3.2449 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=122 \[ \frac{3895 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}+\frac{25 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^3}-\frac{15235 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^3) + (25*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(588*(2 + 3*x)^2) + (3895*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (15
235*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.223571, 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 \[ \frac{3895 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}+\frac{25 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^3}-\frac{15235 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^3) + (25*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(588*(2 + 3*x)^2) + (3895*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (15
235*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 21.6782, size = 109, normalized size = 0.89 \[ \frac{3895 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8232 \left (3 x + 2\right )} + \frac{25 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{588 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21 \left (3 x + 2\right )^{3}} - \frac{15235 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3895*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8232*(3*x + 2)) + 25*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(588*(3*x + 2)**2) - sqrt(-2*x + 1)*sqrt(5*x + 3)/(21*(3*x + 2)**3) - 15235
*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208

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Mathematica [A]  time = 0.0856345, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (11685 x^2+15930 x+5296\right )}{(3 x+2)^3}-15235 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5296 + 15930*x + 11685*x^2))/(2 + 3*x)^3 - 152
35*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/38416

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Maple [B]  time = 0.02, size = 202, normalized size = 1.7 \[{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 411345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+822690\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+163590\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+121880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +223020\,x\sqrt{-10\,{x}^{2}-x+3}+74144\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(411345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^3+822690*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^2+548460*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x+163590*x^2*(-10*x^2-x+3)^(1/2)+121880*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))+223020*x*(-10*x^2-x+3)^(1/2)+74144*(-10*x^2-x+3)^(1/2))/(-
10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.50313, size = 144, normalized size = 1.18 \[ \frac{15235}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{25 \, \sqrt{-10 \, x^{2} - x + 3}}{588 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{3895 \, \sqrt{-10 \, x^{2} - x + 3}}{8232 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15235/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/21*sqr
t(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 25/588*sqrt(-10*x^2 - x + 3)/(
9*x^2 + 12*x + 4) + 3895/8232*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.222604, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (11685 \, x^{2} + 15930 \, x + 5296\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 15235 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38416 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*sqrt(7)*(2*sqrt(7)*(11685*x^2 + 15930*x + 5296)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 15235*(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)

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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)/(2+3*x)**4/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.351742, size = 429, normalized size = 3.52 \[ \frac{3047}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{55 \,{\left (277 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 159040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 20713280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1372 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3047/76832*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)))) - 55/1372*(277*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 - 159040*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)))^3 - 20713280*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(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))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))^2 + 280)^3