3.2581 \(\int \frac{(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{4819}{440} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac{4270537963 \sqrt{1-2 x} (5 x+3)^{3/2}}{3379200}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac{4270537963 \sqrt{1-2 x} \sqrt{5 x+3}}{409600}+\frac{46975917593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{409600 \sqrt{10}} \]

[Out]

(-4270537963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/409600 - (4270537963*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/3379200 - (4819*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/440 - (
439*(2 + 3*x)^3*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^4*(3 + 5*x)^(5/
2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(36714139 + 18161940*x)
)/140800 + (46975917593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(409600*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.320355, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{4819}{440} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac{4270537963 \sqrt{1-2 x} (5 x+3)^{3/2}}{3379200}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac{4270537963 \sqrt{1-2 x} \sqrt{5 x+3}}{409600}+\frac{46975917593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{409600 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^4*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

(-4270537963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/409600 - (4270537963*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/3379200 - (4819*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/440 - (
439*(2 + 3*x)^3*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^4*(3 + 5*x)^(5/
2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(36714139 + 18161940*x)
)/140800 + (46975917593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(409600*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.4328, size = 178, normalized size = 0.96 \[ - \frac{969 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{56} - \frac{43527 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{640} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (\frac{83744836875 x}{4} + \frac{1418769347625}{32}\right )}{25200000} - \frac{4270537963 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{409600} + \frac{46975917593 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4096000} - \frac{439 \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{3}{2}}}{42 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

-969*sqrt(-2*x + 1)*(3*x + 2)**3*(5*x + 3)**(3/2)/56 - 43527*sqrt(-2*x + 1)*(3*x
+ 2)**2*(5*x + 3)**(3/2)/640 - sqrt(-2*x + 1)*(5*x + 3)**(3/2)*(83744836875*x/4
+ 1418769347625/32)/25200000 - 4270537963*sqrt(-2*x + 1)*sqrt(5*x + 3)/409600 +
46975917593*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/4096000 - 439*(3*x + 2)**4
*(5*x + 3)**(3/2)/(42*sqrt(-2*x + 1)) + (3*x + 2)**4*(5*x + 3)**(5/2)/(3*(-2*x +
1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.191504, size = 89, normalized size = 0.48 \[ \frac{140927752779 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (248832000 x^6+1423526400 x^5+4002203520 x^4+8217694800 x^3+18987469764 x^2-58600061024 x+21368105901\right )}{12288000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^4*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

(-10*Sqrt[3 + 5*x]*(21368105901 - 58600061024*x + 18987469764*x^2 + 8217694800*x
^3 + 4002203520*x^4 + 1423526400*x^5 + 248832000*x^6) + 140927752779*Sqrt[10 - 2
0*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(12288000*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 188, normalized size = 1. \[{\frac{1}{24576000\, \left ( -1+2\,x \right ) ^{2}} \left ( -4976640000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-28470528000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-80044070400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-164353896000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-563711011116\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-379749395280\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-427362118020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)

[Out]

1/24576000*(-4976640000*x^6*(-10*x^2-x+3)^(1/2)-28470528000*x^5*(-10*x^2-x+3)^(1
/2)-80044070400*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/1
1)*x^2-164353896000*x^3*(-10*x^2-x+3)^(1/2)-563711011116*10^(1/2)*arcsin(20/11*x
+1/11)*x-379749395280*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)-427362118020*(-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 = 1.5196, size = 478, normalized size = 2.57 \[ -\frac{81}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{891}{256} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11872553}{2048} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{514294407}{8192000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{139491}{5120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{189 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (2 \, x - 1\right )}} - \frac{4250367}{20480} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{89257707}{409600} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{800415}{512} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{132055 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{384 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{56595 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{64 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{24255 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (2 \, x - 1\right )}} + \frac{1452605 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{15827735 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

-81/160*(-10*x^2 - x + 3)^(5/2) + 891/256*(-10*x^2 - x + 3)^(3/2)*x + 11872553/2
048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 514294407/8192000*I*sqrt(5)*sqrt(2)
*arcsin(20/11*x - 21/11) + 139491/5120*(-10*x^2 - x + 3)^(3/2) - 2401/32*(-10*x^
2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 1029/16*(-10*x^2 - x + 3
)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 441/16*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x
+ 1) - 189/32*(-10*x^2 - x + 3)^(5/2)/(2*x - 1) - 4250367/20480*sqrt(10*x^2 - 2
1*x + 8)*x + 89257707/409600*sqrt(10*x^2 - 21*x + 8) - 800415/512*sqrt(-10*x^2 -
x + 3) - 132055/384*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 56595/
64*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 24255/128*(-10*x^2 - x + 3)^(3/2)
/(2*x - 1) + 1452605/768*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 15827735/768*
sqrt(-10*x^2 - x + 3)/(2*x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.230598, size = 140, normalized size = 0.75 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (248832000 \, x^{6} + 1423526400 \, x^{5} + 4002203520 \, x^{4} + 8217694800 \, x^{3} + 18987469764 \, x^{2} - 58600061024 \, x + 21368105901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 140927752779 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{24576000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/24576000*sqrt(10)*(2*sqrt(10)*(248832000*x^6 + 1423526400*x^5 + 4002203520*x^
4 + 8217694800*x^3 + 18987469764*x^2 - 58600061024*x + 21368105901)*sqrt(5*x + 3
)*sqrt(-2*x + 1) - 140927752779*(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)

_______________________________________________________________________________________

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)**4*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.25127, size = 166, normalized size = 0.89 \[ \frac{46975917593}{4096000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \,{\left (72 \,{\left (4 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 509 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 20743 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 18487133 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4270537963 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 469759175930 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7751026402845 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{768000000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

46975917593/4096000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/768000000*(
4*(3*(12*(72*(4*(48*sqrt(5)*(5*x + 3) + 509*sqrt(5))*(5*x + 3) + 20743*sqrt(5))*
(5*x + 3) + 18487133*sqrt(5))*(5*x + 3) + 4270537963*sqrt(5))*(5*x + 3) - 469759
175930*sqrt(5))*(5*x + 3) + 7751026402845*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)
/(2*x - 1)^2