Optimal. Leaf size=58 \[ \frac {8 x \sqrt {a+\frac {b}{x^2}}}{3 a^3}-\frac {4 x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {x}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {8 x \sqrt {a+\frac {b}{x^2}}}{3 a^3}-\frac {4 x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {x}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx &=-\frac {x}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {4 \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {x}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {4 x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {8 \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx}{3 a^2}\\ &=-\frac {x}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {4 x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {8 \sqrt {a+\frac {b}{x^2}} x}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.88 \[ \frac {3 a^2 x^4+12 a b x^2+8 b^2}{3 a^3 x \sqrt {a+\frac {b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 63, normalized size = 1.09 \[ \frac {{\left (3 \, a^{2} x^{5} + 12 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \]
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 = 50, normalized size = 0.86 \[ \frac {\left (a \,x^{2}+b \right ) \left (3 a^{2} x^{4}+12 a b \,x^{2}+8 b^{2}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 51, normalized size = 0.88 \[ \frac {\sqrt {a + \frac {b}{x^{2}}} x}{a^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{2}}\right )} b x^{2} - b^{2}}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 43, normalized size = 0.74 \[ \frac {x\,{\left (\frac {a\,x^2}{b}+1\right )}^{5/2}\,\sqrt {x^{10}}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},3;\ 4;\ -\frac {a\,x^2}{b}\right )}{6\,{\left (a\,x^2+b\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.74, size = 163, normalized size = 2.81 \[ \frac {3 a^{2} b^{\frac {9}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac {8 b^{\frac {13}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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