3.2587 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{2 x+3}}-\frac{1}{210} (136-2493 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}-\frac{4427 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{84 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{2411 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{60 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((136 - 2493*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/210 - ((47 + x)*(2 + 5*x +
3*x^2)^(3/2))/(7*Sqrt[3 + 2*x]) + (2411*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(60*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (4427*Sqrt[-2
- 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(84*Sqrt[3]*Sqrt[2
+ 5*x + 3*x^2])

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Rubi [A]  time = 0.321841, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{2 x+3}}-\frac{1}{210} (136-2493 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}-\frac{4427 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{84 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{2411 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{60 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((136 - 2493*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/210 - ((47 + x)*(2 + 5*x +
3*x^2)^(3/2))/(7*Sqrt[3 + 2*x]) + (2411*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(60*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (4427*Sqrt[-2
- 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(84*Sqrt[3]*Sqrt[2
+ 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 45.7929, size = 163, normalized size = 0.94 \[ - \frac{\left (- 2493 x + 136\right ) \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{210} - \frac{\left (x + 47\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{7 \sqrt{2 x + 3}} + \frac{2411 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{180 \sqrt{3 x^{2} + 5 x + 2}} - \frac{4427 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{252 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

-(-2493*x + 136)*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/210 - (x + 47)*(3*x**2 + 5
*x + 2)**(3/2)/(7*sqrt(2*x + 3)) + 2411*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin
(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(180*sqrt(3*x**2 + 5*x + 2)) - 4427*sqrt(-9*x**
2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(252*sqrt(3*x**2 +
5*x + 2))

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Mathematica [A]  time = 0.58154, size = 192, normalized size = 1.11 \[ \frac{-1620 x^5+8208 x^4+18846 x^3+53340 x^2+73094 x-3596 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+16877 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+28772}{1260 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(28772 + 73094*x + 53340*x^2 + 18846*x^3 + 8208*x^4 - 1620*x^5 + 16877*Sqrt[5]*S
qrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSi
n[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 3596*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x
)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5
])/(1260*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 152, normalized size = 0.9 \[ -{\frac{1}{75600\,{x}^{3}+239400\,{x}^{2}+239400\,x+75600}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 5258\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +16877\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +16200\,{x}^{5}-82080\,{x}^{4}-188460\,{x}^{3}+479220\,{x}^{2}+956760\,x+387360 \right ) } \]

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)^(3/2),x)

[Out]

-1/12600*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(5258*(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))+16877
*(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))+16200*x^5-82080*x^4-188460*x^3+479220*x^2+956760*x+38
7360)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx \]

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)**(3/2),x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x) -
Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x)
- Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)
), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x +
3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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