3.2000 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]

[Out]

(-121*Sqrt[1 - 2*x])/4 + (55*(1 - 2*x)^(3/2))/6 - (5*(1 - 2*x)^(5/2))/4

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Rubi [A]  time = 0.0289431, antiderivative size = 40, 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 \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-121*Sqrt[1 - 2*x])/4 + (55*(1 - 2*x)^(3/2))/6 - (5*(1 - 2*x)^(5/2))/4

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Rubi in Sympy [A]  time = 5.15926, size = 34, normalized size = 0.85 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{5}{2}}}{4} + \frac{55 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6} - \frac{121 \sqrt{- 2 x + 1}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(5/2)/4 + 55*(-2*x + 1)**(3/2)/6 - 121*sqrt(-2*x + 1)/4

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Mathematica [A]  time = 0.0207163, size = 23, normalized size = 0.57 \[ -\frac{1}{3} \sqrt{1-2 x} \left (15 x^2+40 x+67\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(67 + 40*x + 15*x^2))/3

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Maple [A]  time = 0.004, size = 20, normalized size = 0.5 \[ -{\frac{15\,{x}^{2}+40\,x+67}{3}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2/(1-2*x)^(1/2),x)

[Out]

-1/3*(15*x^2+40*x+67)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.34789, size = 38, normalized size = 0.95 \[ -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{121}{4} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(-2*x + 1)^(5/2) + 55/6*(-2*x + 1)^(3/2) - 121/4*sqrt(-2*x + 1)

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Fricas [A]  time = 0.213117, size = 26, normalized size = 0.65 \[ -\frac{1}{3} \,{\left (15 \, x^{2} + 40 \, x + 67\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(15*x^2 + 40*x + 67)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.1714, size = 134, normalized size = 3.35 \[ \begin{cases} - \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5} - \frac{22 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{15} - \frac{242 \sqrt{5} i \sqrt{10 x - 5}}{75} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2} - \frac{22 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{15} - \frac{242 \sqrt{5} \sqrt{- 10 x + 5}}{75} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5) - 22*sqrt(5)*I*(x + 3/5)*sqrt(
10*x - 5)/15 - 242*sqrt(5)*I*sqrt(10*x - 5)/75, 10*Abs(x + 3/5)/11 > 1), (-sqrt(
5)*sqrt(-10*x + 5)*(x + 3/5)**2 - 22*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/15 - 242*
sqrt(5)*sqrt(-10*x + 5)/75, True))

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GIAC/XCAS [A]  time = 0.209799, size = 47, normalized size = 1.18 \[ -\frac{5}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{121}{4} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(2*x - 1)^2*sqrt(-2*x + 1) + 55/6*(-2*x + 1)^(3/2) - 121/4*sqrt(-2*x + 1)