Optimal. Leaf size=34 \[ -\frac {a}{b^2 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {a}{b^2 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^5} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {a}{b^2 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.82 \[ \frac {-2 a x^2-b}{b^2 x^2 \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 37, normalized size = 1.09 \[ -\frac {{\left (2 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a b^{2} x^{2} + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 28, normalized size = 0.82 \[ -\frac {\frac {2 \, a x^{2}}{b^{2}} + \frac {1}{b}}{\sqrt {a x^{4} + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 1.09 \[ -\frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}+b \right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 30, normalized size = 0.88 \[ -\frac {\sqrt {a + \frac {b}{x^{2}}}}{b^{2}} - \frac {a}{\sqrt {a + \frac {b}{x^{2}}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 36, normalized size = 1.06 \[ -\frac {x\,\sqrt {a+\frac {b}{x^2}}\,\left (\frac {1}{b}+\frac {2\,a\,x^2}{b^2}\right )}{a\,x^3+b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.38, size = 48, normalized size = 1.41 \[ \begin {cases} - \frac {2 a}{b^{2} \sqrt {a + \frac {b}{x^{2}}}} - \frac {1}{b x^{2} \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{4 a^{\frac {3}{2}} x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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