3.2482 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^2}{55 \sqrt{5 x+3}}-\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (300 x+979)}{4400}+\frac{2493 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*Sqrt[3 + 5*x]) - (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x]*(979 + 300*x))/4400 + (2493*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.119971, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^2}{55 \sqrt{5 x+3}}-\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (300 x+979)}{4400}+\frac{2493 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*Sqrt[3 + 5*x]) - (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x]*(979 + 300*x))/4400 + (2493*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.7642, size = 76, normalized size = 0.9 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{55 \sqrt{5 x + 3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (1125 x + \frac{14685}{4}\right )}{5500} + \frac{2493 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**2/(55*sqrt(5*x + 3)) - sqrt(-2*x + 1)*sqrt(5*x + 3)
*(1125*x + 14685/4)/5500 + 2493*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/4000

_______________________________________________________________________________________

Mathematica [A]  time = 0.139273, size = 60, normalized size = 0.71 \[ -\frac{\sqrt{1-2 x} \left (5940 x^2+19305 x+9451\right )}{4400 \sqrt{5 x+3}}-\frac{2493 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(9451 + 19305*x + 5940*x^2))/(4400*Sqrt[3 + 5*x]) - (2493*ArcSin
[Sqrt[5/11]*Sqrt[1 - 2*x]])/(400*Sqrt[10])

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 99, normalized size = 1.2 \[{\frac{1}{88000} \left ( 137115\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-118800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+82269\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -386100\,x\sqrt{-10\,{x}^{2}-x+3}-189020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/88000*(137115*10^(1/2)*arcsin(20/11*x+1/11)*x-118800*x^2*(-10*x^2-x+3)^(1/2)+8
2269*10^(1/2)*arcsin(20/11*x+1/11)-386100*x*(-10*x^2-x+3)^(1/2)-189020*(-10*x^2-
x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51555, size = 88, normalized size = 1.05 \[ \frac{2493}{8000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{27}{100} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1431}{2000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{1375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2493/8000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 27/100*sqrt(-10*x^2 - x + 3)*
x - 1431/2000*sqrt(-10*x^2 - x + 3) - 2/1375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.230768, size = 100, normalized size = 1.19 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (5940 \, x^{2} + 19305 \, x + 9451\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 27423 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{88000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/88000*sqrt(10)*(2*sqrt(10)*(5940*x^2 + 19305*x + 9451)*sqrt(5*x + 3)*sqrt(-2*
x + 1) - 27423*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*
x + 1))))/(5*x + 3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.245574, size = 150, normalized size = 1.79 \[ -\frac{27}{10000} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 41 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{2493}{4000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{13750 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{6875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/10000*(4*sqrt(5)*(5*x + 3) + 41*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 249
3/4000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/13750*sqrt(10)*(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/6875*sqrt(10)*sqrt(5*x + 3)/(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))