Optimal. Leaf size=71 \[ \frac{2 x}{\sqrt [4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0138555, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {229, 227, 196} \[ \frac{2 x}{\sqrt [4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx &=\frac{\sqrt [4]{1+\frac{b x^2}{a}} \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{2 x}{\sqrt [4]{a+b x^2}}-\frac{\sqrt [4]{1+\frac{b x^2}{a}} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{2 x}{\sqrt [4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0060906, size = 46, normalized size = 0.65 \[ \frac{x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\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{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{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{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.678049, size = 24, normalized size = 0.34 \begin{align*} \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{\sqrt [4]{a}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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