Optimal. Leaf size=104 \[ \frac{2 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.0562071, antiderivative size = 104, 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 = {275, 321, 229, 227, 196} \[ \frac{2 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt [4]{a+b x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{5 b}\\ &=\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{\left (a \sqrt [4]{1+\frac{b x^4}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx,x,x^2\right )}{5 b \sqrt [4]{a+b x^4}}\\ &=-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}+\frac{\left (a \sqrt [4]{1+\frac{b x^4}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{5 b \sqrt [4]{a+b x^4}}\\ &=-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}+\frac{2 a^{3/2} \sqrt [4]{1+\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0187022, size = 64, normalized size = 0.62 \[ \frac{x^2 \left (-a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+a+b x^4\right )}{5 b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5}{\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^{5}}{{\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^{5}}{{\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 = 1.18742, size = 27, normalized size = 0.26 \begin{align*} \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 \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^{5}}{{\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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