Optimal. Leaf size=83 \[ \frac{x \sqrt [4]{a+b x^4}}{2 b}+\frac{\sqrt{a} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a+b x^4\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.102876, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sqrt [4]{a+b x^4}}{2 b}+\frac{\sqrt{a} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{b} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^4)^(3/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.8845, size = 70, normalized size = 0.84 \[ \frac{\sqrt{a} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{2 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x \sqrt [4]{a + b x^{4}}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.042783, size = 62, normalized size = 0.75 \[ \frac{x \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{2 b \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^4)^(3/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.038, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^4+a)^(3/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.44469, size = 37, normalized size = 0.45 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/4),x, algorithm="giac")
[Out]