3.2334 \(\int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac{187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac{2057 (1-2 x)^{5/2} \sqrt{5 x+3}}{1024}+\frac{22627 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480}+\frac{746691 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}+\frac{8213601 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]

[Out]

(746691*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800 + (22627*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/20480 - (2057*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1024 - (187*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/256 - (17*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/80 - ((1 - 2*x)^(5/2)*(
3 + 5*x)^(7/2))/20 + (8213601*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800*Sqrt[10]
)

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Rubi [A]  time = 0.168638, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac{187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac{2057 (1-2 x)^{5/2} \sqrt{5 x+3}}{1024}+\frac{22627 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480}+\frac{746691 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}+\frac{8213601 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(746691*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800 + (22627*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/20480 - (2057*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1024 - (187*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/256 - (17*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/80 - ((1 - 2*x)^(5/2)*(
3 + 5*x)^(7/2))/20 + (8213601*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800*Sqrt[10]
)

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Rubi in Sympy [A]  time = 14.2216, size = 144, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{20} + \frac{17 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{200} + \frac{561 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{8000} - \frac{2057 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{32000} - \frac{22627 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{51200} - \frac{746691 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{204800} + \frac{8213601 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2048000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(7/2)/20 + 17*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/2
00 + 561*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/8000 - 2057*sqrt(-2*x + 1)*(5*x + 3)**(
5/2)/32000 - 22627*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/51200 - 746691*sqrt(-2*x + 1)
*sqrt(5*x + 3)/204800 + 8213601*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2048000

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Mathematica [A]  time = 0.101311, size = 75, normalized size = 0.47 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (5120000 x^5+8448000 x^4+1456000 x^3-4238560 x^2-2224900 x+555399\right )-8213601 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2048000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(555399 - 2224900*x - 4238560*x^2 + 1456000*x^3
+ 8448000*x^4 + 5120000*x^5) - 8213601*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]
])/2048000

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Maple [A]  time = 0.011, size = 138, normalized size = 0.9 \[{\frac{1}{4096000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -102400000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-168960000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-29120000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8213601\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-11107980\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4096000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-102400000*x^5*(-10*x^2-x+3)^(1/2)-168960
000*x^4*(-10*x^2-x+3)^(1/2)-29120000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x
^2-x+3)^(1/2)+8213601*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/
2)-11107980*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50926, size = 134, normalized size = 0.84 \[ -\frac{1}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{29}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{187}{128} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{187}{2560} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{67881}{10240} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{8213601}{4096000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{67881}{204800} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(-10*x^2 - x + 3)^(5/2)*x - 29/80*(-10*x^2 - x + 3)^(5/2) + 187/128*(-10*x^
2 - x + 3)^(3/2)*x + 187/2560*(-10*x^2 - x + 3)^(3/2) + 67881/10240*sqrt(-10*x^2
- x + 3)*x - 8213601/4096000*sqrt(10)*arcsin(-20/11*x - 1/11) + 67881/204800*sq
rt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222223, size = 104, normalized size = 0.65 \[ -\frac{1}{4096000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (5120000 \, x^{5} + 8448000 \, x^{4} + 1456000 \, x^{3} - 4238560 \, x^{2} - 2224900 \, x + 555399\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 8213601 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4096000*sqrt(10)*(2*sqrt(10)*(5120000*x^5 + 8448000*x^4 + 1456000*x^3 - 42385
60*x^2 - 2224900*x + 555399)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 8213601*arctan(1/20*
sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256429, size = 427, normalized size = 2.67 \[ -\frac{1}{51200000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{41}{38400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{17}{960000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{17}{8000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/51200000*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))) - 41/38400000*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))) - 17/960000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 412
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3))) + 17/8000*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*(2
0*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5
*x + 3)))