3.2321 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac{3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac{34111 \sqrt{5 x+3} (1-2 x)^{5/2}}{25600}+\frac{375221 \sqrt{5 x+3} (1-2 x)^{3/2}}{512000}+\frac{12382293 \sqrt{5 x+3} \sqrt{1-2 x}}{5120000}+\frac{136205223 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

[Out]

(12382293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120000 + (375221*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/512000 - (34111*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/25600 - (3101*(1 - 2*x)^(
5/2)*(3 + 5*x)^(3/2))/6400 - (259*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/2000 - ((1 -
2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/20 + (136205223*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(5120000*Sqrt[10])

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Rubi [A]  time = 0.19511, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac{3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac{34111 \sqrt{5 x+3} (1-2 x)^{5/2}}{25600}+\frac{375221 \sqrt{5 x+3} (1-2 x)^{3/2}}{512000}+\frac{12382293 \sqrt{5 x+3} \sqrt{1-2 x}}{5120000}+\frac{136205223 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(12382293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120000 + (375221*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/512000 - (34111*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/25600 - (3101*(1 - 2*x)^(
5/2)*(3 + 5*x)^(3/2))/6400 - (259*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/2000 - ((1 -
2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/20 + (136205223*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(5120000*Sqrt[10])

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Rubi in Sympy [A]  time = 15.6421, size = 150, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (9 x + 6\right )}{60} - \frac{259 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{2000} + \frac{3101 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{16000} - \frac{34111 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{64000} - \frac{1125663 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{512000} + \frac{12382293 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5120000} + \frac{136205223 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{51200000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)*(9*x + 6)/60 - 259*(-2*x + 1)**(5/2)*(5*x +
3)**(5/2)/2000 + 3101*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/16000 - 34111*(-2*x + 1
)**(3/2)*(5*x + 3)**(3/2)/64000 - 1125663*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/512000
+ 12382293*sqrt(-2*x + 1)*sqrt(5*x + 3)/5120000 + 136205223*sqrt(10)*asin(sqrt(
22)*sqrt(5*x + 3)/11)/51200000

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Mathematica [A]  time = 0.121773, size = 75, normalized size = 0.45 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (76800000 x^5+132864000 x^4+27804800 x^3-66492960 x^2-37288220 x+8705457\right )-136205223 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{51200000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8705457 - 37288220*x - 66492960*x^2 + 27804800
*x^3 + 132864000*x^4 + 76800000*x^5) - 136205223*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt
[1 - 2*x]])/51200000

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Maple [A]  time = 0.012, size = 138, normalized size = 0.8 \[{\frac{1}{102400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -1536000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-2657280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-556096000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-174109140\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/102400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-1536000000*x^5*(-10*x^2-x+3)^(1/2)-265
7280000*x^4*(-10*x^2-x+3)^(1/2)-556096000*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)-174109140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50457, size = 134, normalized size = 0.81 \[ -\frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{459}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3101}{3200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{3101}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1125663}{256000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{136205223}{102400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1125663}{5120000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/20*(-10*x^2 - x + 3)^(5/2)*x - 459/2000*(-10*x^2 - x + 3)^(5/2) + 3101/3200*(
-10*x^2 - x + 3)^(3/2)*x + 3101/64000*(-10*x^2 - x + 3)^(3/2) + 1125663/256000*s
qrt(-10*x^2 - x + 3)*x - 136205223/102400000*sqrt(10)*arcsin(-20/11*x - 1/11) +
1125663/5120000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222, size = 104, normalized size = 0.63 \[ -\frac{1}{102400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (76800000 \, x^{5} + 132864000 \, x^{4} + 27804800 \, x^{3} - 66492960 \, x^{2} - 37288220 \, x + 8705457\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 136205223 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/102400000*sqrt(10)*(2*sqrt(10)*(76800000*x^5 + 132864000*x^4 + 27804800*x^3 -
66492960*x^2 - 37288220*x + 8705457)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 136205223*a
rctan(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)**2*(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257455, size = 427, normalized size = 2.59 \[ -\frac{3}{256000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{43}{64000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{76800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{750} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/256000000*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))) - 43/64000000*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))) - 1/76800*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(22)*sqrt(5*x +
3))) + 1/750*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))) + 3/100*sqrt(5)*(2*(20*x
+ 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)))