3.2059 \(\int \frac{3+5 x}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{2} \sqrt{1-2 x}+\frac{11}{2 \sqrt{1-2 x}} \]

[Out]

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

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Rubi [A]  time = 0.0223937, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{5}{2} \sqrt{1-2 x}+\frac{11}{2 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.91019, size = 22, normalized size = 0.81 \[ \frac{5 \sqrt{- 2 x + 1}}{2} + \frac{11}{2 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(-2*x + 1)/2 + 11/(2*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.00675004, size = 15, normalized size = 0.56 \[ \frac{8-5 x}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(8 - 5*x)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[ -{(-8+5\,x){\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^(3/2),x)

[Out]

-(-8+5*x)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.33933, size = 26, normalized size = 0.96 \[ \frac{5}{2} \, \sqrt{-2 \, x + 1} + \frac{11}{2 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)

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Fricas [A]  time = 0.219632, size = 19, normalized size = 0.7 \[ -\frac{5 \, x - 8}{\sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5*x - 8)/sqrt(-2*x + 1)

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Sympy [A]  time = 0.682001, size = 20, normalized size = 0.74 \[ - \frac{5 x}{\sqrt{- 2 x + 1}} + \frac{8}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*x/sqrt(-2*x + 1) + 8/sqrt(-2*x + 1)

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GIAC/XCAS [A]  time = 0.228574, size = 26, normalized size = 0.96 \[ \frac{5}{2} \, \sqrt{-2 \, x + 1} + \frac{11}{2 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)