Optimal. Leaf size=83 \[ -\frac{1}{2} a^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+a \sqrt [4]{a+b x^4}+\frac{1}{5} \left (a+b x^4\right )^{5/4} \]
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Rubi [A] time = 0.13528, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{1}{2} a^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+a \sqrt [4]{a+b x^4}+\frac{1}{5} \left (a+b x^4\right )^{5/4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(5/4)/x,x]
[Out]
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Rubi in Sympy [A] time = 12.868, size = 70, normalized size = 0.84 \[ - \frac{a^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} - \frac{a^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} + a \sqrt [4]{a + b x^{4}} + \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(5/4)/x,x)
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Mathematica [C] time = 0.0546093, size = 76, normalized size = 0.92 \[ \frac{3 \left (6 a^2+7 a b x^4+b^2 x^8\right )-5 a^2 \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )}{15 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(5/4)/x,x]
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Maple [F] time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(5/4)/x,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)^(5/4)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281554, size = 163, normalized size = 1.96 \[ \frac{1}{5} \,{\left (b x^{4} + 6 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} +{\left (a^{5}\right )}^{\frac{1}{4}} \arctan \left (\frac{{\left (a^{5}\right )}^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a + \sqrt{\sqrt{b x^{4} + a} a^{2} + \sqrt{a^{5}}}}\right ) - \frac{1}{4} \,{\left (a^{5}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a +{\left (a^{5}\right )}^{\frac{1}{4}}\right ) + \frac{1}{4} \,{\left (a^{5}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a -{\left (a^{5}\right )}^{\frac{1}{4}}\right ) \]
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 = 7.07708, size = 48, normalized size = 0.58 \[ - \frac{b^{\frac{5}{4}} x^{5} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{5}{4} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (- \frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(5/4)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.223816, size = 270, normalized size = 3.25 \[ -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} a \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} a \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} a{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} a{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{5} \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x,x, algorithm="giac")
[Out]