Optimal. Leaf size=101 \[ -\frac{2 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{21 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}+\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b} \]
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Rubi [A] time = 0.0623637, antiderivative size = 101, 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, 279, 321, 233, 231} \[ -\frac{2 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}+\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b} \]
Antiderivative was successfully verified.
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Rule 275
Rule 279
Rule 321
Rule 233
Rule 231
Rubi steps
\begin{align*} \int x^5 \sqrt [4]{a+b x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt [4]{a+b x^2} \, dx,x,x^2\right )\\ &=\frac{1}{7} x^6 \sqrt [4]{a+b x^4}+\frac{1}{14} a \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{21 b}\\ &=\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}-\frac{\left (a^2 \left (1+\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{21 b \left (a+b x^4\right )^{3/4}}\\ &=\frac{a x^2 \sqrt [4]{a+b x^4}}{21 b}+\frac{1}{7} x^6 \sqrt [4]{a+b x^4}-\frac{2 a^{5/2} \left (1+\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0588342, size = 64, normalized size = 0.63 \[ \frac{x^2 \sqrt [4]{a+b x^4} \left (-\frac{a \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}}+a+b x^4\right )}{7 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{5}\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{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{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 ({\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.30473, size = 29, normalized size = 0.29 \begin{align*} \frac{\sqrt [4]{a} 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} \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{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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