Optimal. Leaf size=85 \[ \frac{\sqrt{b} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.113596, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{b} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a - b*x^4)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 13.5898, size = 68, normalized size = 0.8 \[ - \frac{\sqrt [4]{a - b x^{4}}}{2 a x^{2}} + \frac{\sqrt{b} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \left (a - b x^{4}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(-b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0489164, size = 70, normalized size = 0.82 \[ \frac{b x^4 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-2 a+2 b x^4}{4 a x^2 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a - b*x^4)^(3/4)),x]
[Out]
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Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(-b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.27207, size = 32, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(-b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(3/4)*x^3),x, algorithm="giac")
[Out]