Optimal. Leaf size=131 \[ -\frac{7 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{5/2} \sqrt [4]{a-b x^2}}-\frac{7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac{7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac{\left (a-b x^2\right )^{3/4}}{5 a x^5} \]
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Rubi [A] time = 0.144018, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{7 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{5/2} \sqrt [4]{a-b x^2}}-\frac{7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac{7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac{\left (a-b x^2\right )^{3/4}}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(a - b*x^2)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 18.6961, size = 114, normalized size = 0.87 \[ - \frac{\left (a - b x^{2}\right )^{\frac{3}{4}}}{5 a x^{5}} - \frac{7 b \left (a - b x^{2}\right )^{\frac{3}{4}}}{30 a^{2} x^{3}} - \frac{7 b^{2} \left (a - b x^{2}\right )^{\frac{3}{4}}}{20 a^{3} x} - \frac{7 b^{\frac{5}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{20 a^{\frac{5}{2}} \sqrt [4]{a - b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(-b*x**2+a)**(1/4),x)
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Mathematica [C] time = 0.0682722, size = 95, normalized size = 0.73 \[ \frac{-24 a^3-4 a^2 b x^2-21 b^3 x^6 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-14 a b^2 x^4+42 b^3 x^6}{120 a^3 x^5 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^6*(a - b*x^2)^(1/4)),x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(-b*x^2+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(1/4)*x^6),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(1/4)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.22515, size = 34, normalized size = 0.26 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 \sqrt [4]{a} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(-b*x**2+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(1/4)*x^6),x, algorithm="giac")
[Out]