3.2535 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=84 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (10380 x+25003)}{8800}-\frac{56421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

[Out]

(7*(2 + 3*x)^2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
]*(25003 + 10380*x))/8800 - (56421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[1
0])

_______________________________________________________________________________________

Rubi [A]  time = 0.119773, 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 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^2}{11 \sqrt{1-2 x}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (10380 x+25003)}{8800}-\frac{56421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
]*(25003 + 10380*x))/8800 - (56421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[1
0])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.2705, size = 76, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (7785 x + \frac{75009}{4}\right )}{2200} - \frac{56421 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8000} + \frac{7 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*sqrt(5*x + 3)*(7785*x + 75009/4)/2200 - 56421*sqrt(10)*asin(sqrt(
22)*sqrt(5*x + 3)/11)/8000 + 7*(3*x + 2)**2*sqrt(5*x + 3)/(11*sqrt(-2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0865199, size = 64, normalized size = 0.76 \[ \frac{620631 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (11880 x^2+51678 x-97409\right )}{88000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-97409 + 51678*x + 11880*x^2) + 620631*Sqrt[10 - 20*x]*ArcSi
n[Sqrt[5/11]*Sqrt[1 - 2*x]])/(88000*Sqrt[1 - 2*x])

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 106, normalized size = 1.3 \[ -{\frac{1}{-176000+352000\,x} \left ( 1241262\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-237600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-620631\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1033560\,x\sqrt{-10\,{x}^{2}-x+3}+1948180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/176000*(1241262*10^(1/2)*arcsin(20/11*x+1/11)*x-237600*x^2*(-10*x^2-x+3)^(1/2
)-620631*10^(1/2)*arcsin(20/11*x+1/11)-1033560*x*(-10*x^2-x+3)^(1/2)+1948180*(-1
0*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50145, size = 88, normalized size = 1.05 \[ -\frac{56421}{16000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{27}{40} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2619}{800} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \, \sqrt{-10 \, x^{2} - x + 3}}{44 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-56421/16000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 27/40*sqrt(-10*x^2 - x + 3
)*x + 2619/800*sqrt(-10*x^2 - x + 3) - 343/44*sqrt(-10*x^2 - x + 3)/(2*x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.230667, size = 100, normalized size = 1.19 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (11880 \, x^{2} + 51678 \, x - 97409\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 620631 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{176000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/176000*sqrt(10)*(2*sqrt(10)*(11880*x^2 + 51678*x - 97409)*sqrt(5*x + 3)*sqrt(-
2*x + 1) - 620631*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(
-2*x + 1))))/(2*x - 1)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231089, size = 96, normalized size = 1.14 \[ -\frac{56421}{8000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (594 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 63 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 620695 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{220000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-56421/8000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/220000*(594*(4*sqrt
(5)*(5*x + 3) + 63*sqrt(5))*(5*x + 3) - 620695*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x
+ 5)/(2*x - 1)