3.2595 \(\int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=234 \[ -\frac{2}{45} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac{\sqrt{2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac{\sqrt{2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac{(287729-2667537 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{14594580}+\frac{5021353 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5837832 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2742319 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{4169880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((287729 - 2667537*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/14594580 - (Sqrt[3 +
2*x]*(15076 + 34643*x)*(2 + 5*x + 3*x^2)^(3/2))/162162 + (Sqrt[3 + 2*x]*(15467 +
17193*x)*(2 + 5*x + 3*x^2)^(5/2))/19305 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7
/2))/45 - (2742319*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(4169880*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5021353*Sqrt[-2 - 5*x - 3*x^2
]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5837832*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])

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Rubi [A]  time = 0.486497, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{45} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac{\sqrt{2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac{\sqrt{2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac{(287729-2667537 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{14594580}+\frac{5021353 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5837832 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2742319 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{4169880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((287729 - 2667537*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/14594580 - (Sqrt[3 +
2*x]*(15076 + 34643*x)*(2 + 5*x + 3*x^2)^(3/2))/162162 + (Sqrt[3 + 2*x]*(15467 +
17193*x)*(2 + 5*x + 3*x^2)^(5/2))/19305 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7
/2))/45 - (2742319*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(4169880*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5021353*Sqrt[-2 - 5*x - 3*x^2
]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5837832*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])

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Rubi in Sympy [A]  time = 63.7765, size = 230, normalized size = 0.98 \[ \frac{\left (- \frac{8002611 x}{2} + \frac{863187}{2}\right ) \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{21891870} + \frac{2 \sqrt{2 x + 3} \left (\frac{17193 x}{2} + \frac{15467}{2}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{19305} - \frac{\sqrt{2 x + 3} \left (\frac{311787 x}{2} + 67842\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{729729} - \frac{2 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{45} - \frac{2742319 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{12509640 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5021353 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{17513496 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-8002611*x/2 + 863187/2)*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/21891870 + 2*sqrt
(2*x + 3)*(17193*x/2 + 15467/2)*(3*x**2 + 5*x + 2)**(5/2)/19305 - sqrt(2*x + 3)*
(311787*x/2 + 67842)*(3*x**2 + 5*x + 2)**(3/2)/729729 - 2*sqrt(2*x + 3)*(3*x**2
+ 5*x + 2)**(7/2)/45 - 2742319*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*
sqrt(6*x + 6)/2), -2/3)/(12509640*sqrt(3*x**2 + 5*x + 2)) + 5021353*sqrt(-9*x**2
- 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(17513496*sqrt(3*x*
*2 + 5*x + 2))

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Mathematica [A]  time = 0.587848, size = 218, normalized size = 0.93 \[ -\frac{2 \left (315242928 x^9+468822816 x^8-6333945660 x^7-30512259036 x^6-63978029658 x^5-76896556902 x^4-56607962679 x^3-25296672765 x^2-6298405666 x-666434848\right ) \sqrt{2 x+3}-4132174 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )[phone] \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{87567480 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-666434848 - 6298405666*x - 25296672765*x^2 - 56607962679*x^3
- 76896556902*x^4 - 63978029658*x^5 - 30512259036*x^6 - 6333945660*x^7 + 468822
816*x^8 + 315242928*x^9) + 19196233*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] - 4132
174*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ellipt
icF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(87567480*(3 + 2*x)*Sqrt[2 + 5*x + 3*
x^2])

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Maple [A]  time = 0.016, size = 172, normalized size = 0.7 \[{\frac{1}{5254048800\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+5254048800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -6304858560\,{x}^{9}-9376456320\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+5910532\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+14096546280 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/875674800*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-6304858560*x^9-9376456320*x^8+12
6678913200*x^7+610245180720*x^6+1279560593160*x^5+5910532*(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))[phone]*(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+1132159253580*x^3+50708
5229280*x^2+127887736620*x+14096546280)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*sqrt(3*x^2 + 5*x + 2)
*sqrt(2*x + 3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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