Optimal. Leaf size=123 \[ \frac{b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{12 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{b^2 \sqrt [4]{a+b x^2}}{12 a^2 x}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}-\frac{\sqrt [4]{a+b x^2}}{5 x^5} \]
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Rubi [A] time = 0.0463112, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {277, 325, 233, 231} \[ \frac{b^2 \sqrt [4]{a+b x^2}}{12 a^2 x}+\frac{b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}-\frac{\sqrt [4]{a+b x^2}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^2}}{x^6} \, dx &=-\frac{\sqrt [4]{a+b x^2}}{5 x^5}+\frac{1}{10} b \int \frac{1}{x^4 \left (a+b x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{a+b x^2}}{5 x^5}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}-\frac{b^2 \int \frac{1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx}{12 a}\\ &=-\frac{\sqrt [4]{a+b x^2}}{5 x^5}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}+\frac{b^2 \sqrt [4]{a+b x^2}}{12 a^2 x}+\frac{b^3 \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{24 a^2}\\ &=-\frac{\sqrt [4]{a+b x^2}}{5 x^5}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}+\frac{b^2 \sqrt [4]{a+b x^2}}{12 a^2 x}+\frac{\left (b^3 \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{24 a^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a+b x^2}}{5 x^5}-\frac{b \sqrt [4]{a+b x^2}}{30 a x^3}+\frac{b^2 \sqrt [4]{a+b x^2}}{12 a^2 x}+\frac{b^{5/2} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0091232, size = 51, normalized size = 0.41 \[ -\frac{\sqrt [4]{a+b x^2} \, _2F_1\left (-\frac{5}{2},-\frac{1}{4};-\frac{3}{2};-\frac{b x^2}{a}\right )}{5 x^5 \sqrt [4]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}\sqrt [4]{b{x}^{2}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.18949, size = 34, normalized size = 0.28 \begin{align*} - \frac{\sqrt [4]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 x^{5}} \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{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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