3.2507 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=57 \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) + (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(7*Sqrt[7])

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Rubi [A]  time = 0.0859833, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) + (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 7.81994, size = 53, normalized size = 0.93 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0847609, size = 68, normalized size = 1.19 \[ \frac{\sqrt{7} (2 x-1) \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-14 \sqrt{1-2 x} \sqrt{5 x+3}}{98 x-49} \]

Antiderivative was successfully verified.

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

[Out]

(-14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] + Sqrt[7]*(-1 + 2*x)*ArcTan[(-20 - 37*x)/(2*Sqr
t[7 - 14*x]*Sqrt[3 + 5*x])])/(-49 + 98*x)

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Maple [B]  time = 0.02, size = 108, normalized size = 1.9 \[ -{\frac{1}{-49+98\,x} \left ( 2\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-\sqrt{7}\arctan \left ({\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \right ) +14\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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),x)

[Out]

-1/49*(2*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+14*(-10*x^2-x+3)^(1/2))*(1-2*x)
^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50003, size = 78, normalized size = 1.37 \[ -\frac{1}{49} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{10 \, x}{7 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6}{7 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/49*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 10/7*x/sqrt(-1
0*x^2 - x + 3) + 6/7/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.22779, size = 85, normalized size = 1.49 \[ -\frac{\sqrt{7}{\left ({\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/49*sqrt(7)*((2*x - 1)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*
x + 1))) + 2*sqrt(7)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 x + 3}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )}\, dx \]

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),x)

[Out]

Integral(sqrt(5*x + 3)/((-2*x + 1)**(3/2)*(3*x + 2)), x)

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GIAC/XCAS [A]  time = 0.248905, size = 135, normalized size = 2.37 \[ -\frac{1}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{35 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/490*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)
))) - 2/35*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)