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

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

[Out]

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

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Rubi [A]  time = 0.137957, antiderivative size = 51, 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]*Sq
rt[1 + (e*x)/d])

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Rubi in Sympy [A]  time = 17.0643, size = 48, normalized size = 0.94 \[ \frac{2 \sqrt{3} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{2} \right )}\middle | - \frac{2 e}{3 d}\right )}{3 \sqrt{1 + \frac{e x}{d}}} \]

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 = 0.515927, size = 117, normalized size = 2.29 \[ \frac{2 (3 x-2) \sqrt{-\frac{d}{e}} (d+e x)-2 d \sqrt{9-\frac{6}{x}} x^{3/2} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|-\frac{2 e}{3 d}\right )}{3 \sqrt{-x (3 x-2)} \sqrt{-\frac{d}{e}} \sqrt{d+e x}} \]

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 = 0.075, size = 215, normalized size = 4.2 \[ -{\frac{2\,d}{3\,ex \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }\sqrt{ex+d}\sqrt{-x \left ( -2+3\,x \right ) }\sqrt{{\frac{ex+d}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{-{\frac{ex}{d}}} \left ( 3\,d{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) +2\,{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e-3\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) d-2\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e \right ) } \]

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)