3.2567 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}} \]

[Out]

(2*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2))

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Rubi [A]  time = 0.0155742, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2))

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Rubi in Sympy [A]  time = 2.89888, size = 19, normalized size = 0.86 \[ \frac{2 \left (5 x + 3\right )^{\frac{3}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(5*x + 3)**(3/2)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0248704, size = 22, normalized size = 1. \[ \frac{2 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.005, size = 17, normalized size = 0.8 \[{\frac{2}{33} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*(3+5*x)^(3/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.48675, size = 65, normalized size = 2.95 \[ \frac{\sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 5/33*sqrt(-10*x^2 - x + 3)/(2*x -
1)

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Fricas [A]  time = 0.218012, size = 38, normalized size = 1.73 \[ \frac{2 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{33 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*(5*x + 3)^(3/2)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 5.0071, size = 82, normalized size = 3.73 \[ \begin{cases} \frac{250 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{330 \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5} - 363 \sqrt{10 x - 5}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{250 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{330 \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right ) - 363 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((250*I*(x + 3/5)**(3/2)/(330*(x + 3/5)*sqrt(10*x - 5) - 363*sqrt(10*x
- 5)), 10*Abs(x + 3/5)/11 > 1), (-250*(x + 3/5)**(3/2)/(330*sqrt(-10*x + 5)*(x +
3/5) - 363*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.224908, size = 35, normalized size = 1.59 \[ \frac{2 \, \sqrt{5}{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-10 \, x + 5}}{165 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/165*sqrt(5)*(5*x + 3)^(3/2)*sqrt(-10*x + 5)/(2*x - 1)^2