3.2577 \(\int (5-x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=170 \[ -\frac{2}{21} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{945} \sqrt{2 x+3} (2169 x+2327) \sqrt{3 x^2+5 x+2}+\frac{1039 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{378 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{697 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{270 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(Sqrt[3 + 2*x]*(2327 + 2169*x)*Sqrt[2 + 5*x + 3*x^2])/945 - (2*Sqrt[3 + 2*x]*(2
+ 5*x + 3*x^2)^(3/2))/21 - (697*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(270*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (1039*Sqrt[-2 - 5*x -
3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(378*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi [A]  time = 0.325755, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{21} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{945} \sqrt{2 x+3} (2169 x+2327) \sqrt{3 x^2+5 x+2}+\frac{1039 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{378 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{697 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{270 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 2*x]*(2327 + 2169*x)*Sqrt[2 + 5*x + 3*x^2])/945 - (2*Sqrt[3 + 2*x]*(2
+ 5*x + 3*x^2)^(3/2))/21 - (697*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(270*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (1039*Sqrt[-2 - 5*x -
3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(378*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi in Sympy [A]  time = 46.7768, size = 167, normalized size = 0.98 \[ \frac{2 \sqrt{2 x + 3} \left (\frac{2169 x}{2} + \frac{2327}{2}\right ) \sqrt{3 x^{2} + 5 x + 2}}{945} - \frac{2 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{21} - \frac{697 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{810 \sqrt{3 x^{2} + 5 x + 2}} + \frac{1039 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{1134 \sqrt{3 x^{2} + 5 x + 2}} \]

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]

2*sqrt(2*x + 3)*(2169*x/2 + 2327/2)*sqrt(3*x**2 + 5*x + 2)/945 - 2*sqrt(2*x + 3)
*(3*x**2 + 5*x + 2)**(3/2)/21 - 697*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqr
t(2)*sqrt(6*x + 6)/2), -2/3)/(810*sqrt(3*x**2 + 5*x + 2)) + 1039*sqrt(-9*x**2 -
15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(1134*sqrt(3*x**2 + 5*
x + 2))

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Mathematica [A]  time = 0.454898, size = 198, normalized size = 1.16 \[ -\frac{2 \left (4860 x^5-15552 x^4-121239 x^3-200865 x^2-128926 x-28888\right ) \sqrt{2 x+3}-1762 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+4879 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{5670 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-28888 - 128926*x - 200865*x^2 - 121239*x^3 - 15552*x^4 + 486
0*x^5) + 4879*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] - 1762*Sqrt[5]*Sqrt[(1 + x)/
(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt
[3 + 2*x]], 3/5])/(5670*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.015, size = 152, normalized size = 0.9 \[{\frac{1}{340200\,{x}^{3}+1077300\,{x}^{2}+1077300\,x+340200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -97200\,{x}^{5}+316\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +4879\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +311040\,{x}^{4}+2424780\,{x}^{3}+4310040\,{x}^{2}+3066420\,x+772920 \right ) } \]

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)^(1/2),x)

[Out]

1/56700*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-97200*x^5+316*(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))+4879*(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))+311040*x^4+2424780*x^3+4310040*x^2+3066420*x+
772920)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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. \[ \int -\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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