3.2104 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{15 \sqrt{1-2 x}}{121 (5 x+3)}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (15*Sqrt[1 - 2*x])/(121*(3 + 5*x)) - (6*Sqrt[5/
11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0609609, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{15 \sqrt{1-2 x}}{121 (5 x+3)}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (15*Sqrt[1 - 2*x])/(121*(3 + 5*x)) - (6*Sqrt[5/
11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 5.93499, size = 56, normalized size = 0.8 \[ - \frac{15 \sqrt{- 2 x + 1}}{121 \left (5 x + 3\right )} - \frac{6 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{2}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(-2*x + 1)/(121*(5*x + 3)) - 6*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11
)/1331 + 2/(11*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.0863881, size = 56, normalized size = 0.8 \[ \frac{-\frac{11 \sqrt{1-2 x} (30 x+7)}{10 x^2+x-3}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

((-11*Sqrt[1 - 2*x]*(7 + 30*x))/(-3 + x + 10*x^2) - 6*Sqrt[55]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/1331

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Maple [A]  time = 0.015, size = 45, normalized size = 0.6 \[{\frac{4}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{121}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{6\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/121/(1-2*x)^(1/2)+2/121*(1-2*x)^(1/2)/(-6/5-2*x)-6/1331*arctanh(1/11*55^(1/2)*
(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52113, size = 88, normalized size = 1.26 \[ \frac{3}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 2/121*(30*x + 7)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.250521, size = 104, normalized size = 1.49 \[ \frac{\sqrt{11}{\left (3 \, \sqrt{5}{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (30 \, x + 7\right )}\right )}}{1331 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1331*sqrt(11)*(3*sqrt(5)*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11
*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(30*x + 7))/((5*x + 3)*sqrt(-2*x
+ 1))

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Sympy [A]  time = 4.32505, size = 177, normalized size = 2.53 \[ \begin{cases} - \frac{6 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} + \frac{3 \sqrt{2}}{121 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{110 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{6 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} - \frac{3 \sqrt{2} i}{121 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{110 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/1331 + 3*sqrt(2)/(121
*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(110*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(3/2)), 11*Abs(1/(x + 3/5))/10 > 1), (6*sqrt(55)*I*asin(sqrt
(110)/(10*sqrt(x + 3/5)))/1331 - 3*sqrt(2)*I/(121*sqrt(1 - 11/(10*(x + 3/5)))*sq
rt(x + 3/5)) + sqrt(2)*I/(110*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), Tru
e))

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GIAC/XCAS [A]  time = 0.217487, size = 92, normalized size = 1.31 \[ \frac{3}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 2/121*(30*x + 7)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))