3.2344 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3 \sqrt{5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac{3}{50} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{51373 \sqrt{5 x+3} (1-2 x)^{3/2}}{320000}+\frac{1695309 \sqrt{5 x+3} \sqrt{1-2 x}}{3200000}+\frac{18648399 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200000 \sqrt{10}} \]

[Out]

(1695309*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200000 + (51373*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/320000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/50 - (3*(1 - 2*x)^(
5/2)*Sqrt[3 + 5*x]*(14629 + 11580*x))/80000 + (18648399*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(3200000*Sqrt[10])

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Rubi [A]  time = 0.16505, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3 \sqrt{5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac{3}{50} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{51373 \sqrt{5 x+3} (1-2 x)^{3/2}}{320000}+\frac{1695309 \sqrt{5 x+3} \sqrt{1-2 x}}{3200000}+\frac{18648399 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1695309*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200000 + (51373*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/320000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/50 - (3*(1 - 2*x)^(
5/2)*Sqrt[3 + 5*x]*(14629 + 11580*x))/80000 + (18648399*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(3200000*Sqrt[10])

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Rubi in Sympy [A]  time = 14.8214, size = 117, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{50} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3} \left (26055 x + \frac{131661}{4}\right )}{60000} + \frac{51373 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{320000} + \frac{1695309 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3200000} + \frac{18648399 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{32000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)*(3*x + 2)**2*sqrt(5*x + 3)/50 - (-2*x + 1)**(5/2)*sqrt(5*x
+ 3)*(26055*x + 131661/4)/60000 + 51373*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/320000 +
1695309*sqrt(-2*x + 1)*sqrt(5*x + 3)/3200000 + 18648399*sqrt(10)*asin(sqrt(22)*
sqrt(5*x + 3)/11)/32000000

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Mathematica [A]  time = 0.12897, size = 70, normalized size = 0.55 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+7862400 x^3-2952480 x^2-5372860 x+314441\right )-18648399 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{32000000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(314441 - 5372860*x - 2952480*x^2 + 7862400*x^3
+ 6912000*x^4) - 18648399*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/32000000

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Maple [A]  time = 0.016, size = 121, normalized size = 1. \[{\frac{1}{64000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-157248000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-6288820\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-15724
8000*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)-6288820*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51402, size = 124, normalized size = 0.97 \[ -\frac{54}{25} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{2457}{1000} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{18453}{20000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{268643}{160000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{18648399}{64000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{314441}{3200000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54/25*sqrt(-10*x^2 - x + 3)*x^4 - 2457/1000*sqrt(-10*x^2 - x + 3)*x^3 + 18453/2
0000*sqrt(-10*x^2 - x + 3)*x^2 + 268643/160000*sqrt(-10*x^2 - x + 3)*x - 1864839
9/64000000*sqrt(10)*arcsin(-20/11*x - 1/11) - 314441/3200000*sqrt(-10*x^2 - x +
3)

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Fricas [A]  time = 0.223875, size = 97, normalized size = 0.76 \[ -\frac{1}{64000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} + 7862400 \, x^{3} - 2952480 \, x^{2} - 5372860 \, x + 314441\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 18648399 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64000000*sqrt(10)*(2*sqrt(10)*(6912000*x^4 + 7862400*x^3 - 2952480*x^2 - 5372
860*x + 314441)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 18648399*arctan(1/20*sqrt(10)*(20
*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 97.1458, size = 593, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-343*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x
+ 1)/968 - sqrt(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)))/8 + 441*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)))/8 - 189*sqr
t(2)*Piecewise((14641*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/399
3 + 7*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)/1
874048 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55)*sqrt(-2*x +
1)/11)/128)/3125, (x <= 1/2) & (x > -3/5)))/8 + 27*sqrt(2)*Piecewise((161051*sq
rt(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)/2662 + 15*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20
*x + 1)/7744 + 5*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(12100*x - 2000*(-2*x + 1
)**3 + 6600*(-2*x + 1)**2 - 4719)/3748096 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6
)/22 + 63*asin(sqrt(55)*sqrt(-2*x + 1)/11)/256)/15625, (x <= 1/2) & (x > -3/5)))
/8

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GIAC/XCAS [A]  time = 0.273547, size = 371, normalized size = 2.9 \[ -\frac{9}{160000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 203\right )}{\left (5 \, x + 3\right )} + 19073\right )}{\left (5 \, x + 3\right )} - 506185\right )}{\left (5 \, x + 3\right )} + 4031895\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 10392195 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{27}{3200000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{3}{20000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{4}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/160000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 50
6185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3))) - 27/3200000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x
+ 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*
arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 3/20000*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)*sqr
t(5*x + 3))) + 1/100*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))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1
/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))