3.2615 \(\int \frac{(5-x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{2 \sqrt{2 x+3} (35 x+29)}{\sqrt{3 x^2+5 x+2}}-\frac{94 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{70 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[3 + 2*x]*(29 + 35*x))/Sqrt[2 + 5*x + 3*x^2] + (70*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])
- (94*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.266136, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 \sqrt{2 x+3} (35 x+29)}{\sqrt{3 x^2+5 x+2}}-\frac{94 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{70 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 2*x]*(29 + 35*x))/Sqrt[2 + 5*x + 3*x^2] + (70*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])
- (94*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 39.5445, size = 136, normalized size = 0.99 \[ - \frac{2 \sqrt{2 x + 3} \left (35 x + 29\right )}{\sqrt{3 x^{2} + 5 x + 2}} + \frac{70 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{94 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

-2*sqrt(2*x + 3)*(35*x + 29)/sqrt(3*x**2 + 5*x + 2) + 70*sqrt(-9*x**2 - 15*x - 6
)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3*sqrt(3*x**2 + 5*x + 2)) - 9
4*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3*sq
rt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.512691, size = 178, normalized size = 1.3 \[ -\frac{10 (139 x+121) \sqrt{2 x+3}+68 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-350 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{15 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[3 + 2*x]*(121 + 139*x) - 350*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)
^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 6
8*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Elliptic
F[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(15*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.025, size = 137, normalized size = 1. \[ -{\frac{1}{90\,{x}^{3}+285\,{x}^{2}+285\,x+90}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 12\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +35\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +2100\,{x}^{2}+4890\,x+2610 \right ) } \]

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)^(3/2),x)

[Out]

-1/15*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(12*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2
)*(-30*x-20)^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+35*(3+2*x)
^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(
1/2),1/3*15^(1/2))+2100*x^2+4890*x+2610)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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