3.2285 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{1}{10} (3 x+2) \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{23}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{277}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3047 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

[Out]

(277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (23*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 -
((1 - 2*x)^(3/2)*(2 + 3*x)*Sqrt[3 + 5*x])/10 + (3047*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(800*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.116456, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{1}{10} (3 x+2) \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{23}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{277}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3047 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (23*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 -
((1 - 2*x)^(3/2)*(2 + 3*x)*Sqrt[3 + 5*x])/10 + (3047*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(800*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.33415, size = 88, normalized size = 0.89 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3} \left (9 x + 6\right )}{30} - \frac{23 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{80} + \frac{277 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{800} + \frac{3047 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8000} \]

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]

-(-2*x + 1)**(3/2)*sqrt(5*x + 3)*(9*x + 6)/30 - 23*(-2*x + 1)**(3/2)*sqrt(5*x +
3)/80 + 277*sqrt(-2*x + 1)*sqrt(5*x + 3)/800 + 3047*sqrt(10)*asin(sqrt(22)*sqrt(
5*x + 3)/11)/8000

_______________________________________________________________________________________

Mathematica [A]  time = 0.0855087, size = 60, normalized size = 0.61 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (480 x^2+540 x-113\right )-3047 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{8000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-113 + 540*x + 480*x^2) - 3047*Sqrt[10]*ArcSin[
Sqrt[5/11]*Sqrt[1 - 2*x]])/8000

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 87, normalized size = 0.9 \[{\frac{1}{16000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 9600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3047\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +10800\,x\sqrt{-10\,{x}^{2}-x+3}-2260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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]

1/16000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(9600*x^2*(-10*x^2-x+3)^(1/2)+3047*10^(1/2)*
arcsin(20/11*x+1/11)+10800*x*(-10*x^2-x+3)^(1/2)-2260*(-10*x^2-x+3)^(1/2))/(-10*
x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.5062, size = 78, normalized size = 0.79 \[ \frac{3047}{16000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{123}{200} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{31}{800} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3047/16000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 3/50*(-10*x^2 - x + 3)^(3/2)
+ 123/200*sqrt(-10*x^2 - x + 3)*x + 31/800*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.214987, size = 84, normalized size = 0.85 \[ \frac{1}{16000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (480 \, x^{2} + 540 \, x - 113\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3047 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16000*sqrt(10)*(2*sqrt(10)*(480*x^2 + 540*x - 113)*sqrt(5*x + 3)*sqrt(-2*x + 1
) + 3047*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [A]  time = 10.9095, size = 291, normalized size = 2.94 \[ - \frac{49 \sqrt{2} \left (\begin{cases} \frac{11 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{2}\right )}{25} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} + \frac{21 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} - \frac{9 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} \]

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]

-49*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + a
sin(sqrt(55)*sqrt(-2*x + 1)/11)/2)/25, (x <= 1/2) & (x > -3/5)))/4 + 21*sqrt(2)*
Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/968 - s
qrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(-2*x + 1)/11)/8)/
125, (x <= 1/2) & (x > -3/5)))/2 - 9*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*
(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 + 3*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x +
6)*(20*x + 1)/1936 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 5*asin(sqrt(55)*
sqrt(-2*x + 1)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.242749, size = 189, normalized size = 1.91 \[ \frac{3}{40000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{2}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5
) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/500*sqrt(5)*(2*(20*x -
23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*
x + 3)*sqrt(-10*x + 5))