3.2114 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=90 \[ -\frac{75 \sqrt{1-2 x}}{2662 (5 x+3)}-\frac{25 \sqrt{1-2 x}}{242 (5 x+3)^2}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{15 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (25*Sqrt[1 - 2*x])/(242*(3 + 5*x)^2) - (75*Sq
rt[1 - 2*x])/(2662*(3 + 5*x)) - (15*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/1331

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Rubi [A]  time = 0.0799638, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{75 \sqrt{1-2 x}}{2662 (5 x+3)}-\frac{25 \sqrt{1-2 x}}{242 (5 x+3)^2}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{15 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (25*Sqrt[1 - 2*x])/(242*(3 + 5*x)^2) - (75*Sq
rt[1 - 2*x])/(2662*(3 + 5*x)) - (15*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/1331

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Rubi in Sympy [A]  time = 7.76296, size = 76, normalized size = 0.84 \[ - \frac{75 \sqrt{- 2 x + 1}}{2662 \left (5 x + 3\right )} - \frac{25 \sqrt{- 2 x + 1}}{242 \left (5 x + 3\right )^{2}} - \frac{15 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{2}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75*sqrt(-2*x + 1)/(2662*(5*x + 3)) - 25*sqrt(-2*x + 1)/(242*(5*x + 3)**2) - 15*
sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/14641 + 2/(11*sqrt(-2*x + 1)*(5*x + 3
)**2)

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Mathematica [A]  time = 0.113239, size = 58, normalized size = 0.64 \[ \frac{\frac{11 \left (750 x^2+625 x-16\right )}{\sqrt{1-2 x} (5 x+3)^2}-30 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{29282} \]

Antiderivative was successfully verified.

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

[Out]

((11*(-16 + 625*x + 750*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 30*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/29282

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Maple [A]  time = 0.017, size = 57, normalized size = 0.6 \[{\frac{8}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{1000}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{7}{40} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{99}{200}\sqrt{1-2\,x}} \right ) }-{\frac{15\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/1331/(1-2*x)^(1/2)+1000/1331*(7/40*(1-2*x)^(3/2)-99/200*(1-2*x)^(1/2))/(-6-10*
x)^2-15/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49476, size = 112, normalized size = 1.24 \[ \frac{15}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{375 \,{\left (2 \, x - 1\right )}^{2} + 2750 \, x - 407}{1331 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 1/1331*(375*(2*x - 1)^2 + 2750*x - 407)/(25*(-2*x + 1)^(5/2) - 110*(-2*x +
1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.22237, size = 124, normalized size = 1.38 \[ \frac{\sqrt{11}{\left (15 \, \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (750 \, x^{2} + 625 \, x - 16\right )}\right )}}{29282 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/29282*sqrt(11)*(15*sqrt(5)*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(11)*(5
*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(750*x^2 + 625*x - 16
))/((25*x^2 + 30*x + 9)*sqrt(-2*x + 1))

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Sympy [A]  time = 6.83444, size = 233, normalized size = 2.59 \[ \begin{cases} - \frac{15 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{14641} + \frac{15 \sqrt{2}}{2662 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{484 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{\sqrt{2}}{1100 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{15 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{14641} - \frac{15 \sqrt{2} i}{2662 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{484 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} i}{1100 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-15*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/14641 + 15*sqrt(2)/(
2662*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(484*sqrt(-1 + 11/(10
*(x + 3/5)))*(x + 3/5)**(3/2)) - sqrt(2)/(1100*sqrt(-1 + 11/(10*(x + 3/5)))*(x +
3/5)**(5/2)), 11*Abs(1/(x + 3/5))/10 > 1), (15*sqrt(55)*I*asin(sqrt(110)/(10*sq
rt(x + 3/5)))/14641 - 15*sqrt(2)*I/(2662*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/
5)) + sqrt(2)*I/(484*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + sqrt(2)*I/(
1100*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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GIAC/XCAS [A]  time = 0.222277, size = 104, normalized size = 1.16 \[ \frac{15}{29282} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8}{1331 \, \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (35 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 99 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/29282*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 8/1331/sqrt(-2*x + 1) + 5/5324*(35*(-2*x + 1)^(3/2) - 99*sqrt(-2*
x + 1))/(5*x + 3)^2