Optimal. Leaf size=80 \[ \frac{1}{2} x \sqrt [4]{a+b x^4}-\frac{\sqrt{a} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a+b x^4\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0894522, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{1}{2} x \sqrt [4]{a+b x^4}-\frac{\sqrt{a} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(1/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.97951, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{a} \sqrt{b} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{2 \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x \sqrt [4]{a + b x^{4}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.038468, size = 58, normalized size = 0.72 \[ \frac{x \left (a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{2 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(1/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int \sqrt [4]{b{x}^{4}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(1/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.29438, size = 37, normalized size = 0.46 \[ \frac{\sqrt [4]{a} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(1/4),x, algorithm="giac")
[Out]