Optimal. Leaf size=75 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}} \]
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Rubi [A] time = 0.0456584, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 5.31439, size = 70, normalized size = 0.93 \[ \frac{3 a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 \sqrt [4]{b}} + \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 \sqrt [4]{b}} + \frac{x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(3/4),x)
[Out]
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Mathematica [A] time = 0.0608566, size = 94, normalized size = 1.25 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{16 \sqrt [4]{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(3/4),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)^(3/4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272914, size = 242, normalized size = 3.23 \[ \frac{1}{4} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x + \frac{3}{4} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{\frac{\sqrt{\frac{a^{4}}{b}} a^{4} b x^{2} + \sqrt{b x^{4} + a} a^{6}}{x^{2}}}}\right ) + \frac{3}{16} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + \left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) - \frac{3}{16} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} - \left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.58564, size = 37, normalized size = 0.49 \[ \frac{a^{\frac{3}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [A] time = 0.235747, size = 302, normalized size = 4.03 \[ \frac{1}{32} \,{\left (\frac{6 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} - \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} + \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{a}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4),x, algorithm="giac")
[Out]