Optimal. Leaf size=108 \[ \frac{5 b^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a-b x^4}}{12 a^2 x^2}-\frac{\sqrt [4]{a-b x^4}}{6 a x^6} \]
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Rubi [A] time = 0.158393, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 b^{3/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a-b x^4}}{12 a^2 x^2}-\frac{\sqrt [4]{a-b x^4}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a - b*x^4)^(3/4)),x]
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Rubi in Sympy [A] time = 17.8162, size = 92, normalized size = 0.85 \[ - \frac{\sqrt [4]{a - b x^{4}}}{6 a x^{6}} - \frac{5 b \sqrt [4]{a - b x^{4}}}{12 a^{2} x^{2}} + \frac{5 b^{\frac{3}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{12 a^{\frac{3}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(-b*x**4+a)**(3/4),x)
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Mathematica [C] time = 0.0575134, size = 84, normalized size = 0.78 \[ \frac{-4 a^2+5 b^2 x^8 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-6 a b x^4+10 b^2 x^8}{24 a^2 x^6 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(a - b*x^4)^(3/4)),x]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(-b*x^4+a)^(3/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^7),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{7}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.97555, size = 34, normalized size = 0.31 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{6 a^{\frac{3}{4}} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(-b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^7),x, algorithm="giac")
[Out]