3.2296 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) - (2*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/5

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Rubi [A]  time = 0.0446696, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) - (2*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/5

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Rubi in Sympy [A]  time = 5.21063, size = 46, normalized size = 0.88 \[ - \frac{2 \sqrt{- 2 x + 1}}{5 \sqrt{5 x + 3}} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)/(5*sqrt(5*x + 3)) - 2*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)
/25

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Mathematica [A]  time = 0.0868619, size = 49, normalized size = 0.94 \[ \frac{2}{25} \left (\sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{5 \sqrt{1-2 x}}{\sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-5*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]
))/25

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{1\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50837, size = 49, normalized size = 0.94 \[ -\frac{1}{25} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{5 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/5*sqrt(-10*x^2 - x + 3)/(5*x +
3)

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Fricas [A]  time = 0.217918, size = 93, normalized size = 1.79 \[ -\frac{\sqrt{5}{\left (\sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*sqrt(5)*(sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*
x + 3)*sqrt(-2*x + 1))) + 2*sqrt(5)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 2.70376, size = 153, normalized size = 2.94 \[ \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} - \frac{\sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{25} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{25} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} + \frac{2 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{25} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/25 - sqrt(10)*I*log(1/(x + 3
/5))/25 - sqrt(10)*I*log(x + 3/5)/25 - 2*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/1
1)/25, 11*Abs(1/(x + 3/5))/10 > 1), (-2*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/2
5 - sqrt(10)*I*log(1/(x + 3/5))/25 + 2*sqrt(10)*I*log(sqrt(1 - 11/(10*(x + 3/5))
) + 1)/25, True))

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GIAC/XCAS [A]  time = 0.227145, size = 112, normalized size = 2.15 \[ -\frac{1}{50} \, \sqrt{5}{\left (4 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{2} \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/50*sqrt(5)*(4*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + sqrt(2)*(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(2)*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))