3.1036 \(\int (2-5 x) x^{3/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=205 \[ \frac{136}{189} \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}-\frac{4 \sqrt{x} (1035 x+779) \sqrt{3 x^2+5 x+2}}{1701}+\frac{2360 \sqrt{x} (3 x+2)}{5103 \sqrt{3 x^2+5 x+2}}+\frac{668 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{1701 \sqrt{3 x^2+5 x+2}}-\frac{2360 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}-\frac{10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2} \]

[Out]

(2360*Sqrt[x]*(2 + 3*x))/(5103*Sqrt[2 + 5*x + 3*x^2]) - (4*Sqrt[x]*(779 + 1035*x
)*Sqrt[2 + 5*x + 3*x^2])/1701 + (136*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/189 - (10*
x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 - (2360*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 +
x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(5103*Sqrt[2 + 5*x + 3*x^2]) + (668*Sqrt[2
]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(1701*Sqrt[2
+ 5*x + 3*x^2])

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Rubi [A]  time = 0.348024, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{136}{189} \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}-\frac{4 \sqrt{x} (1035 x+779) \sqrt{3 x^2+5 x+2}}{1701}+\frac{2360 \sqrt{x} (3 x+2)}{5103 \sqrt{3 x^2+5 x+2}}+\frac{668 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{1701 \sqrt{3 x^2+5 x+2}}-\frac{2360 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}-\frac{10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(2360*Sqrt[x]*(2 + 3*x))/(5103*Sqrt[2 + 5*x + 3*x^2]) - (4*Sqrt[x]*(779 + 1035*x
)*Sqrt[2 + 5*x + 3*x^2])/1701 + (136*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/189 - (10*
x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 - (2360*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 +
x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(5103*Sqrt[2 + 5*x + 3*x^2]) + (668*Sqrt[2
]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(1701*Sqrt[2
+ 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 37.3821, size = 192, normalized size = 0.94 \[ - \frac{10 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{27} + \frac{1180 \sqrt{x} \left (6 x + 4\right )}{5103 \sqrt{3 x^{2} + 5 x + 2}} - \frac{16 \sqrt{x} \left (\frac{15525 x}{4} + \frac{11685}{4}\right ) \sqrt{3 x^{2} + 5 x + 2}}{25515} + \frac{136 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{189} - \frac{590 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{5103 \sqrt{3 x^{2} + 5 x + 2}} + \frac{167 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{1701 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*x**(3/2)*(3*x**2 + 5*x + 2)**(3/2)/27 + 1180*sqrt(x)*(6*x + 4)/(5103*sqrt(3*
x**2 + 5*x + 2)) - 16*sqrt(x)*(15525*x/4 + 11685/4)*sqrt(3*x**2 + 5*x + 2)/25515
+ 136*sqrt(x)*(3*x**2 + 5*x + 2)**(3/2)/189 - 590*sqrt((6*x + 4)/(x + 1))*(4*x
+ 4)*elliptic_e(atan(sqrt(x)), -1/2)/(5103*sqrt(3*x**2 + 5*x + 2)) + 167*sqrt((6
*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(1701*sqrt(3*x**2 + 5
*x + 2))

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Mathematica [C]  time = 0.252552, size = 165, normalized size = 0.8 \[ \frac{-356 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2360 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-17010 x^6-23652 x^5+2970 x^4+7920 x^3+1380 x^2+7792 x+4720}{5103 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(4720 + 7792*x + 1380*x^2 + 7920*x^3 + 2970*x^4 - 23652*x^5 - 17010*x^6 + (2360*
I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/
Sqrt[x]], 3/2] - (356*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*Elliptic
F[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(5103*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.027, size = 133, normalized size = 0.7 \[ -{\frac{2}{15309} \left ( 25515\,{x}^{6}+35478\,{x}^{5}+768\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -590\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -4455\,{x}^{4}-11880\,{x}^{3}+8550\,{x}^{2}+6012\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15309/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(25515*x^6+35478*x^5+768*(6*x+4)^(1/2)*(3+3
*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-590*
(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/
2),I*2^(1/2))-4455*x^4-11880*x^3+8550*x^2+6012*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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