Optimal. Leaf size=94 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.192396, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/Sqrt[b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.9861, size = 83, normalized size = 0.88 \[ \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{\sqrt{c} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 1.12414, size = 121, normalized size = 1.29 \[ -\frac{2 \sqrt{x} \left (\frac{b}{x}+c\right ) \sqrt{d+e x} \left (\frac{d \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )}{\sqrt{-\frac{d}{e}} \sqrt{\frac{b}{c x}+1} \left (\frac{d}{x}+e\right )}-\sqrt{x}\right )}{c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/Sqrt[b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.022, size = 121, normalized size = 1.3 \[ -2\,{\frac{\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }b \left ( be-cd \right ) }{{c}^{2}x \left ( ce{x}^{2}+bex+cdx+bd \right ) }\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(c*x^2+b*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]