3.2258 \(\int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}-\frac{7 (1-2 x)^{3/2} (2256 x+3821) (5 x+3)^{5/2}}{32000}-\frac{953981 (1-2 x)^{3/2} (5 x+3)^{3/2}}{384000}-\frac{10493791 (1-2 x)^{3/2} \sqrt{5 x+3}}{1024000}+\frac{115431701 \sqrt{1-2 x} \sqrt{5 x+3}}{10240000}+\frac{1269748711 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10240000 \sqrt{10}} \]

[Out]

(115431701*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/10240000 - (10493791*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/1024000 - (953981*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/384000 - ((1 - 2*
x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 - (7*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)*(3
821 + 2256*x))/32000 + (1269748711*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10240000*S
qrt[10])

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Rubi [A]  time = 0.18408, antiderivative size = 150, 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 \[ -\frac{1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}-\frac{7 (1-2 x)^{3/2} (2256 x+3821) (5 x+3)^{5/2}}{32000}-\frac{953981 (1-2 x)^{3/2} (5 x+3)^{3/2}}{384000}-\frac{10493791 (1-2 x)^{3/2} \sqrt{5 x+3}}{1024000}+\frac{115431701 \sqrt{1-2 x} \sqrt{5 x+3}}{10240000}+\frac{1269748711 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10240000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(115431701*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/10240000 - (10493791*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/1024000 - (953981*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/384000 - ((1 - 2*
x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 - (7*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)*(3
821 + 2256*x))/32000 + (1269748711*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10240000*S
qrt[10])

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Rubi in Sympy [A]  time = 16.525, size = 136, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{5}{2}}}{20} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (59220 x + \frac{401205}{4}\right )}{120000} + \frac{953981 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{960000} - \frac{10493791 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{7680000} - \frac{115431701 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10240000} + \frac{1269748711 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{102400000} \]

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*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3)**(5/2)/20 - (-2*x + 1)**(3/2)*(5*x + 3
)**(5/2)*(59220*x + 401205/4)/120000 + 953981*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/96
0000 - 10493791*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/7680000 - 115431701*sqrt(-2*x +
1)*sqrt(5*x + 3)/10240000 + 1269748711*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/
102400000

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Mathematica [A]  time = 0.119117, size = 75, normalized size = 0.5 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (691200000 x^5+2163456000 x^4+2600899200 x^3+1349400160 x^2+21761620 x-483864147\right )-3809246133 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{307200000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-483864147 + 21761620*x + 1349400160*x^2 + 2600
899200*x^3 + 2163456000*x^4 + 691200000*x^5) - 3809246133*Sqrt[10]*ArcSin[Sqrt[5
/11]*Sqrt[1 - 2*x]])/307200000

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Maple [A]  time = 0.013, size = 138, normalized size = 0.9 \[{\frac{1}{614400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-9677282940\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/614400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)+432
69120000*x^4*(-10*x^2-x+3)^(1/2)[phone]*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)-9677282940*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49218, size = 140, normalized size = 0.93 \[ -\frac{9}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{2727}{400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{270711}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2147273}{384000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{10493791}{512000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1269748711}{204800000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{10493791}{10240000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 2727/400*(-10*x^2 - x + 3)^(3/2)*x^2 - 270711
/32000*(-10*x^2 - x + 3)^(3/2)*x - 2147273/384000*(-10*x^2 - x + 3)^(3/2) + 1049
3791/512000*sqrt(-10*x^2 - x + 3)*x - 1269748711/204800000*sqrt(10)*arcsin(-20/1
1*x - 1/11) + 10493791/10240000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.221153, size = 104, normalized size = 0.69 \[ \frac{1}{614400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (691200000 \, x^{5} + 2163456000 \, x^{4} + 2600899200 \, x^{3} + 1349400160 \, x^{2} + 21761620 \, x - 483864147\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3809246133 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/614400000*sqrt(10)*(2*sqrt(10)*(691200000*x^5 + 2163456000*x^4 + 2600899200*x^
3 + 1349400160*x^2 + 21761620*x - 483864147)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3809
246133*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 48.1956, size = 694, normalized size = 4.63 \[ \text{result too large to display} \]

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]

-3773*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)))/3
2 + 3283*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(sqr
t(55)*sqrt(-2*x + 1)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/16 - 1071*sqrt(2)*Pi
ecewise((14641*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sq
rt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(-2*x + 1)*sqr
t(10*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)))/8 + 621*
sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)
/322102 - 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)*(
12100*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 - 135*sqrt(2)*Pie
cewise((1771561*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/161051 +
5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 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)/15488 - 13*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(12100*x - 2000*(
-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sqrt(-2*x
+ 1)/11)/1024)/15625, (x <= 1/2) & (x > -3/5)))/32

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GIAC/XCAS [A]  time = 0.267317, size = 427, normalized size = 2.85 \[ \frac{9}{512000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{117}{64000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{57}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{37}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{50} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/512000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) -
318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 117/64000000*sqrt(5)*
(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 6055
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) + 57/320000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 41
25)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) + 37/6000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-1
0*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/50*sqrt(5)*(2*(2
0*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5
*x + 3)))