3.2093 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{11 \sqrt{1-2 x}}-\frac{2}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

2/(11*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

_______________________________________________________________________________________

Rubi [A]  time = 0.0397083, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2}{11 \sqrt{1-2 x}}-\frac{2}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2/(11*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.30472, size = 36, normalized size = 0.84 \[ - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} + \frac{2}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121 + 2/(11*sqrt(-2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0557951, size = 41, normalized size = 0.95 \[ \frac{1}{121} \left (\frac{22}{\sqrt{1-2 x}}-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(22/Sqrt[1 - 2*x] - 2*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 29, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{11}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(3/2)/(3+5*x),x)

[Out]

-2/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/11/(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50677, size = 62, normalized size = 1.44 \[ \frac{1}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{11 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 2/11/sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.231363, size = 81, normalized size = 1.88 \[ \frac{\sqrt{11}{\left (\sqrt{5} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2 \, \sqrt{11}\right )}}{121 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/121*sqrt(11)*(sqrt(5)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt
(-2*x + 1))/(5*x + 3)) + 2*sqrt(11))/sqrt(-2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 2.68139, size = 830, normalized size = 19.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((20*sqrt(5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(110*sqrt(1
1)*(x + 3/5) - 121*sqrt(11)) - 10*sqrt(5)*(x + 3/5)*log(110)/(110*sqrt(11)*(x +
3/5) - 121*sqrt(11)) - 10*sqrt(5)*(x + 3/5)*log(11)/(110*sqrt(11)*(x + 3/5) - 12
1*sqrt(11)) - 20*sqrt(5)*(x + 3/5)*log(2)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)
) + 10*sqrt(5)*(x + 3/5)*log(10)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 20*sq
rt(5)*(x + 3/5)*log(22)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 2*sqrt(55)*I*s
qrt(10*x - 5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 22*sqrt(5)*I*asin(sqrt(1
10)/(10*sqrt(x + 3/5)))/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 22*sqrt(5)*log
(22)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 11*sqrt(5)*log(10)/(110*sqrt(11)*
(x + 3/5) - 121*sqrt(11)) + 22*sqrt(5)*log(2)/(110*sqrt(11)*(x + 3/5) - 121*sqrt
(11)) + 11*sqrt(5)*log(11)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 11*sqrt(5)*
log(110)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (-2*s
qrt(55)*sqrt(-10*x + 5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqrt(5)*(x
+ 3/5)*log(x + 3/5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 20*sqrt(5)*(x + 3/
5)*log(sqrt(-10*x/11 + 5/11) + 1)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 10*s
qrt(5)*(x + 3/5)*log(11)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqrt(5)*(x
+ 3/5)*log(10)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqrt(5)*I*pi*(x + 3
/5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 11*sqrt(5)*log(x + 3/5)/(110*sqrt(
11)*(x + 3/5) - 121*sqrt(11)) + 22*sqrt(5)*log(sqrt(-10*x/11 + 5/11) + 1)/(110*s
qrt(11)*(x + 3/5) - 121*sqrt(11)) - 11*sqrt(5)*log(10)/(110*sqrt(11)*(x + 3/5) -
121*sqrt(11)) + 11*sqrt(5)*log(11)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 11
*sqrt(5)*I*pi/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217001, size = 66, normalized size = 1.53 \[ \frac{1}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{11 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 2/11/sqrt(-2*x + 1)