3.2606 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{400 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}+\frac{40}{363 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{2}{33 \sqrt{5 x+3} (1-2 x)^{3/2}} \]

[Out]

2/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 40/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (4
00*Sqrt[1 - 2*x])/(3993*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.052833, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{400 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}+\frac{40}{363 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{2}{33 \sqrt{5 x+3} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

2/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 40/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (4
00*Sqrt[1 - 2*x])/(3993*Sqrt[3 + 5*x])

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Rubi in Sympy [A]  time = 5.77613, size = 60, normalized size = 0.9 \[ - \frac{400 \sqrt{- 2 x + 1}}{3993 \sqrt{5 x + 3}} + \frac{40}{363 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{2}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-400*sqrt(-2*x + 1)/(3993*sqrt(5*x + 3)) + 40/(363*sqrt(-2*x + 1)*sqrt(5*x + 3))
+ 2/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.0332568, size = 32, normalized size = 0.48 \[ \frac{-1600 x^2+720 x+282}{3993 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(282 + 720*x - 1600*x^2)/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

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Maple [A]  time = 0.005, size = 27, normalized size = 0.4 \[ -{\frac{1600\,{x}^{2}-720\,x-282}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(800*x^2-360*x-141)/(3+5*x)^(1/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.34233, size = 86, normalized size = 1.28 \[ \frac{800 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{40}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{33 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

800/3993*x/sqrt(-10*x^2 - x + 3) + 40/3993/sqrt(-10*x^2 - x + 3) - 2/33/(2*sqrt(
-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.219621, size = 58, normalized size = 0.87 \[ -\frac{2 \,{\left (800 \, x^{2} - 360 \, x - 141\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(800*x^2 - 360*x - 141)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7
*x + 3)

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Sympy [A]  time = 39.9921, size = 231, normalized size = 3.45 \[ \begin{cases} - \frac{8000 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} + \frac{13200 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} - \frac{3630 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{8000 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} + \frac{13200 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} - \frac{3630 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-8000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x +
399300*(x + 3/5)**2 - 43923) + 13200*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x +
3/5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(10)*sqrt(-1 + 11/(10
*(x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), 11*Abs(1/(x + 3/5))/10 >
1), (-8000*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x + 399
300*(x + 3/5)**2 - 43923) + 13200*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/
5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(10)*I*sqrt(1 - 11/(10*(
x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), True))

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GIAC/XCAS [A]  time = 0.232218, size = 135, normalized size = 2.01 \[ -\frac{5 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2662 \, \sqrt{5 \, x + 3}} - \frac{8 \,{\left (5 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{19965 \,{\left (2 \, x - 1\right )}^{2}} + \frac{10 \, \sqrt{10} \sqrt{5 \, x + 3}}{1331 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2662*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/19965*(5
*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 10/
1331*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))