3.2058 \(\int \frac{(2+3 x) (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{5}{4} (1-2 x)^{3/2}+17 \sqrt{1-2 x}+\frac{77}{4 \sqrt{1-2 x}} \]

[Out]

77/(4*Sqrt[1 - 2*x]) + 17*Sqrt[1 - 2*x] - (5*(1 - 2*x)^(3/2))/4

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Rubi [A]  time = 0.0391016, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{5}{4} (1-2 x)^{3/2}+17 \sqrt{1-2 x}+\frac{77}{4 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

77/(4*Sqrt[1 - 2*x]) + 17*Sqrt[1 - 2*x] - (5*(1 - 2*x)^(3/2))/4

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Rubi in Sympy [A]  time = 5.95551, size = 32, normalized size = 0.84 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4} + 17 \sqrt{- 2 x + 1} + \frac{77}{4 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(3/2)/4 + 17*sqrt(-2*x + 1) + 77/(4*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0101844, size = 20, normalized size = 0.53 \[ \frac{-5 x^2-29 x+35}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(35 - 29*x - 5*x^2)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.004, size = 20, normalized size = 0.5 \[ -{(5\,{x}^{2}+29\,x-35){\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5*x^2+29*x-35)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.33205, size = 38, normalized size = 1. \[ -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 17 \, \sqrt{-2 \, x + 1} + \frac{77}{4 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)

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Fricas [A]  time = 0.237529, size = 26, normalized size = 0.68 \[ -\frac{5 \, x^{2} + 29 \, x - 35}{\sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5*x^2 + 29*x - 35)/sqrt(-2*x + 1)

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Sympy [A]  time = 4.84898, size = 160, normalized size = 4.21 \[ - \frac{19 x}{\sqrt{- 2 x + 1}} + 15 \left (\begin{cases} \frac{i x^{2} \sqrt{2 x - 1}}{6 x - 3} + \frac{2 i x \sqrt{2 x - 1}}{6 x - 3} - \frac{4 x}{6 x - 3} - \frac{2 i \sqrt{2 x - 1}}{6 x - 3} + \frac{2}{6 x - 3} & \text{for}\: 2 \left |{x}\right | > 1 \\\frac{x^{2} \sqrt{- 2 x + 1}}{6 x - 3} + \frac{2 x \sqrt{- 2 x + 1}}{6 x - 3} - \frac{4 x}{6 x - 3} - \frac{2 \sqrt{- 2 x + 1}}{6 x - 3} + \frac{2}{6 x - 3} & \text{otherwise} \end{cases}\right ) + \frac{25}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19*x/sqrt(-2*x + 1) + 15*Piecewise((I*x**2*sqrt(2*x - 1)/(6*x - 3) + 2*I*x*sqrt
(2*x - 1)/(6*x - 3) - 4*x/(6*x - 3) - 2*I*sqrt(2*x - 1)/(6*x - 3) + 2/(6*x - 3),
2*Abs(x) > 1), (x**2*sqrt(-2*x + 1)/(6*x - 3) + 2*x*sqrt(-2*x + 1)/(6*x - 3) -
4*x/(6*x - 3) - 2*sqrt(-2*x + 1)/(6*x - 3) + 2/(6*x - 3), True)) + 25/sqrt(-2*x
+ 1)

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GIAC/XCAS [A]  time = 0.214442, size = 38, normalized size = 1. \[ -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 17 \, \sqrt{-2 \, x + 1} + \frac{77}{4 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)