3.2123 \(\int \frac{3+5 x}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{11}{6 (1-2 x)^{3/2}}-\frac{5}{2 \sqrt{1-2 x}} \]

[Out]

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

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Rubi [A]  time = 0.0220561, 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 \[ \frac{11}{6 (1-2 x)^{3/2}}-\frac{5}{2 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.90308, size = 22, normalized size = 0.81 \[ - \frac{5}{2 \sqrt{- 2 x + 1}} + \frac{11}{6 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/(2*sqrt(-2*x + 1)) + 11/(6*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0086389, size = 18, normalized size = 0.67 \[ \frac{15 x-2}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2 + 15*x)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[{\frac{-2+15\,x}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^(5/2),x)

[Out]

1/3*(-2+15*x)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.33921, size = 19, normalized size = 0.7 \[ \frac{15 \, x - 2}{3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(15*x - 2)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.21014, size = 28, normalized size = 1.04 \[ -\frac{15 \, x - 2}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(15*x - 2)/((2*x - 1)*sqrt(-2*x + 1))

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Sympy [A]  time = 1.04829, size = 48, normalized size = 1.78 \[ - \frac{15 x}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{2}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*x/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) + 2/(6*x*sqrt(-2*x + 1) - 3*sqrt(-
2*x + 1))

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GIAC/XCAS [A]  time = 0.221604, size = 28, normalized size = 1.04 \[ -\frac{15 \, x - 2}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(15*x - 2)/((2*x - 1)*sqrt(-2*x + 1))