3.2474 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{3}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)-\frac{333}{400} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3827 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(-333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x
])/20 + (3827*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

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Rubi [A]  time = 0.0980927, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)-\frac{333}{400} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3827 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x
])/20 + (3827*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

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Rubi in Sympy [A]  time = 7.64595, size = 68, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (9 x + 6\right )}{20} - \frac{333 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{400} + \frac{3827 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*sqrt(5*x + 3)*(9*x + 6)/20 - 333*sqrt(-2*x + 1)*sqrt(5*x + 3)/40
0 + 3827*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/4000

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Mathematica [A]  time = 0.0696942, size = 55, normalized size = 0.71 \[ \frac{-30 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+151)-3827 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4000} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(151 + 60*x) - 3827*Sqrt[10]*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]])/4000

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Maple [A]  time = 0.02, size = 70, normalized size = 0.9 \[{\frac{1}{8000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3827\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3600\,x\sqrt{-10\,{x}^{2}-x+3}-9060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3827*10^(1/2)*arcsin(20/11*x+1/11)-3600*x*(-
10*x^2-x+3)^(1/2)-9060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49715, size = 55, normalized size = 0.71 \[ -\frac{9}{20} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{3827}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{453}{400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

-9/20*sqrt(-10*x^2 - x + 3)*x - 3827/8000*sqrt(10)*arcsin(-20/11*x - 1/11) - 453
/400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.220948, size = 77, normalized size = 1. \[ -\frac{1}{8000} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (60 \, x + 151\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3827 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

-1/8000*sqrt(10)*(6*sqrt(10)*(60*x + 151)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3827*ar
ctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2}}{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**2/(sqrt(-2*x + 1)*sqrt(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.226409, size = 61, normalized size = 0.79 \[ -\frac{1}{4000} \, \sqrt{5}{\left (6 \,{\left (60 \, x + 151\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 3827 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^2/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

-1/4000*sqrt(5)*(6*(60*x + 151)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 3827*sqrt(2)*arc
sin(1/11*sqrt(22)*sqrt(5*x + 3)))