3.2328 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(224*(2 + 3*x)^2) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^4)
+ (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(8*(2 + 3*x)^3) - (43923*ArcTan[Sqrt[1 - 2
*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.216138, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(224*(2 + 3*x)^2) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^4)
+ (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(8*(2 + 3*x)^3) - (43923*ArcTan[Sqrt[1 - 2
*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.1334, size = 136, normalized size = 0.9 \[ - \frac{363 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1568 \left (3 x + 2\right )^{2}} - \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{56 \left (3 x + 2\right )^{3}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{4 \left (3 x + 2\right )^{4}} + \frac{3993 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3136 \left (3 x + 2\right )} - \frac{43923 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-363*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(1568*(3*x + 2)**2) - 11*(-2*x + 1)**(3/2)*
(5*x + 3)**(3/2)/(56*(3*x + 2)**3) + (-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(4*(3*x
+ 2)**4) + 3993*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3136*(3*x + 2)) - 43923*sqrt(7)*at
an(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/21952

_______________________________________________________________________________________

Mathematica [A]  time = 0.101496, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{43904} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 213240*x^2 + 100159*x^3))/(
2 + 3*x)^4 - 43923*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])]
)/43904

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 250, normalized size = 1.7 \[{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3557763\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1402226\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4216608\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2985360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+702768\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2043160\,x\sqrt{-10\,{x}^{2}-x+3}+453600\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43904*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3557763*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x^4+9487368*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^3+9487368*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^2+1402226*x^3*(-10*x^2-x+3)^(1/2)+4216608*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2985360*x^2*(-10*x^2-x+3)^(1/2)+702768*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2043160*x*(-10*x^2-x+3)^(1/2)+4
53600*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

_______________________________________________________________________________________

Maxima [A]  time = 1.50695, size = 251, normalized size = 1.66 \[ \frac{8245}{16464} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4947 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{67155}{10976} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{43923}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{59169}{21952} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{19573 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{65856 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8245/16464*(-10*x^2 - x + 3)^(3/2) + 3/28*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16) + 111/392*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 +
36*x + 8) + 4947/10976*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 67155/10976*
sqrt(-10*x^2 - x + 3)*x + 43923/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/1
1/abs(3*x + 2)) - 59169/21952*sqrt(-10*x^2 - x + 3) + 19573/65856*(-10*x^2 - x +
3)^(3/2)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.222612, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (100159 \, x^{3} + 213240 \, x^{2} + 145940 \, x + 32400\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 43923 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{43904 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43904*sqrt(7)*(2*sqrt(7)*(100159*x^3 + 213240*x^2 + 145940*x + 32400)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) + 43923*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/1
4*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 + 216*x
^2 + 96*x + 16)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.41175, size = 512, normalized size = 3.39 \[ \frac{43923}{439040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 3080 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 862400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65856000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1568 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

43923/439040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 14641/1568*(3*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)))^7 + 3080*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)))^5 - 862400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 -
65856000*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) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 +
280)^4