3.2608 \(\int \frac{5-x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=107 \[ \frac{13 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{\sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])) + (13*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*S
qrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.189365, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{13 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{\sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])) + (13*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*S
qrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 31.9476, size = 105, normalized size = 0.98 \[ - \frac{\sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{13 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3*sqr
t(3*x**2 + 5*x + 2)) + 13*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(
6*x + 6)/2), -2/3)/(3*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.772939, size = 177, normalized size = 1.65 \[ -\frac{30 x^2+50 x+34 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+5 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+20}{15 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(20 + 50*x + 30*x^2 + 5*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2
+ 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 34*Sqrt[5]*
Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcS
in[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(15*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.019, size = 71, normalized size = 0.7 \[{\frac{\sqrt{15}}{30} \left ( 12\,{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +{\it EllipticE} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}},{\frac{\sqrt{15}}{3}} \right ) \right ) \sqrt{-30\,x-20}\sqrt{-2-2\,x}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(12*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+EllipticE(1/5*15^(1/
2)*(3+2*x)^(1/2),1/3*15^(1/2)))*(-30*x-20)^(1/2)*(-2-2*x)^(1/2)*15^(1/2)/(3*x^2+
5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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