3.431 \(\int \frac{1}{\sqrt{d+e x} \sqrt{-2 x-3 x^2}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

(I*Sqrt[6 + 4/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]
], (3*d)/(2*e)])/(Sqrt[-(x*(2 + 3*x))]*Sqrt[d + e*x])

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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]

-2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))*(-e*x/d)^(1/2)*(-(2+
3*x)*e/(3*d-2*e))^(1/2)*((e*x+d)/d)^(1/2)*d*(e*x+d)^(1/2)*(-x*(2+3*x))^(1/2)/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. \[ \int \frac{1}{\sqrt{e x + d} \sqrt{-3 \, x^{2} - 2 \, x}}\,{d x} \]

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]

integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{e x + d} \sqrt{-3 \, x^{2} - 2 \, x}}, x\right ) \]

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]

integral(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x \left (3 x + 2\right )} \sqrt{d + e x}}\, dx \]

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]

Integral(1/(sqrt(-x*(3*x + 2))*sqrt(d + e*x)), x)

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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]

integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 - 2*x)), x)