Optimal. Leaf size=122 \[ \frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]
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Rubi [A] time = 0.0362143, antiderivative size = 122, 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 = {321, 229, 227, 196} \[ \frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt [4]{a+b x^2}} \, dx &=\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b}-\frac{(2 a) \int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx}{3 b}\\ &=-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b}+\frac{\left (4 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{15 b^2}\\ &=-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b}+\frac{\left (4 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{15 b^2 \sqrt [4]{a+b x^2}}\\ &=\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b}-\frac{\left (4 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{15 b^2 \sqrt [4]{a+b x^2}}\\ &=\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b}-\frac{8 a^{5/2} \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 )}{15 b^{5/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0209872, size = 79, normalized size = 0.65 \[ \frac{2 \left (6 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-6 a^2 x-a b x^3+5 b^2 x^5\right )}{45 b^2 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4}{\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{x^{4}}{{\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{x^{4}}{{\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 = 0.765393, size = 27, normalized size = 0.22 \begin{align*} \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 \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{x^{4}}{{\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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