3.2576 \(\int (5-x) (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=197 \[ -\frac{2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac{202}{189} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{\sqrt{2 x+3} (30033 x+27914) \sqrt{3 x^2+5 x+2}}{8505}+\frac{5773 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3402 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{4729 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2430 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(Sqrt[3 + 2*x]*(27914 + 30033*x)*Sqrt[2 + 5*x + 3*x^2])/8505 + (202*Sqrt[3 + 2*x
]*(2 + 5*x + 3*x^2)^(3/2))/189 - (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27
- (4729*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(24
30*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5773*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSi
n[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3402*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.400082, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac{202}{189} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{\sqrt{2 x+3} (30033 x+27914) \sqrt{3 x^2+5 x+2}}{8505}+\frac{5773 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3402 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{4729 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2430 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 2*x]*(27914 + 30033*x)*Sqrt[2 + 5*x + 3*x^2])/8505 + (202*Sqrt[3 + 2*x
]*(2 + 5*x + 3*x^2)^(3/2))/189 - (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27
- (4729*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(24
30*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5773*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSi
n[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3402*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 54.5871, size = 192, normalized size = 0.97 \[ - \frac{2 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{27} + \frac{4 \sqrt{2 x + 3} \left (\frac{90099 x}{4} + \frac{41871}{2}\right ) \sqrt{3 x^{2} + 5 x + 2}}{25515} + \frac{202 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{189} - \frac{4729 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{7290 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5773 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{10206 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**(3/2)*(3*x**2+5*x+2)**(1/2),x)

[Out]

-2*(2*x + 3)**(3/2)*(3*x**2 + 5*x + 2)**(3/2)/27 + 4*sqrt(2*x + 3)*(90099*x/4 +
41871/2)*sqrt(3*x**2 + 5*x + 2)/25515 + 202*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(3
/2)/189 - 4729*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2)
, -2/3)/(7290*sqrt(3*x**2 + 5*x + 2)) + 5773*sqrt(-9*x**2 - 15*x - 6)*elliptic_f
(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(10206*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.54903, size = 203, normalized size = 1.03 \[ -\frac{2 \left (68040 x^6-59940 x^5-1799874 x^4-5185953 x^3-6208230 x^2-3389617 x-695446\right ) \sqrt{2 x+3}-15784 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+33103 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{51030 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-695446 - 3389617*x - 6208230*x^2 - 5185953*x^3 - 1799874*x^4
- 59940*x^5 + 68040*x^6) + 33103*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sq
rt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 15784*
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])/(51030*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.016, size = 157, normalized size = 0.8 \[ -{\frac{1}{3061800\,{x}^{3}+9695700\,{x}^{2}+9695700\,x+3061800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 1360800\,{x}^{6}-1198800\,{x}^{5}+4238\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -33103\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -35997480\,{x}^{4}-103719060\,{x}^{3}-126150780\,{x}^{2}-71102640\,x-15233040 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(1/2),x)

[Out]

-1/510300*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(1360800*x^6-1198800*x^5+4238*(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))-33103*(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))-35997480*x^4-103719060*x^3-1
26150780*x^2-71102640*x-15233040)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x^{2} - 7 \, x - 15\right )} \sqrt{2 \, x + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*(2*x^2 - 7*x - 15)*sqrt(2*x + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 15 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 2 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**(3/2)*(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-15*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-7*x*sqrt(2*x
+ 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(2*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*
x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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