3.2310 \(\int (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=150 \[ -\frac{1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac{339983 \sqrt{5 x+3} (1-2 x)^{5/2}}{384000}+\frac{3739813 \sqrt{5 x+3} (1-2 x)^{3/2}}{7680000}+\frac{41137943 \sqrt{5 x+3} \sqrt{1-2 x}}{25600000}+\frac{452517373 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600000 \sqrt{10}} \]

[Out]

(41137943*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600000 + (3739813*(1 - 2*x)^(3/2)*Sqrt[
3 + 5*x])/7680000 - (339983*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/384000 - ((1 - 2*x)^(
5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/20 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)*(88987 +
63120*x))/160000 + (452517373*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600000*Sqrt[
10])

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Rubi [A]  time = 0.183798, antiderivative size = 150, 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 \[ -\frac{1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac{339983 \sqrt{5 x+3} (1-2 x)^{5/2}}{384000}+\frac{3739813 \sqrt{5 x+3} (1-2 x)^{3/2}}{7680000}+\frac{41137943 \sqrt{5 x+3} \sqrt{1-2 x}}{25600000}+\frac{452517373 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(41137943*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600000 + (3739813*(1 - 2*x)^(3/2)*Sqrt[
3 + 5*x])/7680000 - (339983*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/384000 - ((1 - 2*x)^(
5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/20 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)*(88987 +
63120*x))/160000 + (452517373*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600000*Sqrt[
10])

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Rubi in Sympy [A]  time = 17.8714, size = 136, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{20} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (47340 x + \frac{266961}{4}\right )}{120000} + \frac{339983 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{960000} + \frac{3739813 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6400000} - \frac{41137943 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{25600000} + \frac{452517373 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(3*x + 2)**2*(5*x + 3)**(3/2)/20 - (-2*x + 1)**(5/2)*(5*x + 3
)**(3/2)*(47340*x + 266961/4)/120000 + 339983*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)
/960000 + 3739813*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/6400000 - 41137943*sqrt(-2*x +
1)*sqrt(5*x + 3)/25600000 + 452517373*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/
256000000

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Mathematica [A]  time = 0.13018, size = 75, normalized size = 0.5 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (691200000 x^5+1251072000 x^4+308534400 x^3-623566880 x^2-374573660 x+81405921\right )-1357552119 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{768000000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(81405921 - 374573660*x - 623566880*x^2 + 30853
4400*x^3 + 1251072000*x^4 + 691200000*x^5) - 1357552119*Sqrt[10]*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/768000000

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Maple [A]  time = 0.014, size = 138, normalized size = 0.9 \[{\frac{1}{1536000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-25021440000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6170688000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-1628118420\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-13824000000*x^5*(-10*x^2-x+3)^(1/2)-2
5021440000*x^4*(-10*x^2-x+3)^(1/2)-6170688000*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)-1628118420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50404, size = 140, normalized size = 0.93 \[ \frac{9}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{1539}{1000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{41427}{80000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{385939}{960000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3739813}{1280000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{452517373}{512000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3739813}{25600000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/10*(-10*x^2 - x + 3)^(3/2)*x^3 + 1539/1000*(-10*x^2 - x + 3)^(3/2)*x^2 + 41427
/80000*(-10*x^2 - x + 3)^(3/2)*x - 385939/960000*(-10*x^2 - x + 3)^(3/2) + 37398
13/1280000*sqrt(-10*x^2 - x + 3)*x - 452517373/512000000*sqrt(10)*arcsin(-20/11*
x - 1/11) + 3739813/25600000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.220429, size = 104, normalized size = 0.69 \[ -\frac{1}{1536000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (691200000 \, x^{5} + 1251072000 \, x^{4} + 308534400 \, x^{3} - 623566880 \, x^{2} - 374573660 \, x + 81405921\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1357552119 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1536000000*sqrt(10)*(2*sqrt(10)*(691200000*x^5 + 1251072000*x^4 + 308534400*x
^3 - 623566880*x^2 - 374573660*x + 81405921)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1357
552119*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 49.1937, size = 695, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(3/2)*(2+3*x)**3*(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)))/15625
+ 194*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqr
t(5*x + 3)/1936 - sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + asin(sqrt(2
2)*sqrt(5*x + 3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/15625 + 558*sqrt(5)*Piecew
ise((14641*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 - sq
rt(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
(sqrt(22)*sqrt(5*x + 3)/11)/128)/16, (x >= -3/5) & (x < 1/2)))/15625 + 486*sqrt(
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)*(-1210
0*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/3748096 + 7*asin(sqrt(22)*sqr
t(5*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/15625 - 108*sqrt(5)*Piecewise(
(1771561*sqrt(2)*(sqrt(2)*(-20*x - 1)**3*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/850
34928 - 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)/1024
)/64, (x >= -3/5) & (x < 1/2)))/15625

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GIAC/XCAS [A]  time = 0.270168, size = 427, normalized size = 2.85 \[ -\frac{9}{1280000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{27}{64000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{3}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{1200} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/1280000000*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))) - 27/64000000*sqrt(5)
*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 605
55)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) - 3/320000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 41
25)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) + 1/1200*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(1/11*sqrt(22)*sqrt(5*
x + 3)))