Optimal. Leaf size=55 \[ \frac {a^2}{3 b^3 \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {2 a}{b^3 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 55, 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} \[ \frac {a^2}{3 b^3 \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {2 a}{b^3 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^3} \]
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^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{5/2}}-\frac {2 a}{b^2 (a+b x)^{3/2}}+\frac {1}{b^2 \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a^2}{3 b^3 \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {2 a}{b^3 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.87 \[ -\frac {\sqrt {a+\frac {b}{x^2}} \left (8 a^2 x^4+12 a b x^2+3 b^2\right )}{3 b^3 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 61, normalized size = 1.11 \[ -\frac {{\left (8 \, a^{2} x^{4} + 12 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.91 \[ -\frac {\left (a \,x^{2}+b \right ) \left (8 a^{2} x^{4}+12 a b \,x^{2}+3 b^{2}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 47, normalized size = 0.85 \[ -\frac {\sqrt {a + \frac {b}{x^{2}}}}{b^{3}} - \frac {2 \, a}{\sqrt {a + \frac {b}{x^{2}}} b^{3}} + \frac {a^{2}}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 42, normalized size = 0.76 \[ -\frac {\sqrt {a+\frac {b}{x^2}}\,\left (\frac {8\,a^2\,x^4}{3}+4\,a\,b\,x^2+b^2\right )}{b^3\,{\left (a\,x^2+b\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.36, size = 153, normalized size = 2.78 \[ \begin {cases} - \frac {8 a^{2} x^{4}}{3 a b^{3} x^{4} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{4} x^{2} \sqrt {a + \frac {b}{x^{2}}}} - \frac {12 a b x^{2}}{3 a b^{3} x^{4} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{4} x^{2} \sqrt {a + \frac {b}{x^{2}}}} - \frac {3 b^{2}}{3 a b^{3} x^{4} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{4} x^{2} \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{6 a^{\frac {5}{2}} x^{6}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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