3.2291 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{18083 \sqrt{1-2 x} \sqrt{5 x+3}}{1176 (3 x+2)}+\frac{173 \sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}-\frac{68959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (173*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(84*(2 + 3*x)^2) + (18083*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1176*(2 + 3*x)) - (689
59*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.227188, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{18083 \sqrt{1-2 x} \sqrt{5 x+3}}{1176 (3 x+2)}+\frac{173 \sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}-\frac{68959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (173*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(84*(2 + 3*x)^2) + (18083*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1176*(2 + 3*x)) - (689
59*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.6189, size = 109, normalized size = 0.89 \[ \frac{18083 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1176 \left (3 x + 2\right )} + \frac{173 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{3}} - \frac{68959 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2744} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18083*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1176*(3*x + 2)) + 173*sqrt(-2*x + 1)*sqrt(5*
x + 3)/(84*(3*x + 2)**2) + sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*(3*x + 2)**3) - 68959
*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2744

_______________________________________________________________________________________

Mathematica [A]  time = 0.0863561, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (54249 x^2+74754 x+25856\right )}{(3 x+2)^3}-68959 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{5488} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25856 + 74754*x + 54249*x^2))/(2 + 3*x)^3 - 68
959*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/5488

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 202, normalized size = 1.7 \[{\frac{1}{5488\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1861893\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+3723786\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2482524\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+759486\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+551672\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1046556\,x\sqrt{-10\,{x}^{2}-x+3}+361984\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5488*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1861893*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^3+3723786*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))*x^2+2482524*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))*x+759486*x^2*(-10*x^2-x+3)^(1/2)+551672*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))+1046556*x*(-10*x^2-x+3)^(1/2)+361984*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

_______________________________________________________________________________________

Maxima [A]  time = 1.49859, size = 144, normalized size = 1.18 \[ \frac{68959}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{173 \, \sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{18083 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

68959/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/3*sqrt(
-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 173/84*sqrt(-10*x^2 - x + 3)/(9*
x^2 + 12*x + 4) + 18083/1176*sqrt(-10*x^2 - x + 3)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.221107, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (54249 \, x^{2} + 74754 \, x + 25856\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 68959 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5488*sqrt(7)*(2*sqrt(7)*(54249*x^2 + 74754*x + 25856)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 68959*(27*x^3 + 54*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)**(1/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.330176, size = 425, normalized size = 3.48 \[ \frac{11}{54880} \, \sqrt{5}{\left (6269 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{280 \, \sqrt{2}{\left (13331 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 4674880 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{491489600 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{1965958400 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/54880*sqrt(5)*(6269*sqrt(70)*sqrt(2)*(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)))) + 280*sqrt(2)*(13331*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 4674880*
((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)))^3 + 491489600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 1965958400*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/((
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^3)