Optimal. Leaf size=77 \[ \frac{x^2}{b \sqrt [4]{a+b x^4}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.111006, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^2}{b \sqrt [4]{a+b x^4}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a + b*x^4)^(5/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{x^{2}}{b \sqrt [4]{a + b x^{4}}} + \frac{\int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**4+a)**(5/4),x)
[Out]
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Mathematica [C] time = 0.0510082, size = 54, normalized size = 0.7 \[ \frac{x^2 \left (\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-1\right )}{b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a + b*x^4)^(5/4),x]
[Out]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{5} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^4+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^4 + a)^(5/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^4 + a)^(5/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.63554, size = 27, normalized size = 0.35 \[ \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**4+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^4 + a)^(5/4),x, algorithm="giac")
[Out]