3.430 \(\int \frac{\sqrt{d+e x}}{\sqrt{-2 x-3 x^2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(-2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[-x]], (2*e)/(3*d)])/(Sqrt[3]*S
qrt[1 + (e*x)/d])

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Rubi [A]  time = 0.139648, 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 \[ -\frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],x]

[Out]

(-2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[-x]], (2*e)/(3*d)])/(Sqrt[3]*S
qrt[1 + (e*x)/d])

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Rubi in Sympy [A]  time = 17.4952, size = 49, normalized size = 0.92 \[ - \frac{2 \sqrt{3} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{- x}}{2} \right )}\middle | \frac{2 e}{3 d}\right )}{3 \sqrt{1 + \frac{e x}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(-3*x**2-2*x)**(1/2),x)

[Out]

-2*sqrt(3)*sqrt(d + e*x)*elliptic_e(asin(sqrt(6)*sqrt(-x)/2), 2*e/(3*d))/(3*sqrt
(1 + e*x/d))

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Mathematica [B]  time = 0.356665, size = 117, normalized size = 2.21 \[ \frac{2 (3 x+2) \sqrt{-\frac{d}{e}} (d+e x)-2 d \sqrt{\frac{6}{x}+9} x^{3/2} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{2 e}{3 d}\right )}{3 \sqrt{-x (3 x+2)} \sqrt{-\frac{d}{e}} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],x]

[Out]

(2*Sqrt[-(d/e)]*(2 + 3*x)*(d + e*x) - 2*d*Sqrt[9 + 6/x]*Sqrt[1 + d/(e*x)]*x^(3/2
)*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (2*e)/(3*d)])/(3*Sqrt[-(d/e)]*Sqrt[-(x
*(2 + 3*x))]*Sqrt[d + e*x])

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Maple [B]  time = 0.067, size = 215, normalized size = 4.1 \[ -{\frac{2\,d}{3\,ex \left ( 3\,e{x}^{2}+3\,dx+2\,ex+2\,d \right ) }\sqrt{ex+d}\sqrt{-x \left ( 2+3\,x \right ) }\sqrt{{\frac{ex+d}{d}}}\sqrt{-{\frac{ \left ( 2+3\,x \right ) e}{3\,d-2\,e}}}\sqrt{-{\frac{ex}{d}}} \left ( 3\,d{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) -2\,{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) e-3\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) d+2\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) e \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x)

[Out]

-2/3*(e*x+d)^(1/2)*(-x*(2+3*x))^(1/2)*d*((e*x+d)/d)^(1/2)*(-(2+3*x)*e/(3*d-2*e))
^(1/2)*(-e*x/d)^(1/2)*(3*d*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/
2))-2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))*e-3*EllipticE(((e
*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))*d+2*EllipticE(((e*x+d)/d)^(1/2),3^(1
/2)*(d/(3*d-2*e))^(1/2))*e)/e/x/(3*e*x^2+3*d*x+2*e*x+2*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{-3 \, x^{2} - 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x + d}}{\sqrt{-3 \, x^{2} - 2 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x),x, algorithm="fricas")

[Out]

integral(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{- x \left (3 x + 2\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(-3*x**2-2*x)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt(-x*(3*x + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{-3 \, x^{2} - 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)/sqrt(-3*x^2 - 2*x), x)