3.2583 \(\int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=256 \[ -\frac{2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{13318 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{\sqrt{2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{\sqrt{2 x+3} (7817373 x+6006884) \sqrt{3 x^2+5 x+2}}{21891870}+\frac{1015187 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{8756748 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{207851 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6254820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-(Sqrt[3 + 2*x]*(6006884 + 7817373*x)*Sqrt[2 + 5*x + 3*x^2])/21891870 + (Sqrt[3
+ 2*x]*(534271 + 629153*x)*(2 + 5*x + 3*x^2)^(3/2))/243243 + (13318*Sqrt[3 + 2*x
]*(2 + 5*x + 3*x^2)^(5/2))/11583 + (202*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(5/2))
/351 - (2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/45 - (207851*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6254820*Sqrt[3]*Sqrt[2 + 5
*x + 3*x^2]) + (1015187*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(8756748*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.570885, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{13318 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{\sqrt{2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{\sqrt{2 x+3} (7817373 x+6006884) \sqrt{3 x^2+5 x+2}}{21891870}+\frac{1015187 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{8756748 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{207851 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6254820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(6006884 + 7817373*x)*Sqrt[2 + 5*x + 3*x^2])/21891870 + (Sqrt[3
+ 2*x]*(534271 + 629153*x)*(2 + 5*x + 3*x^2)^(3/2))/243243 + (13318*Sqrt[3 + 2*x
]*(2 + 5*x + 3*x^2)^(5/2))/11583 + (202*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(5/2))
/351 - (2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/45 - (207851*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6254820*Sqrt[3]*Sqrt[2 + 5
*x + 3*x^2]) + (1015187*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(8756748*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 72.3544, size = 252, normalized size = 0.98 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{45} + \frac{202 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{351} + \frac{8 \sqrt{2 x + 3} \left (\frac{28311885 x}{8} + \frac{24042195}{8}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{10945935} - \frac{4 \sqrt{2 x + 3} \left (\frac{117260595 x}{8} + \frac{22525815}{2}\right ) \sqrt{3 x^{2} + 5 x + 2}}{164189025} + \frac{13318 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{11583} - \frac{207851 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{18764460 \sqrt{3 x^{2} + 5 x + 2}} + \frac{1015187 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{26270244 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

-2*(2*x + 3)**(5/2)*(3*x**2 + 5*x + 2)**(5/2)/45 + 202*(2*x + 3)**(3/2)*(3*x**2
+ 5*x + 2)**(5/2)/351 + 8*sqrt(2*x + 3)*(28311885*x/8 + 24042195/8)*(3*x**2 + 5*
x + 2)**(3/2)/10945935 - 4*sqrt(2*x + 3)*(117260595*x/8 + 22525815/2)*sqrt(3*x**
2 + 5*x + 2)/164189025 + 13318*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(5/2)/11583 - 2
07851*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(
18764460*sqrt(3*x**2 + 5*x + 2)) + 1015187*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(a
sin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(26270244*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.579971, size = 218, normalized size = 0.85 \[ -\frac{2 \left (630485856 x^9+1907623872 x^8-11776907520 x^7-82311172272 x^6-217661096106 x^5-319887585072 x^4-283276026729 x^3-150475882830 x^2-44206631441 x-5523159638\right ) \sqrt{2 x+3}+1590604 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1454957 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{131351220 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-5523159638 - 44206631441*x - 150475882830*x^2 - 283276026729
*x^3 - 319887585072*x^4 - 217661096106*x^5 - 82311172272*x^6 - 11776907520*x^7 +
1907623872*x^8 + 630485856*x^9) + 1454957*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]
+ 1590604*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])/(131351220*(3 + 2*x)*Sqrt[2 +
5*x + 3*x^2])

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Maple [A]  time = 0.039, size = 172, normalized size = 0.7 \[{\frac{1}{7881073200\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+7881073200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -12609717120\,{x}^{9}-38152477440\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+3620978\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +1454957\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+110521391040 \right ) } \]

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)^(3/2),x)

[Out]

1/1313512200*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-12609717120*x^9-38152477440*x^8
[phone]*x^7+1646223445440*x^6+4353221922120*x^5+3620978*(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))+1454957*(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+5665520534580*x^3+30
09604954020*x^2+884278124520*x+110521391040)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (12 \, x^{5} - 4 \, x^{4} - 185 \, x^{3} - 406 \, x^{2} - 327 \, x - 90\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(12*x^5 - 4*x^4 - 185*x^3 - 406*x^2 - 327*x - 90)*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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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