3.2372 \(\int (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]

[Out]

(9568559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800000 + (869869*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/3840000 + (79079*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/960000 - (7189*(1 - 2*x)
^(7/2)*Sqrt[3 + 5*x])/32000 - (193*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/2000 - ((1 -
2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/20 + (105254149*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(12800000*Sqrt[10])

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Rubi [A]  time = 0.190511, 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 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(9568559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800000 + (869869*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/3840000 + (79079*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/960000 - (7189*(1 - 2*x)
^(7/2)*Sqrt[3 + 5*x])/32000 - (193*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/2000 - ((1 -
2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/20 + (105254149*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(12800000*Sqrt[10])

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Rubi in Sympy [A]  time = 15.8673, size = 150, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (9 x + 6\right )}{60} - \frac{193 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{2000} + \frac{7189 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{80000} - \frac{79079 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{192000} + \frac{869869 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3840000} + \frac{9568559 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12800000} + \frac{105254149 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)*(5*x + 3)**(3/2)*(9*x + 6)/60 - 193*(-2*x + 1)**(7/2)*(5*x +
3)**(3/2)/2000 + 7189*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/80000 - 79079*(-2*x + 1
)**(5/2)*sqrt(5*x + 3)/192000 + 869869*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/3840000 +
9568559*sqrt(-2*x + 1)*sqrt(5*x + 3)/12800000 + 105254149*sqrt(10)*asin(sqrt(22
)*sqrt(5*x + 3)/11)/128000000

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Mathematica [A]  time = 0.120139, size = 75, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (230400000 x^5+94464000 x^4-237187200 x^3-61262560 x^2+102523580 x+9303927\right )-315762447 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{384000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9303927 + 102523580*x - 61262560*x^2 - 23718720
0*x^3 + 94464000*x^4 + 230400000*x^5) - 315762447*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqr
t[1 - 2*x]])/384000000

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Maple [A]  time = 0.014, size = 138, normalized size = 0.8 \[{\frac{1}{768000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4608000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4743744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1225251200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4608000000*x^5*(-10*x^2-x+3)^(1/2)+1889
280000*x^4*(-10*x^2-x+3)^(1/2)-4743744000*x^3*(-10*x^2-x+3)^(1/2)-1225251200*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)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5234, size = 140, normalized size = 0.85 \[ -\frac{3}{5} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{93}{500} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{18251}{40000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27893}{480000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{869869}{640000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{105254149}{256000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{869869}{12800000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/5*(-10*x^2 - x + 3)^(3/2)*x^3 - 93/500*(-10*x^2 - x + 3)^(3/2)*x^2 + 18251/40
000*(-10*x^2 - x + 3)^(3/2)*x + 27893/480000*(-10*x^2 - x + 3)^(3/2) + 869869/64
0000*sqrt(-10*x^2 - x + 3)*x - 105254149/256000000*sqrt(10)*arcsin(-20/11*x - 1/
11) + 869869/12800000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.214086, size = 104, normalized size = 0.63 \[ \frac{1}{768000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 315762447 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768000000*sqrt(10)*(2*sqrt(10)*(230400000*x^5 + 94464000*x^4 - 237187200*x^3 -
61262560*x^2 + 102523580*x + 9303927)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 315762447*
arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 166.381, size = 695, normalized size = 4.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

242*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)))/1562
5 + 1364*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*s
qrt(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)))/15625 + 1658*sqrt(5)*Pie
cewise((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*a
sin(sqrt(22)*sqrt(5*x + 3)/11)/128)/16, (x >= -3/5) & (x < 1/2)))/15625 - 744*sq
rt(5)*Piecewise((161051*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x +
3)/7744 + 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)*(-1
2100*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)))/15625 + 72*sqrt(5)*Piecewis
e((1771561*sqrt(2)*(sqrt(2)*(-20*x - 1)**3*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/8
5034928 - sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/15488 + 4*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 - 13*sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)
**3 + 1056*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/10
24)/64, (x >= -3/5) & (x < 1/2)))/15625

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GIAC/XCAS [A]  time = 0.264288, size = 427, normalized size = 2.59 \[ \frac{3}{640000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{16000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{23}{1920000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/640000000*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))) + 1/16000000*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))) - 23/1920000*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))) - 1/6000*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))) + 1/100*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)))