3.2586 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\)

Optimal. Leaf size=175 \[ \frac{1}{63} (52-7 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (12429 x+107) \sqrt{3 x^2+5 x+2}}{5670}+\frac{20501 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2268 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{11123 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1620 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-(Sqrt[3 + 2*x]*(107 + 12429*x)*Sqrt[2 + 5*x + 3*x^2])/5670 + ((52 - 7*x)*Sqrt[3
+ 2*x]*(2 + 5*x + 3*x^2)^(3/2))/63 - (11123*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[Ar
cSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1620*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (20501*
Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2268*Sqrt[
3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.331413, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{1}{63} (52-7 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (12429 x+107) \sqrt{3 x^2+5 x+2}}{5670}+\frac{20501 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2268 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{11123 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1620 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(107 + 12429*x)*Sqrt[2 + 5*x + 3*x^2])/5670 + ((52 - 7*x)*Sqrt[3
+ 2*x]*(2 + 5*x + 3*x^2)^(3/2))/63 - (11123*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[Ar
cSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1620*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (20501*
Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2268*Sqrt[
3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 45.8754, size = 165, normalized size = 0.94 \[ \frac{\left (- 21 x + 156\right ) \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{189} - \frac{\sqrt{2 x + 3} \left (12429 x + 107\right ) \sqrt{3 x^{2} + 5 x + 2}}{5670} - \frac{11123 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{4860 \sqrt{3 x^{2} + 5 x + 2}} + \frac{20501 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{6804 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(1/2),x)

[Out]

(-21*x + 156)*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(3/2)/189 - sqrt(2*x + 3)*(12429
*x + 107)*sqrt(3*x**2 + 5*x + 2)/5670 - 11123*sqrt(-9*x**2 - 15*x - 6)*elliptic_
e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(4860*sqrt(3*x**2 + 5*x + 2)) + 20501*sqr
t(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(6804*sqrt
(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.537897, size = 203, normalized size = 1.16 \[ -\frac{2 \left (34020 x^6-88290 x^5-687798 x^4-1306791 x^3-1043385 x^2-312914 x-10832\right ) \sqrt{2 x+3}-16358 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+77861 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{34020 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-10832 - 312914*x - 1043385*x^2 - 1306791*x^3 - 687798*x^4 -
88290*x^5 + 34020*x^6) + 77861*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] - 16358*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])/(34020*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.016, size = 157, normalized size = 0.9 \[{\frac{1}{2041200\,{x}^{3}+6463800\,{x}^{2}+6463800\,x+2041200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -680400\,{x}^{6}+1765800\,{x}^{5}+24644\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +77861\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3331080 \right ) } \]

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)^(1/2),x)

[Out]

1/340200*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-680400*x^6+1765800*x^5+24644*(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))+77861*(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+26135820*x^3+25
539360*x^2+14044380*x+3331080)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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