3.2081 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{25}{8} (1-2 x)^{5/2}-\frac{275}{8} (1-2 x)^{3/2}+\frac{1815}{8} \sqrt{1-2 x}+\frac{1331}{8 \sqrt{1-2 x}} \]

[Out]

1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*
(1 - 2*x)^(5/2))/8

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Rubi [A]  time = 0.0340452, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{25}{8} (1-2 x)^{5/2}-\frac{275}{8} (1-2 x)^{3/2}+\frac{1815}{8} \sqrt{1-2 x}+\frac{1331}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*
(1 - 2*x)^(5/2))/8

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Rubi in Sympy [A]  time = 5.87251, size = 46, normalized size = 0.87 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{275 \left (- 2 x + 1\right )^{\frac{3}{2}}}{8} + \frac{1815 \sqrt{- 2 x + 1}}{8} + \frac{1331}{8 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(5/2)/8 - 275*(-2*x + 1)**(3/2)/8 + 1815*sqrt(-2*x + 1)/8 + 1331/
(8*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0309894, size = 25, normalized size = 0.47 \[ \frac{-25 x^3-100 x^2-335 x+362}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(362 - 335*x - 100*x^2 - 25*x^3)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{(25\,{x}^{3}+100\,{x}^{2}+335\,x-362){\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^3/(1-2*x)^(3/2),x)

[Out]

-(25*x^3+100*x^2+335*x-362)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.3444, size = 50, normalized size = 0.94 \[ \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{275}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1815}{8} \, \sqrt{-2 \, x + 1} + \frac{1331}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/8*(-2*x + 1)^(5/2) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1331/8/
sqrt(-2*x + 1)

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Fricas [A]  time = 0.217145, size = 32, normalized size = 0.6 \[ -\frac{25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362}{\sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(25*x^3 + 100*x^2 + 335*x - 362)/sqrt(-2*x + 1)

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Sympy [A]  time = 3.38117, size = 435, normalized size = 8.21 \[ \begin{cases} \frac{125 \sqrt{55} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{275 \sqrt{55} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{1210 \sqrt{55} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{26620 \sqrt{5} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{2662 \sqrt{55} i \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{29282 \sqrt{5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{125 \sqrt{55} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{3}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{275 \sqrt{55} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{1210 \sqrt{55} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{2662 \sqrt{55} \sqrt{- 10 x + 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{26620 \sqrt{5} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{29282 \sqrt{5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((125*sqrt(55)*I*(x + 3/5)**3*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 5
5*sqrt(11)) + 275*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5)
- 55*sqrt(11)) + 1210*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5)
- 55*sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11))
- 2662*sqrt(55)*I*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*s
qrt(5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (125*sqrt
(55)*sqrt(-10*x + 5)*(x + 3/5)**3/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 275*sq
rt(55)*sqrt(-10*x + 5)*(x + 3/5)**2/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210
*sqrt(55)*sqrt(-10*x + 5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662
*sqrt(55)*sqrt(-10*x + 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 26620*sqrt(5)*
(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x
+ 3/5) - 55*sqrt(11)), True))

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GIAC/XCAS [A]  time = 0.221699, size = 59, normalized size = 1.11 \[ \frac{25}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{275}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1815}{8} \, \sqrt{-2 \, x + 1} + \frac{1331}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/8*(2*x - 1)^2*sqrt(-2*x + 1) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1)
+ 1331/8/sqrt(-2*x + 1)