3.2399 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{247}{324} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{1453}{288} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{155777 \sqrt{1-2 x} \sqrt{5 x+3}}{31104}-\frac{660959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{93312 \sqrt{10}}-\frac{1295}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/31104 + (1453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/288 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/324 - (5*(1 - 2*x)^(3/2)*(3 + 5*x)^
(5/2))/18 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) - (660959*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/(93312*Sqrt[10]) - (1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/729

_______________________________________________________________________________________

Rubi [A]  time = 0.457604, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{247}{324} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{1453}{288} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{155777 \sqrt{1-2 x} \sqrt{5 x+3}}{31104}-\frac{660959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{93312 \sqrt{10}}-\frac{1295}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/31104 + (1453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/288 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/324 - (5*(1 - 2*x)^(3/2)*(3 + 5*x)^
(5/2))/18 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) - (660959*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/(93312*Sqrt[10]) - (1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/729

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.1979, size = 162, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (3 x + 2\right )} - \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{18} + \frac{1235 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{648} - \frac{11045 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5184} - \frac{9983 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{31104} - \frac{660959 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{933120} - \frac{1295 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{729} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(3*(3*x + 2)) - 5*(-2*x + 1)**(3/2)*(5*x + 3
)**(5/2)/18 + 1235*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/648 - 11045*(-2*x + 1)**(3
/2)*sqrt(5*x + 3)/5184 - 9983*sqrt(-2*x + 1)*sqrt(5*x + 3)/31104 - 660959*sqrt(1
0)*asin(sqrt(22)*sqrt(5*x + 3)/11)/933120 - 1295*sqrt(7)*atan(sqrt(7)*sqrt(-2*x
+ 1)/(7*sqrt(5*x + 3)))/729

_______________________________________________________________________________________

Mathematica [A]  time = 0.213163, size = 122, normalized size = 0.68 \[ \frac{\frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (259200 x^4-214560 x^3-60348 x^2+72849 x-45658\right )}{3 x+2}-1657600 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-660959 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1866240} \]

Antiderivative was successfully verified.

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

[Out]

((60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-45658 + 72849*x - 60348*x^2 - 214560*x^3 + 25
9200*x^4))/(2 + 3*x) - 1657600*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqr
t[3 + 5*x])] - 660959*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x
])])/1866240

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 197, normalized size = 1.1 \[{\frac{1}{3732480+5598720\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 15552000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-12873600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4972800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-1982877\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3620880\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3315200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1321918\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +4370940\,x\sqrt{-10\,{x}^{2}-x+3}-2739480\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1866240*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15552000*x^4*(-10*x^2-x+3)^(1/2)-12873600
*x^3*(-10*x^2-x+3)^(1/2)+4972800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x-1982877*10^(1/2)*arcsin(20/11*x+1/11)*x-3620880*x^2*(-10*x^2-x+3)^
(1/2)+3315200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1321918
*10^(1/2)*arcsin(20/11*x+1/11)+4370940*x*(-10*x^2-x+3)^(1/2)-2739480*(-10*x^2-x+
3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.51234, size = 161, normalized size = 0.9 \[ -\frac{25}{18} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{695}{648} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3 \,{\left (3 \, x + 2\right )}} + \frac{11045}{2592} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{660959}{1866240} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1295}{1458} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{76253}{31104} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/18*(-10*x^2 - x + 3)^(3/2)*x + 695/648*(-10*x^2 - x + 3)^(3/2) - 1/3*(-10*x^
2 - x + 3)^(5/2)/(3*x + 2) + 11045/2592*sqrt(-10*x^2 - x + 3)*x - 660959/1866240
*sqrt(10)*arcsin(20/11*x + 1/11) + 1295/1458*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) - 76253/31104*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.230635, size = 165, normalized size = 0.92 \[ \frac{\sqrt{10}{\left (165760 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{10}{\left (259200 \, x^{4} - 214560 \, x^{3} - 60348 \, x^{2} + 72849 \, x - 45658\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 660959 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1866240 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1866240*sqrt(10)*(165760*sqrt(10)*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x
+ 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(10)*(259200*x^4 - 214560*x^3 - 60
348*x^2 + 72849*x - 45658)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 660959*(3*x + 2)*arcta
n(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(3*x + 2)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.456466, size = 429, normalized size = 2.4 \[ \frac{259}{2916} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{777600} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 593 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 26185 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 622085 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{660959}{1866240} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{243 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

259/2916*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2
)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2)))) + 1/777600*(12*(8*(36*sqrt(5)*(5*x + 3) - 593*sqrt(5))*(5*x + 3) + 26185*s
qrt(5))*(5*x + 3) - 622085*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 660959/18662
40*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1078/243*sqrt(10)*
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)
- 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)