Optimal. Leaf size=97 \[ -\frac{5 a^{3/2} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{12 \left (a+b x^4\right )^{3/4}}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{5}{12} a x \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.0383726, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {195, 237, 335, 275, 231} \[ -\frac{5 a^{3/2} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{5}{12} a x \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{1}{6} (5 a) \int \sqrt [4]{a+b x^4} \, dx\\ &=\frac{5}{12} a x \sqrt [4]{a+b x^4}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{1}{12} \left (5 a^2\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{5}{12} a x \sqrt [4]{a+b x^4}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{\left (5 a^2 \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}{12 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5}{12} a x \sqrt [4]{a+b x^4}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}-\frac{\left (5 a^2 \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 )}{12 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5}{12} a x \sqrt [4]{a+b x^4}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}-\frac{\left (5 a^2 \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 )}{24 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5}{12} a x \sqrt [4]{a+b x^4}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}-\frac{5 a^{3/2} \sqrt{b} \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 )}{12 \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0050164, size = 47, normalized size = 0.48 \[ \frac{a x \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \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}}\,{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{5}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.29669, size = 37, normalized size = 0.38 \begin{align*} \frac{a^{\frac{5}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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