Optimal. Leaf size=80 \[ \frac{x^3}{2 \sqrt [4]{a+b x^4}}+\frac{\sqrt{a} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.040962, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {310, 281, 335, 275, 196} \[ \frac{x^3}{2 \sqrt [4]{a+b x^4}}+\frac{\sqrt{a} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 310
Rule 281
Rule 335
Rule 275
Rule 196
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt [4]{a+b x^4}} \, dx &=\frac{x^3}{2 \sqrt [4]{a+b x^4}}-\frac{1}{2} a \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx\\ &=\frac{x^3}{2 \sqrt [4]{a+b x^4}}-\frac{\left (a \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{2 b \sqrt [4]{a+b x^4}}\\ &=\frac{x^3}{2 \sqrt [4]{a+b x^4}}+\frac{\left (a \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{2 b \sqrt [4]{a+b x^4}}\\ &=\frac{x^3}{2 \sqrt [4]{a+b x^4}}+\frac{\left (a \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{4 b \sqrt [4]{a+b x^4}}\\ &=\frac{x^3}{2 \sqrt [4]{a+b x^4}}+\frac{\sqrt{a} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0087901, size = 51, normalized size = 0.64 \[ \frac{x^3 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{3 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt [4]{b{x}^{4}+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{x^{2}}{{\left (b x^{4} + 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{x^{2}}{{\left (b x^{4} + 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 = 0.896892, size = 37, normalized size = 0.46 \begin{align*} \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{7}{4}\right )} \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^{2}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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