3.2245 \(\int \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=157 \[ -\frac{1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac{333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac{7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120000}+\frac{374762311 \sqrt{1-2 x} \sqrt{5 x+3}}{51200000}+\frac{4122385421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

[Out]

(374762311*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 - (34069301*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/5120000 - (333*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/2000 - (
(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 - (7*(1 - 2*x)^(3/2)*(3 + 5*x)^(
3/2)*(231223 + 140652*x))/640000 + (4122385421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(51200000*Sqrt[10])

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Rubi [A]  time = 0.230687, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac{333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac{7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120000}+\frac{374762311 \sqrt{1-2 x} \sqrt{5 x+3}}{51200000}+\frac{4122385421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x],x]

[Out]

(374762311*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 - (34069301*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/5120000 - (333*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/2000 - (
(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 - (7*(1 - 2*x)^(3/2)*(3 + 5*x)^(
3/2)*(231223 + 140652*x))/640000 + (4122385421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(51200000*Sqrt[10])

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Rubi in Sympy [A]  time = 22.1848, size = 144, normalized size = 0.92 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{20} - \frac{333 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{2000} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (\frac{11076345 x}{2} + \frac{72835245}{8}\right )}{3600000} + \frac{34069301 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{12800000} - \frac{374762311 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{51200000} + \frac{4122385421 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512000000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)

[Out]

-(-2*x + 1)**(3/2)*(3*x + 2)**3*(5*x + 3)**(3/2)/20 - 333*(-2*x + 1)**(3/2)*(3*x
+ 2)**2*(5*x + 3)**(3/2)/2000 - (-2*x + 1)**(3/2)*(5*x + 3)**(3/2)*(11076345*x/
2 + 72835245/8)/3600000 + 34069301*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/12800000 - 37
4762311*sqrt(-2*x + 1)*sqrt(5*x + 3)/51200000 + 4122385421*sqrt(10)*asin(sqrt(22
)*sqrt(5*x + 3)/11)/512000000

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Mathematica [A]  time = 0.135338, size = 75, normalized size = 0.48 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (691200000 x^5+2218752000 x^4+2739830400 x^3+1468973920 x^2+45781940 x-518122939\right )-4122385421 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{512000000} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x],x]

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-518122939 + 45781940*x + 1468973920*x^2 + 2739
830400*x^3 + 2218752000*x^4 + 691200000*x^5) - 4122385421*Sqrt[10]*ArcSin[Sqrt[5
/11]*Sqrt[1 - 2*x]])/512000000

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Maple [A]  time = 0.024, size = 138, normalized size = 0.9 \[{\frac{1}{1024000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-10362458780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4*(1-2*x)^(1/2)*(3+5*x)^(1/2),x)

[Out]

1/1024000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)+44
375040000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)[phone]
0*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)-10362458780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5053, size = 140, normalized size = 0.89 \[ -\frac{27}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{8397}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{853821}{160000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2300801}{640000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{34069301}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4122385421}{1024000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{34069301}{51200000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

-27/20*(-10*x^2 - x + 3)^(3/2)*x^3 - 8397/2000*(-10*x^2 - x + 3)^(3/2)*x^2 - 853
821/160000*(-10*x^2 - x + 3)^(3/2)*x - 2300801/640000*(-10*x^2 - x + 3)^(3/2) +
34069301/2560000*sqrt(-10*x^2 - x + 3)*x - 4122385421/1024000000*sqrt(10)*arcsin
(-20/11*x - 1/11) + 34069301/51200000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.220717, size = 104, normalized size = 0.66 \[ \frac{1}{1024000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (691200000 \, x^{5} + 2218752000 \, x^{4} + 2739830400 \, x^{3} + 1468973920 \, x^{2} + 45781940 \, x - 518122939\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4122385421 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

1/1024000000*sqrt(10)*(2*sqrt(10)*(691200000*x^5 + 2218752000*x^4 + 2739830400*x
^3 + 1468973920*x^2 + 45781940*x - 518122939)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 412
2385421*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)**4*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277797, size = 427, normalized size = 2.72 \[ \frac{27}{2560000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{8000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{80000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{250} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="giac")

[Out]

27/2560000000*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))) + 9/8000000*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))) + 9/80000*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
))) + 1/250*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))) + 1/25*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)))