3.2071 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{12} (1-2 x)^{3/2}+\frac{55}{2} \sqrt{1-2 x}+\frac{121}{4 \sqrt{1-2 x}} \]

[Out]

121/(4*Sqrt[1 - 2*x]) + (55*Sqrt[1 - 2*x])/2 - (25*(1 - 2*x)^(3/2))/12

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Rubi [A]  time = 0.0281371, 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 \[ -\frac{25}{12} (1-2 x)^{3/2}+\frac{55}{2} \sqrt{1-2 x}+\frac{121}{4 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

121/(4*Sqrt[1 - 2*x]) + (55*Sqrt[1 - 2*x])/2 - (25*(1 - 2*x)^(3/2))/12

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Rubi in Sympy [A]  time = 5.04754, size = 34, normalized size = 0.85 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12} + \frac{55 \sqrt{- 2 x + 1}}{2} + \frac{121}{4 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(3/2)/12 + 55*sqrt(-2*x + 1)/2 + 121/(4*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0258466, size = 27, normalized size = 0.68 \[ \frac{\sqrt{1-2 x} \left (25 x^2+140 x-167\right )}{6 x-3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-167 + 140*x + 25*x^2))/(-3 + 6*x)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.5 \[ -{\frac{25\,{x}^{2}+140\,x-167}{3}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2/(1-2*x)^(3/2),x)

[Out]

-1/3*(25*x^2+140*x-167)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.34777, size = 38, normalized size = 0.95 \[ -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{55}{2} \, \sqrt{-2 \, x + 1} + \frac{121}{4 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/12*(-2*x + 1)^(3/2) + 55/2*sqrt(-2*x + 1) + 121/4/sqrt(-2*x + 1)

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Fricas [A]  time = 0.241737, size = 26, normalized size = 0.65 \[ -\frac{25 \, x^{2} + 140 \, x - 167}{3 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(25*x^2 + 140*x - 167)/sqrt(-2*x + 1)

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Sympy [A]  time = 2.20332, size = 352, normalized size = 8.8 \[ \begin{cases} \frac{25 \sqrt{55} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{110 \sqrt{55} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{2420 \sqrt{5} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{242 \sqrt{55} i \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{2662 \sqrt{5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{25 \sqrt{55} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{110 \sqrt{55} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{242 \sqrt{55} \sqrt{- 10 x + 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{2420 \sqrt{5} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{2662 \sqrt{5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((25*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33
*sqrt(11)) + 110*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33
*sqrt(11)) - 2420*sqrt(5)*(x + 3/5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 242*
sqrt(55)*I*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) + 2662*sqrt(5)/(
30*sqrt(11)*(x + 3/5) - 33*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (25*sqrt(55)*sqrt
(-10*x + 5)*(x + 3/5)**2/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) + 110*sqrt(55)*sq
rt(-10*x + 5)*(x + 3/5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 242*sqrt(55)*sqr
t(-10*x + 5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 2420*sqrt(5)*(x + 3/5)/(30*
sqrt(11)*(x + 3/5) - 33*sqrt(11)) + 2662*sqrt(5)/(30*sqrt(11)*(x + 3/5) - 33*sqr
t(11)), True))

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GIAC/XCAS [A]  time = 0.211795, size = 38, normalized size = 0.95 \[ -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{55}{2} \, \sqrt{-2 \, x + 1} + \frac{121}{4 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/12*(-2*x + 1)^(3/2) + 55/2*sqrt(-2*x + 1) + 121/4/sqrt(-2*x + 1)