3.2593 \(\int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=288 \[ -\frac{2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{430}{969} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{2350 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2907}+\frac{25 \sqrt{2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}}{1247103}-\frac{125 \sqrt{2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}}{52378326}+\frac{25 \sqrt{2 x+3} (216603 x+749099) \sqrt{3 x^2+5 x+2}}{942809868}+\frac{142149125 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1885619736 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{16503475 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{269374248 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(25*Sqrt[3 + 2*x]*(749099 + 216603*x)*Sqrt[2 + 5*x + 3*x^2])/942809868 - (125*Sq
rt[3 + 2*x]*(64006 + 79583*x)*(2 + 5*x + 3*x^2)^(3/2))/52378326 + (25*Sqrt[3 + 2
*x]*(72737 + 86493*x)*(2 + 5*x + 3*x^2)^(5/2))/1247103 + (2350*Sqrt[3 + 2*x]*(2
+ 5*x + 3*x^2)^(7/2))/2907 + (430*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(7/2))/969 -
(2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(7/2))/57 - (16503475*Sqrt[-2 - 5*x - 3*x^
2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(269374248*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2]) + (142149125*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(1885619736*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.651579, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{430}{969} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{2350 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2907}+\frac{25 \sqrt{2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}}{1247103}-\frac{125 \sqrt{2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}}{52378326}+\frac{25 \sqrt{2 x+3} (216603 x+749099) \sqrt{3 x^2+5 x+2}}{942809868}+\frac{142149125 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1885619736 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{16503475 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{269374248 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(25*Sqrt[3 + 2*x]*(749099 + 216603*x)*Sqrt[2 + 5*x + 3*x^2])/942809868 - (125*Sq
rt[3 + 2*x]*(64006 + 79583*x)*(2 + 5*x + 3*x^2)^(3/2))/52378326 + (25*Sqrt[3 + 2
*x]*(72737 + 86493*x)*(2 + 5*x + 3*x^2)^(5/2))/1247103 + (2350*Sqrt[3 + 2*x]*(2
+ 5*x + 3*x^2)^(7/2))/2907 + (430*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(7/2))/969 -
(2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(7/2))/57 - (16503475*Sqrt[-2 - 5*x - 3*x^
2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(269374248*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2]) + (142149125*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(1885619736*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 79.9511, size = 286, normalized size = 0.99 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{57} + \frac{430 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{969} + \frac{8 \sqrt{2 x + 3} \left (\frac{97304625 x}{8} + \frac{81829125}{8}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{56119635} + \frac{2 \sqrt{2 x + 3} \left (\frac{731035125 x}{8} + \frac{2528209125}{8}\right ) \sqrt{3 x^{2} + 5 x + 2}}{31819833045} - \frac{4 \sqrt{2 x + 3} \left (\frac{805777875 x}{8} + \frac{324030375}{4}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{2121322203} + \frac{2350 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{2907} - \frac{16503475 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{808122744 \sqrt{3 x^{2} + 5 x + 2}} + \frac{142149125 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{5656859208 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(5/2)*(3*x**2 + 5*x + 2)**(7/2)/57 + 430*(2*x + 3)**(3/2)*(3*x**2
+ 5*x + 2)**(7/2)/969 + 8*sqrt(2*x + 3)*(97304625*x/8 + 81829125/8)*(3*x**2 + 5*
x + 2)**(5/2)/56119635 + 2*sqrt(2*x + 3)*(731035125*x/8 + 2528209125/8)*sqrt(3*x
**2 + 5*x + 2)/31819833045 - 4*sqrt(2*x + 3)*(805777875*x/8 + 324030375/4)*(3*x*
*2 + 5*x + 2)**(3/2)/2121322203 + 2350*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(7/2)/2
907 - 16503475*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2)
, -2/3)/(808122744*sqrt(3*x**2 + 5*x + 2)) + 142149125*sqrt(-9*x**2 - 15*x - 6)*
elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(5656859208*sqrt(3*x**2 + 5*x +
2))

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Mathematica [A]  time = 0.634218, size = 228, normalized size = 0.79 \[ -\frac{-30234850 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )[phone] \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+2 \left (64309557312 x^{11}[phone] x^{10}-694795413312 x^9-9445976815968 x^8-34294970344572 x^7-69684837178068 x^6-90580760151282 x^5-78460508136978 x^4-45255052994607 x^3-16735272462363 x^2-3595384785664 x-341519551612\right ) \sqrt{2 x+3}}{5656859208 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-341519551612 - 3595384785664*x - 16735272462363*x^2 - 452550
52994607*x^3 - 78460508136978*x^4 - 90580760151282*x^5 - 69684837178068*x^6 - 34
294970344572*x^7 - 9445976815968*x^8 - 694795413312*x^9 + 311460012864*x^10 + 64
309557312*x^11) + 115524325*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] - 30234850*Sqr
t[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[Arc
Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(5656859208*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.04, size = 182, normalized size = 0.6 \[{\frac{1}{67882310496\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+67882310496}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -257238229248\,{x}^{11}-1245840051456\,{x}^{10}[phone]\,{x}^{9}[phone]\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+5324960\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+1367002401048 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11313718416*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-257238229248*x^11-124584005145
6*x^10+2779181653248*x^9+37783907263872*x^8+137179881378288*x^7+278739348712272*
x^6+362323040605128*x^5+5324960*(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+181020211978428*x^3+66942476141352*x^2+14383849
629156*x+1367002401048)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (36 \, x^{7} + 48 \, x^{6} - 551 \, x^{5} - 2151 \, x^{4} - 3381 \, x^{3} - 2717 \, x^{2} - 1104 \, x - 180\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(36*x^7 + 48*x^6 - 551*x^5 - 2151*x^4 - 3381*x^3 - 2717*x^2 - 1104*x -
180)*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+2*x)**(5/2)*(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError