3.2177 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=108 \[ -\frac{2873 \sqrt{1-2 x}}{29282 (5 x+3)}+\frac{2873}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{614}{1815 \sqrt{1-2 x} (5 x+3)^2}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641 \sqrt{55}} \]

[Out]

49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 614/(1815*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 287
3/(19965*Sqrt[1 - 2*x]*(3 + 5*x)) - (2873*Sqrt[1 - 2*x])/(29282*(3 + 5*x)) - (28
73*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(14641*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.136952, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2873 \sqrt{1-2 x}}{29282 (5 x+3)}+\frac{2873}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{614}{1815 \sqrt{1-2 x} (5 x+3)^2}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 614/(1815*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 287
3/(19965*Sqrt[1 - 2*x]*(3 + 5*x)) - (2873*Sqrt[1 - 2*x])/(29282*(3 + 5*x)) - (28
73*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(14641*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.318, size = 83, normalized size = 0.77 \[ - \frac{2873 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{805255} + \frac{2873}{73205 \sqrt{- 2 x + 1}} + \frac{2873}{99825 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{139}{6050 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} - \frac{1}{550 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2873*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/805255 + 2873/(73205*sqrt(-2*x
+ 1)) + 2873/(99825*(-2*x + 1)**(3/2)) - 139/(6050*(-2*x + 1)**(3/2)*(5*x + 3))
- 1/(550*(-2*x + 1)**(3/2)*(5*x + 3)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.113802, size = 66, normalized size = 0.61 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (172380 x^3+57460 x^2-107127 x-47568\right )}{\left (10 x^2+x-3\right )^2}-17238 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4831530} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(-47568 - 107127*x + 57460*x^2 + 172380*x^3))/(-3 + x + 10*x
^2)^2 - 17238*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/4831530

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 66, normalized size = 0.6 \[{\frac{98}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{546}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{143}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{319}{10}\sqrt{1-2\,x}} \right ) }-{\frac{2873\,\sqrt{55}}{805255}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

98/3993/(1-2*x)^(3/2)+546/14641/(1-2*x)^(1/2)+50/14641*(143/10*(1-2*x)^(3/2)-319
/10*(1-2*x)^(1/2))/(-6-10*x)^2-2873/805255*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*
55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50409, size = 124, normalized size = 1.15 \[ \frac{2873}{1610510} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{43095 \,{\left (2 \, x - 1\right )}^{3} + 158015 \,{\left (2 \, x - 1\right )}^{2} + 159236 \, x - 210056}{43923 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2873/1610510*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 1/43923*(43095*(2*x - 1)^3 + 158015*(2*x - 1)^2 + 159236*x - 210056)/(
25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.212277, size = 136, normalized size = 1.26 \[ \frac{\sqrt{55}{\left (8619 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (172380 \, x^{3} + 57460 \, x^{2} - 107127 \, x - 47568\right )}\right )}}{4831530 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4831530*sqrt(55)*(8619*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log((sqrt(5
5)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(172380*x^3 + 57460*x^2
- 107127*x - 47568))/((50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1))

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229237, size = 120, normalized size = 1.11 \[ \frac{2873}{1610510} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{28 \,{\left (117 \, x - 97\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (13 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 29 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2873/1610510*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) + 28/43923*(117*x - 97)/((2*x - 1)*sqrt(-2*x + 1)) + 5/5324*(13
*(-2*x + 1)^(3/2) - 29*sqrt(-2*x + 1))/(5*x + 3)^2