3.2027 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=97 \[ \frac{219 \sqrt{1-2 x}}{98 (3 x+2)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (219*Sqrt[1 - 2*x])/(98*(2 + 3*x)) + (2523*
Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - 50*Sqrt[5/11]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.201161, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{219 \sqrt{1-2 x}}{98 (3 x+2)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (219*Sqrt[1 - 2*x])/(98*(2 + 3*x)) + (2523*
Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - 50*Sqrt[5/11]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.8522, size = 83, normalized size = 0.86 \[ \frac{219 \sqrt{- 2 x + 1}}{98 \left (3 x + 2\right )} + \frac{3 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right )^{2}} + \frac{2523 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{50 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

219*sqrt(-2*x + 1)/(98*(3*x + 2)) + 3*sqrt(-2*x + 1)/(14*(3*x + 2)**2) + 2523*sq
rt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 50*sqrt(55)*atanh(sqrt(55)*sqrt(-2
*x + 1)/11)/11

_______________________________________________________________________________________

Mathematica [A]  time = 0.154241, size = 82, normalized size = 0.85 \[ \frac{9 \sqrt{1-2 x} (73 x+51)}{98 (3 x+2)^2}+\frac{2523}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(9*Sqrt[1 - 2*x]*(51 + 73*x))/(98*(2 + 3*x)^2) + (2523*Sqrt[3/7]*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/49 - 50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 66, normalized size = 0.7 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{73\, \left ( 1-2\,x \right ) ^{3/2}}{882}}-{\frac{25\,\sqrt{1-2\,x}}{126}} \right ) }+{\frac{2523\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{50\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(73/882*(1-2*x)^(3/2)-25/126*(1-2*x)^(1/2))/(-4-6*x)^2+2523/343*arctanh(1/7
*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^
(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.48074, size = 149, normalized size = 1.54 \[ \frac{25}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2523}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9 \,{\left (73 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 175 \, \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 2523/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 9/49*(73*(-2*x + 1)^(3/2) - 175*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x +
7)

_______________________________________________________________________________________

Fricas [A]  time = 0.222315, size = 188, normalized size = 1.94 \[ \frac{\sqrt{11} \sqrt{7}{\left (2450 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2523 \, \sqrt{11} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 9 \, \sqrt{11} \sqrt{7}{\left (73 \, x + 51\right )} \sqrt{-2 \, x + 1}\right )}}{7546 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7546*sqrt(11)*sqrt(7)*(2450*sqrt(7)*sqrt(5)*(9*x^2 + 12*x + 4)*log((sqrt(11)*(
5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 2523*sqrt(11)*sqrt(3)*(9*x^2
+ 12*x + 4)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + 9*sq
rt(11)*sqrt(7)*(73*x + 51)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216804, size = 144, normalized size = 1.48 \[ \frac{25}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2523}{686} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9 \,{\left (73 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 175 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 2523/686*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) - 9/196*(73*(-2*x + 1)^(3/2) - 175*sqrt(-2*x + 1))/(3*x
+ 2)^2