Optimal. Leaf size=98 \[ \frac{5 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{21 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4} \]
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Rubi [A] time = 0.0551144, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 195, 233, 231} \[ \frac{5 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4} \]
Antiderivative was successfully verified.
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Rule 275
Rule 195
Rule 233
Rule 231
Rubi steps
\begin{align*} \int x \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b x^2\right )^{5/4} \, dx,x,x^2\right )\\ &=\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac{1}{14} (5 a) \operatorname{Subst}\left (\int \sqrt [4]{a+b x^2} \, dx,x,x^2\right )\\ &=\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac{1}{42} \left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac{\left (5 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 )}{42 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac{5 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 \sqrt{b} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0099244, size = 52, normalized size = 0.53 \[ \frac{a x^2 \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{5}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )}{2 \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int x \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\,{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^{5} + a x\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 = 2.01137, size = 29, normalized size = 0.3 \begin{align*} \frac{a^{\frac{5}{4}} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \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\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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