3.2575 \(\int (5-x) (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=224 \[ -\frac{2}{33} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{5/2}+\frac{730}{891} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{3/2}+\frac{12130 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{2 x+3}}{6237}+\frac{(280359 x+250447) \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}}{56133}+\frac{168145 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{112266 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{32567 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{16038 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(Sqrt[3 + 2*x]*(250447 + 280359*x)*Sqrt[2 + 5*x + 3*x^2])/56133 + (12130*Sqrt[3
+ 2*x]*(2 + 5*x + 3*x^2)^(3/2))/6237 + (730*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3
/2))/891 - (2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2))/33 - (32567*Sqrt[-2 - 5*x
- 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(16038*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2]) + (168145*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(112266*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.472268, antiderivative size = 224, 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 \[ -\frac{2}{33} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{5/2}+\frac{730}{891} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^{3/2}+\frac{12130 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{2 x+3}}{6237}+\frac{(280359 x+250447) \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}}{56133}+\frac{168145 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{112266 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{32567 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{16038 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 2*x]*(250447 + 280359*x)*Sqrt[2 + 5*x + 3*x^2])/56133 + (12130*Sqrt[3
+ 2*x]*(2 + 5*x + 3*x^2)^(3/2))/6237 + (730*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3
/2))/891 - (2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2))/33 - (32567*Sqrt[-2 - 5*x
- 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(16038*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2]) + (168145*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(112266*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 63.4577, size = 218, normalized size = 0.97 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{33} + \frac{730 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{891} + \frac{8 \sqrt{2 x + 3} \left (\frac{4205385 x}{8} + \frac{3756705}{8}\right ) \sqrt{3 x^{2} + 5 x + 2}}{841995} + \frac{12130 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{6237} - \frac{32567 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{48114 \sqrt{3 x^{2} + 5 x + 2}} + \frac{168145 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{336798 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(1/2),x)

[Out]

-2*(2*x + 3)**(5/2)*(3*x**2 + 5*x + 2)**(3/2)/33 + 730*(2*x + 3)**(3/2)*(3*x**2
+ 5*x + 2)**(3/2)/891 + 8*sqrt(2*x + 3)*(4205385*x/8 + 3756705/8)*sqrt(3*x**2 +
5*x + 2)/841995 + 12130*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(3/2)/6237 - 32567*sqr
t(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(48114*sqr
t(3*x**2 + 5*x + 2)) + 168145*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*s
qrt(6*x + 6)/2), -2/3)/(336798*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.571693, size = 208, normalized size = 0.93 \[ -\frac{2 \left (734832 x^7+789264 x^6-18348768 x^5-80563032 x^4-147414969 x^3-137602437 x^2-64194200 x-11846900\right ) \sqrt{2 x+3}-127082 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+227969 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{336798 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-11846900 - 64194200*x - 137602437*x^2 - 147414969*x^3 - 8056
3032*x^4 - 18348768*x^5 + 789264*x^6 + 734832*x^7) + 227969*Sqrt[5]*Sqrt[(1 + x)
/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqr
t[3 + 2*x]], 3/5] - 127082*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])/(336798*(3 + 2
*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.118, size = 162, normalized size = 0.7 \[ -{\frac{1}{20207880\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+20207880}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 14696640\,{x}^{7}[phone]\,{x}^{6}-366975360\,{x}^{5}+59824\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -227969\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -1611260640\,{x}^{4}-2948299380\,{x}^{3}-2765726880\,{x}^{2}-1306680900\,x-246056760 \right ) } \]

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)^(1/2),x)

[Out]

-1/3367980*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(14696640*x^7+15785280*x^6-36697536
0*x^5+59824*(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))-227969*(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))-161126064
0*x^4-2948299380*x^3-2765726880*x^2-1306680900*x-246056760)/(6*x^3+19*x^2+19*x+6
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(4*x^3 - 8*x^2 - 51*x - 45)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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