3.1978 \(\int \frac{(1-2 x)^{5/2}}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=83 \[ -\frac{(1-2 x)^{5/2}}{10 (5 x+3)^2}+\frac{(1-2 x)^{3/2}}{10 (5 x+3)}+\frac{3}{25} \sqrt{1-2 x}-\frac{3}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(3*Sqrt[1 - 2*x])/25 - (1 - 2*x)^(5/2)/(10*(3 + 5*x)^2) + (1 - 2*x)^(3/2)/(10*(3
+ 5*x)) - (3*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.0737343, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{(1-2 x)^{5/2}}{10 (5 x+3)^2}+\frac{(1-2 x)^{3/2}}{10 (5 x+3)}+\frac{3}{25} \sqrt{1-2 x}-\frac{3}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x])/25 - (1 - 2*x)^(5/2)/(10*(3 + 5*x)^2) + (1 - 2*x)^(3/2)/(10*(3
+ 5*x)) - (3*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi in Sympy [A]  time = 8.19581, size = 66, normalized size = 0.8 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{10 \left (5 x + 3\right )^{2}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{10 \left (5 x + 3\right )} + \frac{3 \sqrt{- 2 x + 1}}{25} - \frac{3 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/(10*(5*x + 3)**2) + (-2*x + 1)**(3/2)/(10*(5*x + 3)) + 3*sqrt
(-2*x + 1)/25 - 3*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.0858566, size = 58, normalized size = 0.7 \[ \frac{1}{250} \left (\frac{5 \sqrt{1-2 x} \left (80 x^2+195 x+64\right )}{(5 x+3)^2}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(64 + 195*x + 80*x^2))/(3 + 5*x)^2 - 6*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/250

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Maple [A]  time = 0.015, size = 57, normalized size = 0.7 \[{\frac{8}{125}\sqrt{1-2\,x}}+{\frac{88}{5\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{9}{40} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{77}{200}\sqrt{1-2\,x}} \right ) }-{\frac{3\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(3+5*x)^3,x)

[Out]

8/125*(1-2*x)^(1/2)+88/5*(-9/40*(1-2*x)^(3/2)+77/200*(1-2*x)^(1/2))/(-6-10*x)^2-
3/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48288, size = 112, normalized size = 1.35 \[ \frac{3}{250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8}{125} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{125 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 8/125*sqrt(-2*x + 1) - 11/125*(45*(-2*x + 1)^(3/2) - 77*sqrt(-2*x + 1))/(25*(
2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.213657, size = 115, normalized size = 1.39 \[ \frac{\sqrt{5}{\left (3 \, \sqrt{11}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (80 \, x^{2} + 195 \, x + 64\right )} \sqrt{-2 \, x + 1}\right )}}{250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/250*sqrt(5)*(3*sqrt(11)*(25*x^2 + 30*x + 9)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11
)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(80*x^2 + 195*x + 64)*sqrt(-2*x + 1))/(25
*x^2 + 30*x + 9)

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Sympy [A]  time = 5.67293, size = 299, normalized size = 3.6 \[ \begin{cases} - \frac{3 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} - \frac{8 \sqrt{2} \sqrt{x + \frac{3}{5}}}{125 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} - \frac{11 \sqrt{2}}{1250 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{1331 \sqrt{2}}{12500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{1331 \sqrt{2}}{62500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{3 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} + \frac{8 \sqrt{2} i \sqrt{x + \frac{3}{5}}}{125 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} + \frac{11 \sqrt{2} i}{1250 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{1331 \sqrt{2} i}{12500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{1331 \sqrt{2} i}{62500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/125 - 8*sqrt(2)*sqrt(
x + 3/5)/(125*sqrt(-1 + 11/(10*(x + 3/5)))) - 11*sqrt(2)/(1250*sqrt(-1 + 11/(10*
(x + 3/5)))*sqrt(x + 3/5)) + 1331*sqrt(2)/(12500*sqrt(-1 + 11/(10*(x + 3/5)))*(x
+ 3/5)**(3/2)) - 1331*sqrt(2)/(62500*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(5
/2)), 11*Abs(1/(x + 3/5))/10 > 1), (3*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5
)))/125 + 8*sqrt(2)*I*sqrt(x + 3/5)/(125*sqrt(1 - 11/(10*(x + 3/5)))) + 11*sqrt(
2)*I/(1250*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - 1331*sqrt(2)*I/(12500*sq
rt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + 1331*sqrt(2)*I/(62500*sqrt(1 - 11/
(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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GIAC/XCAS [A]  time = 0.21234, size = 104, normalized size = 1.25 \[ \frac{3}{250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8}{125} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 8/125*sqrt(-2*x + 1) - 11/500*(45*(-2*x + 1)^(3/2) - 77*sqrt(-2*x +
1))/(5*x + 3)^2