3.2155 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{27}{20} \sqrt{1-2 x}-\frac{784}{121 \sqrt{1-2 x}}+\frac{343}{132 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

[Out]

343/(132*(1 - 2*x)^(3/2)) - 784/(121*Sqrt[1 - 2*x]) - (27*Sqrt[1 - 2*x])/20 - (2
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.0966134, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{27}{20} \sqrt{1-2 x}-\frac{784}{121 \sqrt{1-2 x}}+\frac{343}{132 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

343/(132*(1 - 2*x)^(3/2)) - 784/(121*Sqrt[1 - 2*x]) - (27*Sqrt[1 - 2*x])/20 - (2
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.5398, size = 60, normalized size = 0.9 \[ - \frac{27 \sqrt{- 2 x + 1}}{20} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{33275} - \frac{784}{121 \sqrt{- 2 x + 1}} + \frac{343}{132 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*sqrt(-2*x + 1)/20 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/33275 - 784
/(121*sqrt(-2*x + 1)) + 343/(132*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.120755, size = 51, normalized size = 0.76 \[ \frac{-\frac{55 \left (9801 x^2-33321 x+9494\right )}{(1-2 x)^{3/2}}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{99825} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(9494 - 33321*x + 9801*x^2))/(1 - 2*x)^(3/2) - 6*Sqrt[55]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/99825

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 47, normalized size = 0.7 \[{\frac{343}{132} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{33275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{784}{121}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{27}{20}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

343/132/(1-2*x)^(3/2)-2/33275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-784/
121/(1-2*x)^(1/2)-27/20*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50606, size = 81, normalized size = 1.21 \[ \frac{1}{33275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{20} \, \sqrt{-2 \, x + 1} + \frac{49 \,{\left (384 \, x - 115\right )}}{1452 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/33275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 27/20*sqrt(-2*x + 1) + 49/1452*(384*x - 115)/(-2*x + 1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.22711, size = 103, normalized size = 1.54 \[ \frac{\sqrt{55}{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (9801 \, x^{2} - 33321 \, x + 9494\right )}\right )}}{99825 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/99825*sqrt(55)*(3*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-
2*x + 1))/(5*x + 3)) + sqrt(55)*(9801*x^2 - 33321*x + 9494))/((2*x - 1)*sqrt(-2*
x + 1))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221199, size = 95, normalized size = 1.42 \[ \frac{1}{33275} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{27}{20} \, \sqrt{-2 \, x + 1} - \frac{49 \,{\left (384 \, x - 115\right )}}{1452 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/33275*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 27/20*sqrt(-2*x + 1) - 49/1452*(384*x - 115)/((2*x - 1)*sqrt(-2*x
+ 1))