3.2297 \(\int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ 2 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{2 \sqrt{1-2 x}}{\sqrt{5 x+3}} \]

[Out]

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

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Rubi [A]  time = 0.0823963, antiderivative size = 53, 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 \[ 2 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{2 \sqrt{1-2 x}}{\sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.48982, size = 49, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1}}{\sqrt{5 x + 3}} + 2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.066337, size = 55, normalized size = 1.04 \[ \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{2 \sqrt{1-2 x}}{\sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 100, normalized size = 1.9 \[{1 \left ( -5\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-3\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-5*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-3*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-2*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/
2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.50648, size = 78, normalized size = 1.47 \[ -\sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4 \, x}{\sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{\sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4*x/sqrt(-10*x^2 -
x + 3) - 2/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217067, size = 81, normalized size = 1.53 \[ -\frac{\sqrt{7}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{5 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23499, size = 184, normalized size = 3.47 \[ -\frac{1}{10} \, \sqrt{5}{\left (\sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*sqrt(5)*(sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))) + sqrt(2)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))