Optimal. Leaf size=53 \[ -\frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.145291, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[d + e*x]*Sqrt[-2*x - 3*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.9165, size = 49, normalized size = 0.92 \[ - \frac{2 \sqrt{3} \sqrt{1 + \frac{e x}{d}} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{- x}}{2} \right )}\middle | \frac{2 e}{3 d}\right )}{3 \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(1/2)/(-3*x**2-2*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.168358, size = 82, normalized size = 1.55 \[ \frac{i \sqrt{\frac{4}{x}+6} x^{3/2} \sqrt{\frac{d}{e x}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3 d}{2 e}\right )}{\sqrt{-x (3 x+2)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[d + e*x]*Sqrt[-2*x - 3*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.041, size = 115, normalized size = 2.2 \[ -2\,{\frac{d\sqrt{ex+d}\sqrt{-x \left ( 2+3\,x \right ) }}{ex \left ( 3\,e{x}^{2}+3\,dx+2\,ex+2\,d \right ) }{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) \sqrt{-{\frac{ex}{d}}}\sqrt{-{\frac{ \left ( 2+3\,x \right ) e}{3\,d-2\,e}}}\sqrt{{\frac{ex+d}{d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + d} \sqrt{-3 \, x^{2} - 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{e x + d} \sqrt{-3 \, x^{2} - 2 \, x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x \left (3 x + 2\right )} \sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**(1/2)/(-3*x**2-2*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + d} \sqrt{-3 \, x^{2} - 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)),x, algorithm="giac")
[Out]