3.2246 \(\int \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]

[Out]

(3558401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000 - (323491*(1 - 2*x)^(3/2)*Sqrt[3 +
 5*x])/128000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 - (21*(1 - 2*
x)^(3/2)*(3 + 5*x)^(3/2)*(731 + 444*x))/16000 + (39142411*ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]])/(1280000*Sqrt[10])

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Rubi [A]  time = 0.158578, 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 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(3558401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000 - (323491*(1 - 2*x)^(3/2)*Sqrt[3 +
 5*x])/128000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 - (21*(1 - 2*
x)^(3/2)*(3 + 5*x)^(3/2)*(731 + 444*x))/16000 + (39142411*ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]])/(1280000*Sqrt[10])

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Rubi in Sympy [A]  time = 14.7547, size = 117, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{50} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (34965 x + \frac{230265}{4}\right )}{60000} + \frac{323491 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{320000} - \frac{3558401 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1280000} + \frac{39142411 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{12800000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3)**(3/2)/50 - (-2*x + 1)**(3/2)*(5*x +
 3)**(3/2)*(34965*x + 230265/4)/60000 + 323491*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/3
20000 - 3558401*sqrt(-2*x + 1)*sqrt(5*x + 3)/1280000 + 39142411*sqrt(10)*asin(sq
rt(22)*sqrt(5*x + 3)/11)/12800000

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Mathematica [A]  time = 0.108311, size = 70, normalized size = 0.55 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+17366400 x^3+14946720 x^2+3002540 x-4282349\right )-39142411 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12800000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-4282349 + 3002540*x + 14946720*x^2 + 17366400*
x^3 + 6912000*x^4) - 39142411*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1280000
0

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Maple [A]  time = 0.013, size = 121, normalized size = 1. \[{\frac{1}{25600000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+347328000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+298934400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+39142411\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +60050800\,x\sqrt{-10\,{x}^{2}-x+3}-85646980\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)+347328
000*x^3*(-10*x^2-x+3)^(1/2)+298934400*x^2*(-10*x^2-x+3)^(1/2)+39142411*10^(1/2)*
arcsin(20/11*x+1/11)+60050800*x*(-10*x^2-x+3)^(1/2)-85646980*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49733, size = 117, normalized size = 0.91 \[ -\frac{27}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{5211}{4000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{19191}{16000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{323491}{64000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{39142411}{25600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{323491}{1280000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/50*(-10*x^2 - x + 3)^(3/2)*x^2 - 5211/4000*(-10*x^2 - x + 3)^(3/2)*x - 19191
/16000*(-10*x^2 - x + 3)^(3/2) + 323491/64000*sqrt(-10*x^2 - x + 3)*x - 39142411
/25600000*sqrt(10)*arcsin(-20/11*x - 1/11) + 323491/1280000*sqrt(-10*x^2 - x + 3
)

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Fricas [A]  time = 0.221817, size = 97, normalized size = 0.76 \[ \frac{1}{25600000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 39142411 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25600000*sqrt(10)*(2*sqrt(10)*(6912000*x^4 + 17366400*x^3 + 14946720*x^2 + 300
2540*x - 4282349)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 39142411*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((2+3*x)**3*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275267, size = 317, normalized size = 2.48 \[ \frac{9}{64000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/64000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13640
5)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) + 9/320000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 21
79)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) + 3/2000*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))) + 1
/50*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)))