3.2584 \(\int \frac{(2+3 x) (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{7 (5 x+3)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{239 (5 x+3)^{5/2}}{66 \sqrt{1-2 x}}-\frac{5975}{528} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{5975}{64} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{13145}{64} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-5975*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (5975*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/52
8 - (239*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(7/2))/(33*(1 - 2*x)
^(3/2)) + (13145*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/64

_______________________________________________________________________________________

Rubi [A]  time = 0.119453, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7 (5 x+3)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{239 (5 x+3)^{5/2}}{66 \sqrt{1-2 x}}-\frac{5975}{528} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{5975}{64} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{13145}{64} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5975*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (5975*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/52
8 - (239*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(7/2))/(33*(1 - 2*x)
^(3/2)) + (13145*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/64

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.4912, size = 105, normalized size = 0.89 \[ - \frac{5975 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{528} - \frac{5975 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{64} + \frac{13145 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128} - \frac{239 \left (5 x + 3\right )^{\frac{5}{2}}}{66 \sqrt{- 2 x + 1}} + \frac{7 \left (5 x + 3\right )^{\frac{7}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5975*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/528 - 5975*sqrt(-2*x + 1)*sqrt(5*x + 3)/64
+ 13145*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/128 - 239*(5*x + 3)**(5/2)/(66
*sqrt(-2*x + 1)) + 7*(5*x + 3)**(7/2)/(33*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.122769, size = 74, normalized size = 0.63 \[ \frac{39435 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (3600 x^3+20820 x^2-84064 x+29601\right )}{384 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(29601 - 84064*x + 20820*x^2 + 3600*x^3) + 39435*Sqrt[10 - 20*
x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(384*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 137, normalized size = 1.2 \[{\frac{1}{768\, \left ( -1+2\,x \right ) ^{2}} \left ( 157740\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-14400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-157740\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-83280\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+39435\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +336256\,x\sqrt{-10\,{x}^{2}-x+3}-118404\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768*(157740*10^(1/2)*arcsin(20/11*x+1/11)*x^2-14400*x^3*(-10*x^2-x+3)^(1/2)-15
7740*10^(1/2)*arcsin(20/11*x+1/11)*x-83280*x^2*(-10*x^2-x+3)^(1/2)+39435*10^(1/2
)*arcsin(20/11*x+1/11)+336256*x*(-10*x^2-x+3)^(1/2)-118404*(-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)

_______________________________________________________________________________________

Maxima [A]  time = 1.50239, size = 251, normalized size = 2.13 \[ \frac{13145}{256} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{385 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{48 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{165 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{4235 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{43285 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13145/256*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/4*(-10*x^2 - x + 3)^(5/2)/(
16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^
2 + 6*x - 1) - 385/48*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 165/3
2*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 4235/96*sqrt(-10*x^2 - x + 3)/(4*x
^2 - 4*x + 1) + 43285/192*sqrt(-10*x^2 - x + 3)/(2*x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.229734, size = 128, normalized size = 1.08 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (3600 \, x^{3} + 20820 \, x^{2} - 84064 \, x + 29601\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 39435 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{768 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/768*sqrt(2)*(2*sqrt(2)*(3600*x^3 + 20820*x^2 - 84064*x + 29601)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 39435*sqrt(5)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(2
0*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*x + 1)

_______________________________________________________________________________________

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+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23958, size = 113, normalized size = 0.96 \[ \frac{13145}{128} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 239 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 26290 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 433785 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4800 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13145/128*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/4800*(4*(3*(12*sqrt(5
)*(5*x + 3) + 239*sqrt(5))*(5*x + 3) - 26290*sqrt(5))*(5*x + 3) + 433785*sqrt(5)
)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2