Optimal. Leaf size=64 \[ \frac {4 x^2 \sqrt {a+\frac {b}{x^4}}}{3 a^3}-\frac {2 x^2}{3 a^2 \sqrt {a+\frac {b}{x^4}}}-\frac {x^2}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac {4 x^2 \sqrt {a+\frac {b}{x^4}}}{3 a^3}-\frac {2 x^2}{3 a^2 \sqrt {a+\frac {b}{x^4}}}-\frac {x^2}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx &=-\frac {x^2}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}+\frac {4 \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {x^2}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {2 x^2}{3 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {8 \int \frac {x}{\sqrt {a+\frac {b}{x^4}}} \, dx}{3 a^2}\\ &=-\frac {x^2}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {2 x^2}{3 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {4 \sqrt {a+\frac {b}{x^4}} x^2}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.80 \[ \frac {3 a^2 x^8+12 a b x^4+8 b^2}{6 a^3 x^2 \sqrt {a+\frac {b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 65, normalized size = 1.02 \[ \frac {{\left (3 \, a^{2} x^{10} + 12 \, a b x^{6} + 8 \, b^{2} x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 53, normalized size = 0.83 \[ \frac {\sqrt {a x^{4} + b}}{2 \, a^{3}} - \frac {4 \, \sqrt {b}}{3 \, a^{3}} + \frac {6 \, {\left (a x^{4} + b\right )} b - b^{2}}{6 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.78 \[ \frac {\left (a \,x^{4}+b \right ) \left (3 x^{8} a^{2}+12 a b \,x^{4}+8 b^{2}\right )}{6 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} a^{3} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 54, normalized size = 0.84 \[ \frac {\sqrt {a + \frac {b}{x^{4}}} x^{2}}{2 \, a^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{4}}\right )} b x^{4} - b^{2}}{6 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 38, normalized size = 0.59 \[ \frac {3\,a^2\,x^8+12\,a\,b\,x^4+8\,b^2}{6\,a^3\,x^6\,{\left (a+\frac {b}{x^4}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.66, size = 163, normalized size = 2.55 \[ \frac {3 a^{2} b^{\frac {9}{2}} x^{8} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac {12 a b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} + \frac {8 b^{\frac {13}{2}} \sqrt {\frac {a x^{4}}{b} + 1}}{6 a^{5} b^{4} x^{8} + 12 a^{4} b^{5} x^{4} + 6 a^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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