3.2579 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{\sqrt{3 x^2+5 x+2} (x+21)}{3 \sqrt{2 x+3}}-\frac{161 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{121 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((21 + x)*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[3 + 2*x]) + (121*Sqrt[-2 - 5*x - 3*x^2
]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
) - (161*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.258825, antiderivative size = 141, 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 \[ -\frac{\sqrt{3 x^2+5 x+2} (x+21)}{3 \sqrt{2 x+3}}-\frac{161 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{121 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((21 + x)*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[3 + 2*x]) + (121*Sqrt[-2 - 5*x - 3*x^2
]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
) - (161*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 39.8163, size = 134, normalized size = 0.95 \[ - \frac{\left (x + 21\right ) \sqrt{3 x^{2} + 5 x + 2}}{3 \sqrt{2 x + 3}} + \frac{121 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{18 \sqrt{3 x^{2} + 5 x + 2}} - \frac{161 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{18 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x + 21)*sqrt(3*x**2 + 5*x + 2)/(3*sqrt(2*x + 3)) + 121*sqrt(-9*x**2 - 15*x - 6
)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(18*sqrt(3*x**2 + 5*x + 2)) -
161*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(18
*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.567879, size = 185, normalized size = 1.31 \[ \frac{10 \left (-9 x^3+159 x^2+284 x+116\right )-122 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+605 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{90 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(10*(116 + 284*x + 159*x^2 - 9*x^3) + 605*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] - 122*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*
x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(90*Sqrt[3 + 2*x]*Sqrt[2 +
5*x + 3*x^2])

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Maple [A]  time = 0.046, size = 142, normalized size = 1. \[ -{\frac{1}{1080\,{x}^{3}+3420\,{x}^{2}+3420\,x+1080}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 40\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +121\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +180\,{x}^{3}+4080\,{x}^{2}+6420\,x+2520 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/180*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(40*(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))+121*(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))+180*x^3+4080*x^2+6420*x+2520)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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