Optimal. Leaf size=85 \[ -\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}} \]
[Out]
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Rubi [A] time = 0.112398, 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 \[ -\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a - b*x^4)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 13.4015, size = 70, normalized size = 0.82 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{2 a x^{2}} - \frac{\sqrt{b} \sqrt [4]{1 - \frac{b x^{4}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \sqrt [4]{a - b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(-b*x**4+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0520952, size = 71, normalized size = 0.84 \[ \frac{-b x^4 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^4}{a}\right )-2 a+2 b x^4}{4 a x^2 \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a - b*x^4)^(1/4)),x]
[Out]
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Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(-b*x^4+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(1/4)*x^3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(1/4)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.8398, size = 32, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(-b*x**4+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^4 + a)^(1/4)*x^3),x, algorithm="giac")
[Out]