3.2272 \(\int \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac{30371 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120}+\frac{334081 \sqrt{1-2 x} \sqrt{5 x+3}}{51200}+\frac{3674891 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]

[Out]

(334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200 - (30371*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
])/5120 - (2761*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/1920 - (251*(1 - 2*x)^(3/2)*(3
+ 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/50 + (3674891*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/(51200*Sqrt[10])

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Rubi [A]  time = 0.142083, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac{30371 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120}+\frac{334081 \sqrt{1-2 x} \sqrt{5 x+3}}{51200}+\frac{3674891 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200 - (30371*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
])/5120 - (2761*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/1920 - (251*(1 - 2*x)^(3/2)*(3
+ 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/50 + (3674891*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/(51200*Sqrt[10])

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Rubi in Sympy [A]  time = 12.2577, size = 126, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{50} + \frac{251 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{2000} - \frac{2761 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{24000} - \frac{30371 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{38400} - \frac{334081 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{51200} + \frac{3674891 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/50 + 251*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/2
000 - 2761*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/24000 - 30371*sqrt(-2*x + 1)*(5*x + 3
)**(3/2)/38400 - 334081*sqrt(-2*x + 1)*sqrt(5*x + 3)/51200 + 3674891*sqrt(10)*as
in(sqrt(22)*sqrt(5*x + 3)/11)/512000

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Mathematica [A]  time = 0.114028, size = 70, normalized size = 0.51 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (2304000 x^4+5404800 x^3+4310240 x^2+718340 x-1254087\right )-11024673 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1536000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-1254087 + 718340*x + 4310240*x^2 + 5404800*x^3
+ 2304000*x^4) - 11024673*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1536000

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Maple [A]  time = 0.011, size = 121, normalized size = 0.9 \[{\frac{1}{3072000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 46080000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-25081740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3072000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(46080000*x^4*(-10*x^2-x+3)^(1/2)[phone]
0*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arc
sin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-25081740*(-10*x^2-x+3)^(1/2))/(
-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49448, size = 117, normalized size = 0.85 \[ -\frac{3}{2} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{539}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{1121}{384} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{30371}{2560} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{3674891}{1024000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{30371}{51200} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*(-10*x^2 - x + 3)^(3/2)*x^2 - 539/160*(-10*x^2 - x + 3)^(3/2)*x - 1121/384*
(-10*x^2 - x + 3)^(3/2) + 30371/2560*sqrt(-10*x^2 - x + 3)*x - 3674891/1024000*s
qrt(10)*arcsin(-20/11*x - 1/11) + 30371/51200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.219048, size = 97, normalized size = 0.7 \[ \frac{1}{3072000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 11024673 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3072000*sqrt(10)*(2*sqrt(10)*(2304000*x^4 + 5404800*x^3 + 4310240*x^2 + 718340
*x - 1254087)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 11024673*arctan(1/20*sqrt(10)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 118.613, size = 488, normalized size = 3.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-847*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
+ 1133*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 - 505*sqrt(2)*Piec
ewise((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)/1874048 + 5
*asin(sqrt(55)*sqrt(-2*x + 1)/11)/128)/625, (x <= 1/2) & (x > -3/5)))/16 + 75*sq
rt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/3
22102 - 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)*(12
100*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

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GIAC/XCAS [A]  time = 0.264859, size = 317, normalized size = 2.3 \[ \frac{1}{2560000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{7}{96000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{29}{8000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2560000*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))) + 7/96000*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*sq
rt(22)*sqrt(5*x + 3))) + 29/8000*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))) + 9/
200*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3)))