Optimal. Leaf size=77 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a+b x^4} \]
[Out]
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Rubi [A] time = 0.0759054, antiderivative size = 77, 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 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x^4)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 10.5406, size = 68, normalized size = 0.88 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{3}{4}}} + \frac{a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{3}{4}}} + \frac{x^{3} \sqrt [4]{a + b x^{4}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x**4+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0434783, size = 63, normalized size = 0.82 \[ \frac{x^3 \left (a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+3 \left (a+b x^4\right )\right )}{12 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x^4)^(1/4),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt [4]{b{x}^{4}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x^4+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283194, size = 234, normalized size = 3.04 \[ \frac{1}{4} \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3} - \frac{1}{4} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x}{x \sqrt{\frac{\sqrt{\frac{a^{4}}{b^{3}}} b^{2} x^{2} + \sqrt{b x^{4} + a} a^{2}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) + \frac{1}{16} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) - \frac{1}{16} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.18308, size = 39, normalized size = 0.51 \[ \frac{\sqrt [4]{a} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x**4+a)**(1/4),x)
[Out]
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GIAC/XCAS [A] time = 0.235306, size = 304, normalized size = 3.95 \[ \frac{1}{32} \,{\left (\frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}{a} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} - \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4)*x^2,x, algorithm="giac")
[Out]