3.2060 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]

[Out]

11/(7*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.0502511, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

11/(7*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.47729, size = 36, normalized size = 0.88 \[ \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{147} + \frac{11}{7 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/147 + 11/(7*sqrt(-2*x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0626907, size = 41, normalized size = 1. \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

11/(7*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 29, normalized size = 0.7 \[{\frac{2\,\sqrt{21}}{147}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{11}{7}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/147*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7/(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50929, size = 62, normalized size = 1.51 \[ -\frac{1}{147} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{11}{7 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/147*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) + 11/7/sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.229048, size = 73, normalized size = 1.78 \[ \frac{\sqrt{21}{\left (\sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 11 \, \sqrt{21}\right )}}{147 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/147*sqrt(21)*(sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x
+ 2)) + 11*sqrt(21))/sqrt(-2*x + 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x + 3}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((5*x + 3)/((-2*x + 1)**(3/2)*(3*x + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231853, size = 66, normalized size = 1.61 \[ -\frac{1}{147} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{11}{7 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/147*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 11/7/sqrt(-2*x + 1)