3.2585 \(\int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=202 \[ -\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}+\frac{4153 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-(Sqrt[3 + 2*x]*(1246 + 3987*x)*Sqrt[2 + 5*x + 3*x^2])/8910 + (Sqrt[3 + 2*x]*(11
9 + 127*x)*(2 + 5*x + 3*x^2)^(3/2))/99 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2
))/33 - (15283*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(17820*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (4153*Sqrt[-2 - 5*x - 3*x^2]*Ellipti
cF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3564*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.393753, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}+\frac{4153 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(1246 + 3987*x)*Sqrt[2 + 5*x + 3*x^2])/8910 + (Sqrt[3 + 2*x]*(11
9 + 127*x)*(2 + 5*x + 3*x^2)^(3/2))/99 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2
))/33 - (15283*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(17820*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (4153*Sqrt[-2 - 5*x - 3*x^2]*Ellipti
cF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3564*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 54.3436, size = 197, normalized size = 0.98 \[ \frac{2 \sqrt{2 x + 3} \left (\frac{8001 x}{2} + \frac{7497}{2}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{6237} - \frac{\sqrt{2 x + 3} \left (\frac{83727 x}{2} + 13083\right ) \sqrt{3 x^{2} + 5 x + 2}}{93555} - \frac{2 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{33} - \frac{15283 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{53460 \sqrt{3 x^{2} + 5 x + 2}} + \frac{4153 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{10692 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(2*x + 3)*(8001*x/2 + 7497/2)*(3*x**2 + 5*x + 2)**(3/2)/6237 - sqrt(2*x +
3)*(83727*x/2 + 13083)*sqrt(3*x**2 + 5*x + 2)/93555 - 2*sqrt(2*x + 3)*(3*x**2 +
5*x + 2)**(5/2)/33 - 15283*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt
(6*x + 6)/2), -2/3)/(53460*sqrt(3*x**2 + 5*x + 2)) + 4153*sqrt(-9*x**2 - 15*x -
6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(10692*sqrt(3*x**2 + 5*x + 2)
)

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Mathematica [A]  time = 0.535119, size = 208, normalized size = 1.03 \[ -\frac{2 \left (87480 x^7-48600 x^6-2001510 x^5-6002964 x^4-8112483 x^3-5740860 x^2-2059597 x-293686\right ) \sqrt{2 x+3}-2824 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+15283 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{53460 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-293686 - 2059597*x - 5740860*x^2 - 8112483*x^3 - 6002964*x^4
- 2001510*x^5 - 48600*x^6 + 87480*x^7) + 15283*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] - 2824*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])/(53460*(3 + 2*x)*Sqrt[2 + 5*
x + 3*x^2])

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Maple [A]  time = 0.014, size = 162, normalized size = 0.8 \[{\frac{1}{3207600\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3207600}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -1749600\,{x}^{7}+972000\,{x}^{6}[phone]\,{x}^{5}+5482\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +15283\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+6485040 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/534600*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-1749600*x^7+972000*x^6+40030200*x^5
+5482*(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))+15283*(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))[phone]*x^4+16
2249660*x^3+115734180*x^2+42720240*x+6485040)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*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. \[ - \int \left (- 10 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 23 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 3 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-10*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-23*x*sqrt(2*x
+ 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10*x**2*sqrt(2*x + 3)*sqrt(3*x**2 +
5*x + 2), x) - Integral(3*x**3*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. \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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