3.2583 \(\int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-123745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256 - (123745*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/2112 - (24749*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/2904 + (49*(3 + 5*x)^(7/2))/(66*
(1 - 2*x)^(3/2)) - (1183*(3 + 5*x)^(7/2))/(363*Sqrt[1 - 2*x]) + (272239*Sqrt[5/2
]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/256

_______________________________________________________________________________________

Rubi [A]  time = 0.167187, antiderivative size = 140, 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 \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-123745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256 - (123745*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/2112 - (24749*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/2904 + (49*(3 + 5*x)^(7/2))/(66*
(1 - 2*x)^(3/2)) - (1183*(3 + 5*x)^(7/2))/(363*Sqrt[1 - 2*x]) + (272239*Sqrt[5/2
]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/256

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7156, size = 126, normalized size = 0.9 \[ - \frac{24749 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{2904} - \frac{123745 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{2112} - \frac{123745 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{256} + \frac{272239 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512} - \frac{1183 \left (5 x + 3\right )^{\frac{7}{2}}}{363 \sqrt{- 2 x + 1}} + \frac{49 \left (5 x + 3\right )^{\frac{7}{2}}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24749*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/2904 - 123745*sqrt(-2*x + 1)*(5*x + 3)**(
3/2)/2112 - 123745*sqrt(-2*x + 1)*sqrt(5*x + 3)/256 + 272239*sqrt(10)*asin(sqrt(
22)*sqrt(5*x + 3)/11)/512 - 1183*(5*x + 3)**(7/2)/(363*sqrt(-2*x + 1)) + 49*(5*x
+ 3)**(7/2)/(66*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.153539, size = 79, normalized size = 0.56 \[ \frac{816717 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (28800 x^4+146160 x^3+497868 x^2-1713440 x+617319\right )}{1536 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(617319 - 1713440*x + 497868*x^2 + 146160*x^3 + 28800*x^4) + 8
16717*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1536*(1 - 2*
x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 154, normalized size = 1.1 \[{\frac{1}{3072\, \left ( -1+2\,x \right ) ^{2}} \left ( -115200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-584640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1991472\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+816717\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +6853760\,x\sqrt{-10\,{x}^{2}-x+3}-2469276\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3072*(-115200*x^4*(-10*x^2-x+3)^(1/2)+3266868*10^(1/2)*arcsin(20/11*x+1/11)*x^
2-584640*x^3*(-10*x^2-x+3)^(1/2)-3266868*10^(1/2)*arcsin(20/11*x+1/11)*x-1991472
*x^2*(-10*x^2-x+3)^(1/2)+816717*10^(1/2)*arcsin(20/11*x+1/11)+6853760*x*(-10*x^2
-x+3)^(1/2)-2469276*(-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 = 1.52009, size = 333, normalized size = 2.38 \[ \frac{272239}{1024} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{5445}{256} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{1155 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{165 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{64 \,{\left (2 \, x - 1\right )}} + \frac{29645 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{104335 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

272239/1024*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 49/8*(-10*x^2 - x + 3)^(5/2
)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 21/8*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 1
2*x^2 + 6*x - 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 1) - 5445/256*sqrt
(-10*x^2 - x + 3) - 2695/96*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) +
1155/32*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 165/64*(-10*x^2 - x + 3)^(3
/2)/(2*x - 1) + 29645/192*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 104335/96*sq
rt(-10*x^2 - x + 3)/(2*x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.239262, size = 135, normalized size = 0.96 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (28800 \, x^{4} + 146160 \, x^{3} + 497868 \, x^{2} - 1713440 \, x + 617319\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 816717 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3072 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3072*sqrt(2)*(2*sqrt(2)*(28800*x^4 + 146160*x^3 + 497868*x^2 - 1713440*x + 61
7319)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 816717*sqrt(5)*(4*x^2 - 4*x + 1)*arctan(1/2
0*sqrt(5)*sqrt(2)*(20*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)**2*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257254, size = 131, normalized size = 0.94 \[ \frac{272239}{512} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 107 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 24749 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 2722390 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 44919435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{96000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

272239/512*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/96000*(4*(3*(12*(8*s
qrt(5)*(5*x + 3) + 107*sqrt(5))*(5*x + 3) + 24749*sqrt(5))*(5*x + 3) - 2722390*s
qrt(5))*(5*x + 3) + 44919435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2