3.2495 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0325141, 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 \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*x])

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Rubi in Sympy [A]  time = 4.3171, size = 41, normalized size = 0.91 \[ - \frac{8 \sqrt{- 2 x + 1}}{363 \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1}}{33 \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*sqrt(-2*x + 1)/(363*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)/(33*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.0230932, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (20 x+23)}{363 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(23 + 20*x))/(363*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.5 \[ -{\frac{46+40\,x}{363}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(23+20*x)/(3+5*x)^(3/2)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49888, size = 65, normalized size = 1.44 \[ -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{8 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/33*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 8/363*sqrt(-10*x^2 - x + 3)/(5
*x + 3)

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Fricas [A]  time = 0.214481, size = 45, normalized size = 1. \[ -\frac{2 \,{\left (20 \, x + 23\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(20*x + 23)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 8.11816, size = 104, normalized size = 2.31 \[ \begin{cases} - \frac{8 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{8 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \left (x + \frac{3}{5}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-8*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*sqrt(-1 +
11/(10*(x + 3/5)))/(825*(x + 3/5)), 11*Abs(1/(x + 3/5))/10 > 1), (-8*sqrt(10)*I*
sqrt(1 - 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(825
*(x + 3/5)), True))

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GIAC/XCAS [A]  time = 0.233517, size = 170, normalized size = 3.78 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{29040 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{3 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2420 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{9 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1815 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/29040*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 3/242
0*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/1815*(9*sqrt(1
0)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2
)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3