Optimal. Leaf size=76 \[ -\frac {a^3}{3 b^4 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {3 a^2}{b^4 \sqrt {a+\frac {b}{x^2}}}+\frac {3 a \sqrt {a+\frac {b}{x^2}}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^4} \]
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Rubi [A] time = 0.04, antiderivative size = 76, 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} \[ -\frac {a^3}{3 b^4 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {3 a^2}{b^4 \sqrt {a+\frac {b}{x^2}}}+\frac {3 a \sqrt {a+\frac {b}{x^2}}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^4} \]
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 )^{5/2} x^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{5/2}}+\frac {3 a^2}{b^3 (a+b x)^{3/2}}-\frac {3 a}{b^3 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {a^3}{3 b^4 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {3 a^2}{b^4 \sqrt {a+\frac {b}{x^2}}}+\frac {3 a \sqrt {a+\frac {b}{x^2}}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 62, normalized size = 0.82 \[ \frac {16 a^3 x^6+24 a^2 b x^4+6 a b^2 x^2-b^3}{3 b^4 x^4 \sqrt {a+\frac {b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 76, normalized size = 1.00 \[ \frac {{\left (16 \, a^{3} x^{6} + 24 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b^{4} x^{6} + 2 \, a b^{5} x^{4} + b^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 58, normalized size = 0.76 \[ \frac {2 \, {\left (4 \, x^{2} {\left (\frac {2 \, a^{3} x^{2}}{b^{4}} + \frac {3 \, a^{2}}{b^{3}}\right )} + \frac {3 \, a}{b^{2}}\right )} x^{2} - \frac {1}{b}}{3 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.80 \[ \frac {\left (a \,x^{2}+b \right ) \left (16 a^{3} x^{6}+24 a^{2} b \,x^{4}+6 a \,b^{2} x^{2}-b^{3}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} b^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 64, normalized size = 0.84 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}}}{3 \, b^{4}} + \frac {3 \, \sqrt {a + \frac {b}{x^{2}}} a}{b^{4}} + \frac {3 \, a^{2}}{\sqrt {a + \frac {b}{x^{2}}} b^{4}} - \frac {a^{3}}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 58, normalized size = 0.76 \[ \frac {\sqrt {a+\frac {b}{x^2}}\,\left (16\,a^3\,x^6+24\,a^2\,b\,x^4+6\,a\,b^2\,x^2-b^3\right )}{3\,b^4\,x^2\,{\left (a\,x^2+b\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.15, size = 201, normalized size = 2.64 \[ \begin {cases} \frac {16 a^{3} x^{6}}{3 a b^{4} x^{6} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{5} x^{4} \sqrt {a + \frac {b}{x^{2}}}} + \frac {24 a^{2} b x^{4}}{3 a b^{4} x^{6} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{5} x^{4} \sqrt {a + \frac {b}{x^{2}}}} + \frac {6 a b^{2} x^{2}}{3 a b^{4} x^{6} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{5} x^{4} \sqrt {a + \frac {b}{x^{2}}}} - \frac {b^{3}}{3 a b^{4} x^{6} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{5} x^{4} \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{8 a^{\frac {5}{2}} x^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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