3.2573 \(\int \frac{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]

[Out]

(-4246733*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/14080 - (101*(2 + 3*x)^2*(3 + 5*x)^(3/2))
/(22*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqr
t[1 - 2*x]*(3 + 5*x)^(3/2)*(59719 + 28200*x))/3520 + (4246733*ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]])/(1280*Sqrt[10])

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Rubi [A]  time = 0.212006, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-4246733*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/14080 - (101*(2 + 3*x)^2*(3 + 5*x)^(3/2))
/(22*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqr
t[1 - 2*x]*(3 + 5*x)^(3/2)*(59719 + 28200*x))/3520 + (4246733*ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]])/(1280*Sqrt[10])

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Rubi in Sympy [A]  time = 27.6202, size = 131, normalized size = 0.97 \[ - \frac{711 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{56} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{83845125 x}{4} + \frac{808206525}{16}\right )}{126000} + \frac{4246733 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{12800} - \frac{101 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{14 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-711*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/56 - sqrt(-2*x + 1)*sqrt(5*x + 3)
*(83845125*x/4 + 808206525/16)/126000 + 4246733*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/12800 - 101*(3*x + 2)**3*sqrt(5*x + 3)/(14*sqrt(-2*x + 1)) + (3*x + 2)*
*3*(5*x + 3)**(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.161726, size = 79, normalized size = 0.59 \[ \frac{12740199 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (86400 x^4+447120 x^3+1544724 x^2-5349344 x+1925361\right )}{38400 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(1925361 - 5349344*x + 1544724*x^2 + 447120*x^3 + 86400*x^4)
+ 12740199*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(38400*(
1 - 2*x)^(3/2))

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Maple [A]  time = 0.019, size = 154, normalized size = 1.1 \[{\frac{1}{76800\, \left ( -1+2\,x \right ) ^{2}} \left ( -1728000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-8942400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-50960796\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-30894480\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-38507220\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/76800*(-1728000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)
*x^2-8942400*x^3*(-10*x^2-x+3)^(1/2)-50960796*10^(1/2)*arcsin(20/11*x+1/11)*x-30
894480*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)-38507220*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/
(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52774, size = 285, normalized size = 2.11 \[ \frac{428267}{2560} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{35937}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{9}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{297}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{6237}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{6237}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{48 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{189 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} + \frac{3773 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3479 \, \sqrt{-10 \, x^{2} - x + 3}}{6 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

428267/2560*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 35937/25600*I*sqrt(5)*sqrt(
2)*arcsin(20/11*x - 21/11) + 9/16*(-10*x^2 - x + 3)^(3/2) - 297/64*sqrt(10*x^2 -
21*x + 8)*x + 6237/1280*sqrt(10*x^2 - 21*x + 8) - 6237/128*sqrt(-10*x^2 - x + 3
) - 343/48*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 441/16*(-10*x^2
- x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 189/32*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 37
73/96*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 3479/6*sqrt(-10*x^2 - x + 3)/(2*
x - 1)

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Fricas [A]  time = 0.222786, size = 127, normalized size = 0.94 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (86400 \, x^{4} + 447120 \, x^{3} + 1544724 \, x^{2} - 5349344 \, x + 1925361\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 12740199 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{76800 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/76800*sqrt(10)*(2*sqrt(10)*(86400*x^4 + 447120*x^3 + 1544724*x^2 - 5349344*x
+ 1925361)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 12740199*(4*x^2 - 4*x + 1)*arctan(1/20
*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*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*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234973, size = 131, normalized size = 0.97 \[ \frac{4246733}{12800} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (27 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 111 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8579 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 8493466 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 140142189 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{480000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4246733/12800*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/480000*(4*(27*(4*
(8*sqrt(5)*(5*x + 3) + 111*sqrt(5))*(5*x + 3) + 8579*sqrt(5))*(5*x + 3) - 849346
6*sqrt(5))*(5*x + 3) + 140142189*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1
)^2