Optimal. Leaf size=29 \[ \frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0498754, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[(a*g + e*x - b*g*x^4)/(a + b*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.11, size = 22, normalized size = 0.76 \[ \frac{x \left (2 a g + e x\right )}{2 a \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*g*x**4+a*g+e*x)/(b*x**4+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0414228, size = 29, normalized size = 1. \[ \frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*g + e*x - b*g*x^4)/(a + b*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 24, normalized size = 0.8 \[{\frac{x \left ( 2\,ag+ex \right ) }{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*g*x^4+a*g+e*x)/(b*x^4+a)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.45333, size = 34, normalized size = 1.17 \[ \frac{2 \, a g x + e x^{2}}{2 \, \sqrt{b x^{4} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - a*g - e*x)/(b*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.217029, size = 46, normalized size = 1.59 \[ \frac{\sqrt{b x^{4} + a}{\left (2 \, a g x + e x^{2}\right )}}{2 \,{\left (a b x^{4} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - a*g - e*x)/(b*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 19.287, size = 104, normalized size = 3.59 \[ \frac{g x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} - \frac{b g x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{e x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*g*x**4+a*g+e*x)/(b*x**4+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.214844, size = 43, normalized size = 1.48 \[ -\frac{x{\left (\frac{2 \, g}{a^{2} b^{4}} + \frac{x e}{a^{3} b^{4}}\right )}}{64 \, \sqrt{b x^{4} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - a*g - e*x)/(b*x^4 + a)^(3/2),x, algorithm="giac")
[Out]