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 = 0.019163, antiderivative size = 64, normalized size of antiderivative = 1., 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 273
Rule 264
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*}
Mathematica [A] time = 0.0217474, size = 51, normalized size = 0.8 \[ \frac{3 a^2 x^8+12 a b x^4+8 b^2}{6 a^3 x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 50, normalized size = 0.8 \begin{align*}{\frac{ \left ( a{x}^{4}+b \right ) \left ( 3\,{x}^{8}{a}^{2}+12\,ab{x}^{4}+8\,{b}^{2} \right ) }{6\,{a}^{3}{x}^{10}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945905, size = 73, normalized size = 1.14 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4919, size = 134, normalized size = 2.09 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.50946, size = 163, normalized size = 2.55 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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