Optimal. Leaf size=98 \[ \frac{5 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4} \]
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Rubi [A] time = 0.116676, 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 \[ \frac{5 a^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac{1}{7} x^2 \left (a+b x^4\right )^{5/4} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^4)^(5/4),x]
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Rubi in Sympy [A] time = 9.97542, size = 87, normalized size = 0.89 \[ \frac{5 a^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{21 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 a x^{2} \sqrt [4]{a + b x^{4}}}{21} + \frac{x^{2} \left (a + b x^{4}\right )^{\frac{5}{4}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**4+a)**(5/4),x)
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Mathematica [C] time = 0.0579835, size = 77, normalized size = 0.79 \[ \frac{x^2 \left (5 a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+16 a^2+22 a b x^4+6 b^2 x^8\right )}{42 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^4)^(5/4),x]
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Maple [F] time = 0.035, size = 0, normalized size = 0. \[ \int x \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^4+a)^(5/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)*x,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{5} + a x\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)*x,x, algorithm="fricas")
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Sympy [A] time = 6.1634, size = 29, normalized size = 0.3 \[ \frac{a^{\frac{5}{4}} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**4+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)*x,x, algorithm="giac")
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