3.2473 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{1}{10} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2220 x+5363)}{1600}+\frac{44437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/10 - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(53
63 + 2220*x))/1600 + (44437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi [A]  time = 0.118713, antiderivative size = 84, 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 \[ -\frac{1}{10} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2220 x+5363)}{1600}+\frac{44437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/10 - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(53
63 + 2220*x))/1600 + (44437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi in Sympy [A]  time = 11.4343, size = 75, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{10} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (8325 x + \frac{80445}{4}\right )}{6000} + \frac{44437 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/10 - sqrt(-2*x + 1)*sqrt(5*x + 3)*(83
25*x + 80445/4)/6000 + 44437*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/16000

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Mathematica [A]  time = 0.087835, size = 60, normalized size = 0.71 \[ \frac{-90 \sqrt{1-2 x} \sqrt{5 x+3} \left (160 x^2+460 x+667\right )-44437 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{16000} \]

Antiderivative was successfully verified.

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

[Out]

(-90*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(667 + 460*x + 160*x^2) - 44437*Sqrt[10]*ArcSin
[Sqrt[5/11]*Sqrt[1 - 2*x]])/16000

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Maple [A]  time = 0.028, size = 87, normalized size = 1. \[{\frac{1}{32000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -28800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+44437\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -82800\,x\sqrt{-10\,{x}^{2}-x+3}-120060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-28800*x^2*(-10*x^2-x+3)^(1/2)+44437*10^(1/
2)*arcsin(20/11*x+1/11)-82800*x*(-10*x^2-x+3)^(1/2)-120060*(-10*x^2-x+3)^(1/2))/
(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49255, size = 78, normalized size = 0.93 \[ -\frac{9}{10} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{207}{80} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{44437}{32000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{6003}{1600} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/10*sqrt(-10*x^2 - x + 3)*x^2 - 207/80*sqrt(-10*x^2 - x + 3)*x - 44437/32000*s
qrt(10)*arcsin(-20/11*x - 1/11) - 6003/1600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217762, size = 84, normalized size = 1. \[ -\frac{1}{32000} \, \sqrt{10}{\left (18 \, \sqrt{10}{\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 44437 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32000*sqrt(10)*(18*sqrt(10)*(160*x^2 + 460*x + 667)*sqrt(5*x + 3)*sqrt(-2*x +
1) - 44437*arctan(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. \[ \int \frac{\left (3 x + 2\right )^{3}}{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228928, size = 73, normalized size = 0.87 \[ -\frac{1}{80000} \, \sqrt{5}{\left (18 \,{\left (4 \,{\left (40 \, x + 91\right )}{\left (5 \, x + 3\right )} + 2243\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 222185 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/80000*sqrt(5)*(18*(4*(40*x + 91)*(5*x + 3) + 2243)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 222185*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))