Optimal. Leaf size=123 \[ \frac{a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{24 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.0562202, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 321, 237, 335, 275, 231} \[ \frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int x^4 \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{2} a \int x^4 \sqrt [4]{a+b x^4} \, dx\\ &=\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{12} a^2 \int \frac{x^4}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}-\frac{a^3 \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{24 b}\\ &=\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}-\frac{\left (a^3 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{24 b \left (a+b x^4\right )^{3/4}}\\ &=\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{\left (a^3 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{24 b \left (a+b x^4\right )^{3/4}}\\ &=\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{\left (a^3 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{48 b \left (a+b x^4\right )^{3/4}}\\ &=\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{a^{5/2} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{b} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0537738, size = 67, normalized size = 0.54 \[ \frac{x \sqrt [4]{a+b x^4} \left (\left (a+b x^4\right )^2-\frac{a^2 \, _2F_1\left (-\frac{5}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}}\right )}{10 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, 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{5}{4}} x^{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 ({\left (b x^{8} + a x^{4}\right )}{\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 = 3.36936, size = 39, normalized size = 0.32 \begin{align*} \frac{a^{\frac{5}{4}} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{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{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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