3.2471 \(\int \frac{\sqrt{3+5 x}}{\sqrt{2+5 x-12 x^2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{1}{3} \sqrt{19} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|\frac{55}{76}\right ) \]

[Out]

-(Sqrt[19]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], 55/76])/3

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Rubi [A]  time = 0.071265, antiderivative size = 30, 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 \[ -\frac{1}{3} \sqrt{19} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|\frac{55}{76}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 + 5*x]/Sqrt[2 + 5*x - 12*x^2],x]

[Out]

-(Sqrt[19]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], 55/76])/3

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Rubi in Sympy [A]  time = 18.6858, size = 80, normalized size = 2.67 \[ - \frac{11 \sqrt{5 x + 3} \sqrt{- \frac{144 x^{2}}{121} + \frac{60 x}{121} + \frac{24}{121}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- \frac{24 x}{11} + \frac{16}{11}}}{2} \right )}\middle | \frac{55}{76}\right )}{6 \sqrt{\frac{15 x}{19} + \frac{9}{19}} \sqrt{- 12 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(1/2)/(-12*x**2+5*x+2)**(1/2),x)

[Out]

-11*sqrt(5*x + 3)*sqrt(-144*x**2/121 + 60*x/121 + 24/121)*elliptic_e(asin(sqrt(2
)*sqrt(-24*x/11 + 16/11)/2), 55/76)/(6*sqrt(15*x/19 + 9/19)*sqrt(-12*x**2 + 5*x
+ 2))

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Mathematica [B]  time = 0.126422, size = 86, normalized size = 2.87 \[ \frac{\sqrt{19} \sqrt{-4 x-1} \sqrt{2-3 x} \left (F\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right )|\frac{21}{76}\right )-E\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right )|\frac{21}{76}\right )\right )}{3 \sqrt{-12 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 + 5*x]/Sqrt[2 + 5*x - 12*x^2],x]

[Out]

(Sqrt[19]*Sqrt[-1 - 4*x]*Sqrt[2 - 3*x]*(-EllipticE[ArcSin[(2*Sqrt[3 + 5*x])/Sqrt
[7]], 21/76] + EllipticF[ArcSin[(2*Sqrt[3 + 5*x])/Sqrt[7]], 21/76]))/(3*Sqrt[2 +
5*x - 12*x^2])

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Maple [B]  time = 0.148, size = 89, normalized size = 3. \[ -{\frac{\sqrt{57}}{6840\,{x}^{2}-2850\,x-1140} \left ({\it EllipticF} \left ({\frac{\sqrt{57}}{19}\sqrt{3+5\,x}},{\frac{2\,\sqrt{57}\sqrt{7}}{21}} \right ) -{\it EllipticE} \left ({\frac{\sqrt{57}}{19}\sqrt{3+5\,x}},{\frac{2\,\sqrt{57}\sqrt{7}}{21}} \right ) \right ) \sqrt{190-285\,x}\sqrt{-35-140\,x}\sqrt{-12\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(1/2)/(-12*x^2+5*x+2)^(1/2),x)

[Out]

-1/570*(EllipticF(1/19*57^(1/2)*(3+5*x)^(1/2),2/21*57^(1/2)*7^(1/2))-EllipticE(1
/19*57^(1/2)*(3+5*x)^(1/2),2/21*57^(1/2)*7^(1/2)))*(190-285*x)^(1/2)*(-35-140*x)
^(1/2)*57^(1/2)*(-12*x^2+5*x+2)^(1/2)/(12*x^2-5*x-2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

integral(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 x + 3}}{\sqrt{- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)**(1/2)/(-12*x**2+5*x+2)**(1/2),x)

[Out]

Integral(sqrt(5*x + 3)/sqrt(-(3*x - 2)*(4*x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2), x)