3.2257 \(\int \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac{403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac{52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac{580364059 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480000}+\frac{6384004649 \sqrt{1-2 x} \sqrt{5 x+3}}{204800000}+\frac{70224051139 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800000 \sqrt{10}} \]

[Out]

(6384004649*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800000 - (580364059*(1 - 2*x)^(3/2)*
Sqrt[3 + 5*x])/20480000 - (52760369*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/7680000 - (
403*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(3/2)*(2 +
3*x)^3*(3 + 5*x)^(5/2))/70 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)*(1480103 + 874608*
x))/640000 + (70224051139*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800000*Sqrt[10])

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Rubi [A]  time = 0.252496, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac{403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac{52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac{580364059 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480000}+\frac{6384004649 \sqrt{1-2 x} \sqrt{5 x+3}}{204800000}+\frac{70224051139 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2),x]

[Out]

(6384004649*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800000 - (580364059*(1 - 2*x)^(3/2)*
Sqrt[3 + 5*x])/20480000 - (52760369*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/7680000 - (
403*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(3/2)*(2 +
3*x)^3*(3 + 5*x)^(5/2))/70 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)*(1480103 + 874608*
x))/640000 + (70224051139*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800000*Sqrt[10])

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Rubi in Sympy [A]  time = 24.0542, size = 165, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{5}{2}}}{70} - \frac{403 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{5}{2}}}{2800} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (11479230 x + \frac{155410815}{8}\right )}{8400000} + \frac{52760369 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{19200000} - \frac{580364059 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{153600000} - \frac{6384004649 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{204800000} + \frac{70224051139 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2048000000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4*(3+5*x)**(3/2)*(1-2*x)**(1/2),x)

[Out]

-3*(-2*x + 1)**(3/2)*(3*x + 2)**3*(5*x + 3)**(5/2)/70 - 403*(-2*x + 1)**(3/2)*(3
*x + 2)**2*(5*x + 3)**(5/2)/2800 - (-2*x + 1)**(3/2)*(5*x + 3)**(5/2)*(11479230*
x + 155410815/8)/8400000 + 52760369*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/19200000 - 5
80364059*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/153600000 - 6384004649*sqrt(-2*x + 1)*s
qrt(5*x + 3)/204800000 + 70224051139*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20
48000000

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Mathematica [A]  time = 0.150524, size = 80, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (248832000000 x^6+950400000000 x^5+1480681728000 x^4+1161696585600 x^3+402838062880 x^2-72932734340 x-201521732121\right )-1474705073919 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{43008000000} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2),x]

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-201521732121 - 72932734340*x + 402838062880*x^
2 + 1161696585600*x^3 + 1480681728000*x^4 + 950400000000*x^5 + 248832000000*x^6)
- 1474705073919*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/43008000000

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Maple [A]  time = 0.016, size = 155, normalized size = 0.9 \[{\frac{1}{86016000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4976640000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1458654686800\,x\sqrt{-10\,{x}^{2}-x+3}-4030434642420\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x)

[Out]

1/86016000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4976640000000*x^6*(-10*x^2-x+3)^(1/2)
[phone]*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)+2
3233931712000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)+1474
705073919*10^(1/2)*arcsin(20/11*x+1/11)-1458654686800*x*(-10*x^2-x+3)^(1/2)-4030
434642420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49661, size = 163, normalized size = 0.91 \[ -\frac{81}{14} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{12051}{560} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{1904661}{56000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{134695173}{4480000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{890455739}{53760000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{580364059}{10240000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{70224051139}{4096000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{580364059}{204800000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

-81/14*(-10*x^2 - x + 3)^(3/2)*x^4 - 12051/560*(-10*x^2 - x + 3)^(3/2)*x^3 - 190
4661/56000*(-10*x^2 - x + 3)^(3/2)*x^2 - 134695173/4480000*(-10*x^2 - x + 3)^(3/
2)*x - 890455739/53760000*(-10*x^2 - x + 3)^(3/2) + 580364059/10240000*sqrt(-10*
x^2 - x + 3)*x - 70224051139/4096000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 58036
4059/204800000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.234322, size = 111, normalized size = 0.62 \[ \frac{1}{86016000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (248832000000 \, x^{6} + 950400000000 \, x^{5} + 1480681728000 \, x^{4} + 1161696585600 \, x^{3} + 402838062880 \, x^{2} - 72932734340 \, x - 201521732121\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1474705073919 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

1/86016000000*sqrt(10)*(2*sqrt(10)*(248832000000*x^6 + 950400000000*x^5 + 148068
1728000*x^4 + 1161696585600*x^3 + 402838062880*x^2 - 72932734340*x - 20152173212
1)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1474705073919*arctan(1/20*sqrt(10)*(20*x + 1)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 76.3984, size = 925, normalized size = 5.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**4*(3+5*x)**(3/2)*(1-2*x)**(1/2),x)

[Out]

-26411*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(2
0*x + 1)/121 + asin(sqrt(55)*sqrt(-2*x + 1)/11))/200, (x <= 1/2) & (x > -3/5)))/
64 + 57281*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(s
qrt(55)*sqrt(-2*x + 1)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/64 - 24843*sqrt(2)
*Piecewise((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(10*x + 6)*(12100*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/187404
8 + 5*asin(sqrt(55)*sqrt(-2*x + 1)/11)/128)/625, (x <= 1/2) & (x > -3/5)))/32 +
10773*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)*sqr
t(-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)))/32 - 4671*sqrt
(2)*Piecewise((1771561*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/16
1051 + 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)*sqr
t(-2*x + 1)/11)/1024)/15625, (x <= 1/2) & (x > -3/5)))/64 + 405*sqrt(2)*Piecewis
e((19487171*sqrt(5)*(-125*sqrt(5)*(-2*x + 1)**(7/2)*(10*x + 6)**(7/2)/272820394
+ 15*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/322102 + 25*sqrt(5)*(-2*x + 1)*
*(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/340139712 - 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)/30976 -
25*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(12100*x - 2000*(-2*x + 1)**3 + 6600*(
-2*x + 1)**2 - 4719)/29984768 + 33*asin(sqrt(55)*sqrt(-2*x + 1)/11)/2048)/78125,
(x <= 1/2) & (x > -3/5)))/64

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GIAC/XCAS [A]  time = 0.297389, size = 548, normalized size = 3.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="giac")

[Out]

27/71680000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x
+ 3) - 3968469)*(5*x + 3) + 33617829)*(5*x + 3) - 276044685)*(5*x + 3) + 873561
15)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(
5*x + 3))) + 441/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27
999)*(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))) + 9/100
0000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*
x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3))) + 47/80000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5
*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(2
2)*sqrt(5*x + 3))) + 23/1500*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/25*s
qrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqr
t(22)*sqrt(5*x + 3)))