3.2311 \(\int (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=143 \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]

[Out]

(498883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640000 + (45353*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/192000 - (4123*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/9600 - (567*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/4000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/50 + (548771
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640000*Sqrt[10])

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Rubi [A]  time = 0.1656, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(498883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640000 + (45353*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/192000 - (4123*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/9600 - (567*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/4000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/50 + (548771
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640000*Sqrt[10])

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Rubi in Sympy [A]  time = 14.1562, size = 129, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (9 x + 6\right )}{50} - \frac{567 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4000} + \frac{4123 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{24000} - \frac{45353 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{64000} + \frac{498883 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{640000} + \frac{5487713 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6400000} \]

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)**(1/2),x)

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)*(9*x + 6)/50 - 567*(-2*x + 1)**(5/2)*(5*x +
3)**(3/2)/4000 + 4123*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/24000 - 45353*(-2*x + 1
)**(3/2)*sqrt(5*x + 3)/64000 + 498883*sqrt(-2*x + 1)*sqrt(5*x + 3)/640000 + 5487
713*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/6400000

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Mathematica [A]  time = 0.110383, size = 70, normalized size = 0.49 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+7286400 x^3-3141280 x^2-4872460 x+382101\right )-16463139 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{19200000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(382101 - 4872460*x - 3141280*x^2 + 7286400*x^3
+ 6912000*x^4) - 16463139*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/19200000

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Maple [A]  time = 0.013, size = 121, normalized size = 0.9 \[{\frac{1}{38400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-145728000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-7642020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-14572
8000*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)-7642020*(-10*x^2-x+3)^(1/2))
/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49271, size = 117, normalized size = 0.82 \[ \frac{9}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{687}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2159}{24000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{45353}{32000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{5487713}{12800000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{45353}{640000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 687/2000*(-10*x^2 - x + 3)^(3/2)*x - 2159/240
00*(-10*x^2 - x + 3)^(3/2) + 45353/32000*sqrt(-10*x^2 - x + 3)*x - 5487713/12800
000*sqrt(10)*arcsin(-20/11*x - 1/11) + 45353/640000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.224981, size = 97, normalized size = 0.68 \[ -\frac{1}{38400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 16463139 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/38400000*sqrt(10)*(2*sqrt(10)*(6912000*x^4 + 7286400*x^3 - 3141280*x^2 - 4872
460*x + 382101)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 16463139*arctan(1/20*sqrt(10)*(20
*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 30.8409, size = 490, normalized size = 3.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x
+ 3)/121 + asin(sqrt(22)*sqrt(5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/3125
+ 128*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt
(5*x + 3)/1936 - sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + asin(sqrt(22
)*sqrt(5*x + 3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/3125 + 174*sqrt(5)*Piecewis
e((14641*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 - sqrt
(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x
+ 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(s
qrt(22)*sqrt(5*x + 3)/11)/128)/16, (x >= -3/5) & (x < 1/2)))/3125 - 36*sqrt(5)*P
iecewise((161051*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/774
4 + 2*sqrt(2)*(-10*x + 5)**(5/2)*(5*x + 3)**(5/2)/805255 - sqrt(2)*(-10*x + 5)**
(3/2)*(5*x + 3)**(3/2)/3993 - 3*sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)*(-12100*x
- 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*
x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/3125

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GIAC/XCAS [A]  time = 0.253068, size = 317, normalized size = 2.22 \[ -\frac{3}{32000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{128000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/32000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 1364
05)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3))) - 1/128000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2
179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) + 1/6000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqr
t(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) +
1/100*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)))