}{x^2}}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2\,a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.57, size = 71, normalized size = 1.03 \[ \frac {x^{3}}{2 a \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {3 \sqrt {b} x}{2 a^{2} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{2 a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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