3.2339 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=171 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{331 \sqrt{1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac{39745 \sqrt{1-2 x} \sqrt{5 x+3}}{4536}-\frac{575}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{326717 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{13608 \sqrt{7}} \]

[Out]

(-39745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4536 + (331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(168*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + (181*Sqrt[
1 - 2*x]*(3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (575*Sqrt[10]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/243 - (326717*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1360
8*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.38295, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{331 \sqrt{1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac{39745 \sqrt{1-2 x} \sqrt{5 x+3}}{4536}-\frac{575}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{326717 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{13608 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-39745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4536 + (331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(168*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + (181*Sqrt[
1 - 2*x]*(3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (575*Sqrt[10]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/243 - (326717*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1360
8*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.6133, size = 156, normalized size = 0.91 \[ - \frac{6961 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{10584 \left (3 x + 2\right )} - \frac{181 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{756 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{3}} - \frac{24251 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15876} - \frac{575 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{243} - \frac{326717 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{95256} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6961*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(10584*(3*x + 2)) - 181*(-2*x + 1)**(3/2)*
(5*x + 3)**(3/2)/(756*(3*x + 2)**2) - (-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(9*(3*x
+ 2)**3) - 24251*sqrt(-2*x + 1)*sqrt(5*x + 3)/15876 - 575*sqrt(10)*asin(sqrt(22
)*sqrt(5*x + 3)/11)/243 - 326717*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x
+ 3)))/95256

_______________________________________________________________________________________

Mathematica [A]  time = 0.237735, size = 117, normalized size = 0.68 \[ \frac{-\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (75600 x^3+286791 x^2+275022 x+78416\right )}{(3 x+2)^3}-326717 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-225400 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{190512} \]

Antiderivative was successfully verified.

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

[Out]

((-42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(78416 + 275022*x + 286791*x^2 + 75600*x^3))/(
2 + 3*x)^3 - 326717*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])
] - 225400*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/190512

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 270, normalized size = 1.6 \[{\frac{1}{190512\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 8821359\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-6085800\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12171600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-3175200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-8114400\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-12045222\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2613736\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1803200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -11550924\,x\sqrt{-10\,{x}^{2}-x+3}-3293472\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/190512*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(8821359*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^3-6085800*10^(1/2)*arcsin(20/11*x+1/11)*x^3+17642718*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-12171600*10^(1/2)
*arcsin(20/11*x+1/11)*x^2-3175200*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-8114400*10^(1/2)*arcsin(20/11*x+
1/11)*x-12045222*x^2*(-10*x^2-x+3)^(1/2)+2613736*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))-1803200*10^(1/2)*arcsin(20/11*x+1/11)-11550924*x*(-1
0*x^2-x+3)^(1/2)-3293472*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

_______________________________________________________________________________________

Maxima [A]  time = 1.54101, size = 217, normalized size = 1.27 \[ \frac{865}{2646} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{173 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{588 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{34805}{5292} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{575}{486} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{326717}{190512} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{152917}{31752} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{2507 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3528 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

865/2646*(-10*x^2 - x + 3)^(3/2) - 1/21*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2
+ 36*x + 8) + 173/588*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 34805/5292*s
qrt(-10*x^2 - x + 3)*x - 575/486*sqrt(10)*arcsin(20/11*x + 1/11) + 326717/190512
*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 152917/31752*sqrt(-
10*x^2 - x + 3) + 2507/3528*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.232188, size = 198, normalized size = 1.16 \[ -\frac{\sqrt{7}{\left (32200 \, \sqrt{10} \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (75600 \, x^{3} + 286791 \, x^{2} + 275022 \, x + 78416\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 326717 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{190512 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/190512*sqrt(7)*(32200*sqrt(10)*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/
20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(75600*x^3 +
286791*x^2 + 275022*x + 78416)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 326717*(27*x^3 + 5
4*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))
))/(27*x^3 + 54*x^2 + 36*x + 8)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.502517, size = 545, normalized size = 3.19 \[ \frac{326717}{1905120} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{575}{486} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{10}{81} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \,{\left (2463 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1767360 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 377652800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{756 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

326717/1905120*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(22)))) - 575/486*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(
-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) -
10/81*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/756*(2463*sqrt(10)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^5 + 1767360*sqrt(10)*((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)))^3 + 377652800*
sqrt(10)*((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))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3