3.2259 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=143 \[ -\frac{153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac{9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac{99077 (1-2 x)^{3/2} \sqrt{5 x+3}}{25600}+\frac{1089847 \sqrt{1-2 x} \sqrt{5 x+3}}{256000}+\frac{11988317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

[Out]

(1089847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000 - (99077*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/25600 - (9007*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/9600 - (153*(1 - 2*x)^(3/2)*
(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/50 + (11988
317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.162697, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac{9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac{99077 (1-2 x)^{3/2} \sqrt{5 x+3}}{25600}+\frac{1089847 \sqrt{1-2 x} \sqrt{5 x+3}}{256000}+\frac{11988317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1089847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000 - (99077*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/25600 - (9007*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/9600 - (153*(1 - 2*x)^(3/2)*
(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/50 + (11988
317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.2975, size = 129, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (9 x + 6\right )}{50} - \frac{153 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{800} + \frac{9007 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{24000} - \frac{99077 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{192000} - \frac{1089847 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{256000} + \frac{11988317 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2560000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)*(9*x + 6)/50 - 153*(-2*x + 1)**(3/2)*(5*x +
3)**(5/2)/800 + 9007*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/24000 - 99077*sqrt(-2*x + 1
)*(5*x + 3)**(3/2)/192000 - 1089847*sqrt(-2*x + 1)*sqrt(5*x + 3)/256000 + 119883
17*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2560000

_______________________________________________________________________________________

Mathematica [A]  time = 0.102934, size = 70, normalized size = 0.49 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+16790400 x^3+13913120 x^2+2552540 x-4015809\right )-35964951 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7680000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-4015809 + 2552540*x + 13913120*x^2 + 16790400*
x^3 + 6912000*x^4) - 35964951*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/7680000

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 121, normalized size = 0.9 \[{\frac{1}{15360000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-80316180\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)+335808
000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*
arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-80316180*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50137, size = 117, normalized size = 0.82 \[ -\frac{9}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{1677}{800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{17971}{9600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{99077}{12800} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{11988317}{5120000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{99077}{256000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/10*(-10*x^2 - x + 3)^(3/2)*x^2 - 1677/800*(-10*x^2 - x + 3)^(3/2)*x - 17971/9
600*(-10*x^2 - x + 3)^(3/2) + 99077/12800*sqrt(-10*x^2 - x + 3)*x - 11988317/512
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 99077/256000*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.218071, size = 97, normalized size = 0.68 \[ \frac{1}{15360000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} + 16790400 \, x^{3} + 13913120 \, x^{2} + 2552540 \, x - 4015809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 35964951 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360000*sqrt(10)*(2*sqrt(10)*(6912000*x^4 + 16790400*x^3 + 13913120*x^2 + 255
2540*x - 4015809)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 35964951*arctan(1/20*sqrt(10)*(
20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [A]  time = 30.3169, size = 488, normalized size = 3.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-539*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*
x + 1)/121 + asin(sqrt(55)*sqrt(-2*x + 1)/11))/200, (x <= 1/2) & (x > -3/5)))/16
+ 707*sqrt(2)*Piecewise((1331*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)*
*(3/2)/7986 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/1936 + asin(sqrt(
55)*sqrt(-2*x + 1)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/16 - 309*sqrt(2)*Piece
wise((14641*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(
5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(-2*x + 1)*sqrt(1
0*x + 6)*(12100*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/1874048 + 5*
asin(sqrt(55)*sqrt(-2*x + 1)/11)/128)/625, (x <= 1/2) & (x > -3/5)))/16 + 45*sqr
t(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/32
2102 - 5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(-2*x +
1)*sqrt(10*x + 6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(121
00*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/3748096 + 7*asin(sqrt(55)
*sqrt(-2*x + 1)/11)/256)/3125, (x <= 1/2) & (x > -3/5)))/16

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255463, size = 317, normalized size = 2.22 \[ \frac{3}{12800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{29}{640000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{7}{3000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/12800000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13640
5)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) + 29/640000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2
179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) + 7/3000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqr
t(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) +
3/100*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1
/11*sqrt(22)*sqrt(5*x + 3)))