Optimal. Leaf size=105 \[ -\frac{2 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 a x^5} \]
[Out]
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Rubi [A] time = 0.142548, antiderivative size = 105, 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 \[ -\frac{2 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(a + b*x^4)^(1/4)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{b x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{5 a \sqrt [4]{a + b x^{4}}} + \frac{2 b}{5 a x \sqrt [4]{a + b x^{4}}} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{5 a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(b*x**4+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0565743, size = 83, normalized size = 0.79 \[ \frac{-3 a^2-4 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+3 a b x^4+6 b^2 x^8}{15 a^2 x^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^6*(a + b*x^4)^(1/4)),x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(b*x^4+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.85368, size = 29, normalized size = 0.28 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(b*x**4+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="giac")
[Out]