Optimal. Leaf size=94 \[ -\frac{\left (a+b x^4\right )^{5/4}}{x}-\frac{5}{8} a \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{8} a \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{4} b x^3 \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.101425, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^4\right )^{5/4}}{x}-\frac{5}{8} a \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{8} a \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{4} b x^3 \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(5/4)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 13.2339, size = 87, normalized size = 0.93 \[ - \frac{5 a \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8} + \frac{5 a \sqrt [4]{b} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8} + \frac{5 b x^{3} \sqrt [4]{a + b x^{4}}}{4} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(5/4)/x**2,x)
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Mathematica [C] time = 0.0433878, size = 79, normalized size = 0.84 \[ \frac{-12 a^2+5 a b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-9 a b x^4+3 b^2 x^8}{12 x \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(5/4)/x^2,x]
[Out]
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Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(5/4)/x^2,x)
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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^2,x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.15099, size = 41, normalized size = 0.44 \[ \frac{a^{\frac{5}{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(5/4)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.237999, size = 309, normalized size = 3.29 \[ \frac{1}{32} \,{\left (\frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b x^{3}}{a} + 10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + 10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + 5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - 5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{32 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x^2,x, algorithm="giac")
[Out]