3.2357 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 - (3
3*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi [A]  time = 0.0599891, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 - (3
3*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi in Sympy [A]  time = 6.63255, size = 66, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5 \sqrt{5 x + 3}} - \frac{6 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{25} - \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)/(5*sqrt(5*x + 3)) - 6*sqrt(-2*x + 1)*sqrt(5*x + 3)/25 - 33*
sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/125

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Mathematica [A]  time = 0.08146, size = 57, normalized size = 0.77 \[ \frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{2 \sqrt{1-2 x} (5 x+14)}{25 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(14 + 5*x))/(25*Sqrt[3 + 5*x]) + (33*Sqrt[2/5]*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/25

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{1 \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.5014, size = 84, normalized size = 1.14 \[ -\frac{33}{250} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{5 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/5*(-10*x^2 - x + 3)^(3/2)/(25
*x^2 + 30*x + 9) - 33/25*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.22149, size = 101, normalized size = 1.36 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (5 \, x + 14\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 33 \, \sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{250 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/250*sqrt(5)*(4*sqrt(5)*(5*x + 14)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 33*sqrt(2)*(
5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))
/(5*x + 3)

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Sympy [A]  time = 5.18093, size = 187, normalized size = 2.53 \[ \begin{cases} - \frac{4 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{5 \sqrt{10 x - 5}} - \frac{22 i \sqrt{x + \frac{3}{5}}}{25 \sqrt{10 x - 5}} + \frac{33 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} + \frac{242 i}{125 \sqrt{x + \frac{3}{5}} \sqrt{10 x - 5}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} + \frac{4 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{5 \sqrt{- 10 x + 5}} + \frac{22 \sqrt{x + \frac{3}{5}}}{25 \sqrt{- 10 x + 5}} - \frac{242}{125 \sqrt{- 10 x + 5} \sqrt{x + \frac{3}{5}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*I*(x + 3/5)**(3/2)/(5*sqrt(10*x - 5)) - 22*I*sqrt(x + 3/5)/(25*sqr
t(10*x - 5)) + 33*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/125 + 242*I/(125*
sqrt(x + 3/5)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (-33*sqrt(10)*asin(sqrt(
110)*sqrt(x + 3/5)/11)/125 + 4*(x + 3/5)**(3/2)/(5*sqrt(-10*x + 5)) + 22*sqrt(x
+ 3/5)/(25*sqrt(-10*x + 5)) - 242/(125*sqrt(-10*x + 5)*sqrt(x + 3/5)), True))

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GIAC/XCAS [A]  time = 0.246553, size = 132, normalized size = 1.78 \[ -\frac{2}{125} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{33}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/125*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 33/125*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 11/250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5
*x + 3) + 22/125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))