3.2508 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=93 \[ \frac{4 (5 x+3)^{3/2}}{77 \sqrt{1-2 x} (3 x+2)}-\frac{29 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)}-\frac{29 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

(-29*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(539*(2 + 3*x)) + (4*(3 + 5*x)^(3/2))/(77*Sqrt
[1 - 2*x]*(2 + 3*x)) - (29*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sq
rt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.128351, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{3/2}}{77 \sqrt{1-2 x} (3 x+2)}-\frac{29 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)}-\frac{29 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-29*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(539*(2 + 3*x)) + (4*(3 + 5*x)^(3/2))/(77*Sqrt
[1 - 2*x]*(2 + 3*x)) - (29*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sq
rt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.92505, size = 78, normalized size = 0.84 \[ - \frac{29 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{29 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1}} + \frac{3 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-29*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 - 29*sqrt(5*x + 3
)/(49*sqrt(-2*x + 1)) + 3*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0761256, size = 75, normalized size = 0.81 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3} (18 x+5)}{49 \left (6 x^2+x-2\right )}-\frac{29 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{98 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5 + 18*x))/(49*(-2 + x + 6*x^2)) - (29*ArcTan[(-2
0 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(98*Sqrt[7])

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 161, normalized size = 1.7 \[{\frac{1}{ \left ( 1372+2058\,x \right ) \left ( -1+2\,x \right ) } \left ( 174\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+29\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-58\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -252\,x\sqrt{-10\,{x}^{2}-x+3}-70\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/686*(174*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+29*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-58*7^(1/2)*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-252*x*(-10*x^2-x+3)^(1/2)-70*(-10*x^2-x+
3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50691, size = 124, normalized size = 1.33 \[ \frac{29}{686} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{30 \, x}{49 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{19}{147 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{21 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 30/49*x/sqrt(
-10*x^2 - x + 3) + 19/147/sqrt(-10*x^2 - x + 3) + 1/21/(3*sqrt(-10*x^2 - x + 3)*
x + 2*sqrt(-10*x^2 - x + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.231633, size = 101, normalized size = 1.09 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (18 \, x + 5\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 29 \,{\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{686 \,{\left (6 \, x^{2} + x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/686*sqrt(7)*(2*sqrt(7)*(18*x + 5)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 29*(6*x^2 +
x - 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(6*x^2 +
x - 2)

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.303508, size = 296, normalized size = 3.18 \[ \frac{29}{6860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{245 \,{\left (2 \, x - 1\right )}} - \frac{66 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/6860*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
)))) - 4/245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 66/49*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-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)