Optimal. Leaf size=103 \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.100093, antiderivative size = 103, 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 \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^2)^(3/4)/x^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.8268, size = 82, normalized size = 0.8 \[ - \frac{\left (a - b x^{2}\right )^{\frac{3}{4}}}{3 x^{3}} + \frac{b \left (a - b x^{2}\right )^{\frac{3}{4}}}{2 a x} + \frac{b^{\frac{3}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \sqrt [4]{a - b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+a)**(3/4)/x**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0481872, size = 84, normalized size = 0.82 \[ \frac{-4 a^2+3 b^2 x^4 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+10 a b x^2-6 b^2 x^4}{12 a x^3 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^2)^(3/4)/x^4,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+a)^(3/4)/x^4,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(3/4)/x^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(3/4)/x^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.04885, size = 36, normalized size = 0.35 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**2+a)**(3/4)/x**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(3/4)/x^4,x, algorithm="giac")
[Out]