3.2596 \(\int \frac{1}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{20 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{2 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}} \]

[Out]

(2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (20*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0331464, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{20 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{2 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (20*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi in Sympy [A]  time = 4.18521, size = 39, normalized size = 0.87 \[ \frac{20 \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{2 \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20*sqrt(5*x + 3)/(363*sqrt(-2*x + 1)) + 2*sqrt(5*x + 3)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0254751, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{5 x+3} (20 x-21)}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(-21 + 20*x))/(363*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.5 \[ -{\frac{-42+40\,x}{363}\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(3+5*x)^(1/2)*(-21+20*x)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.4941, size = 65, normalized size = 1.44 \[ \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{20 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) - 20/363*sqrt(-10*x^2 - x + 3)/(2*x
- 1)

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Fricas [A]  time = 0.213241, size = 45, normalized size = 1. \[ -\frac{2 \,{\left (20 \, x - 21\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(20*x - 21)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 8.2289, size = 178, normalized size = 3.96 \[ \begin{cases} \frac{100 \sqrt{10} \left (x + \frac{3}{5}\right )}{3630 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} - \frac{165 \sqrt{10}}{3630 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{100 \sqrt{10} i \left (x + \frac{3}{5}\right )}{3630 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} + \frac{165 \sqrt{10} i}{3630 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((100*sqrt(10)*(x + 3/5)/(3630*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5) -
3993*sqrt(-1 + 11/(10*(x + 3/5)))) - 165*sqrt(10)/(3630*sqrt(-1 + 11/(10*(x + 3
/5)))*(x + 3/5) - 3993*sqrt(-1 + 11/(10*(x + 3/5)))), 11*Abs(1/(x + 3/5))/10 > 1
), (-100*sqrt(10)*I*(x + 3/5)/(3630*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5) - 3993
*sqrt(1 - 11/(10*(x + 3/5)))) + 165*sqrt(10)*I/(3630*sqrt(1 - 11/(10*(x + 3/5)))
*(x + 3/5) - 3993*sqrt(1 - 11/(10*(x + 3/5)))), True))

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GIAC/XCAS [A]  time = 0.245857, size = 53, normalized size = 1.18 \[ -\frac{2 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1815 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1815*(4*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x -
1)^2