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

Optimal. Leaf size=51 \[ \frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]

[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 = 0.100718, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 - 3*x]*Sqrt[x]*Sqrt[d + e*x]),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 = 8.03634, size = 66, normalized size = 1.29 \[ \frac{2 \sqrt{6} \sqrt{1 + \frac{e x}{d}} \sqrt{- \frac{3 x}{2} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{2} \right )}\middle | - \frac{2 e}{3 d}\right )}{3 \sqrt{d + e x} \sqrt{- 3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.297958, size = 72, normalized size = 1.41 \[ -\frac{\sqrt{x} \sqrt{\frac{d+e x}{e (3 x-2)}} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{1-\frac{3 x}{2}}}\right )|\frac{3 d}{2 e}+1\right )}{\sqrt{\frac{x}{6 x-4}} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.089, size = 112, normalized size = 2.2 \[ -2\,{\frac{d\sqrt{2-3\,x}\sqrt{ex+d}}{\sqrt{x}e \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) \sqrt{-{\frac{ex}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{{\frac{ex+d}{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(e*x + d)*sqrt(x)*sqrt(-3*x + 2)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(e*x + d)*sqrt(x)*sqrt(-3*x + 2)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(e*x + d)*sqrt(x)*sqrt(-3*x + 2)),x, algorithm="giac")

[Out]

Exception raised: TypeError