3.2039 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{35845 \sqrt{1-2 x}}{1078 (5 x+3)}+\frac{162 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-35845*Sqrt[1 - 2*x])/(1078*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 +
5*x)) + (162*Sqrt[1 - 2*x])/(49*(2 + 3*x)*(3 + 5*x)) - (22479*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/11

_______________________________________________________________________________________

Rubi [A]  time = 0.262432, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{35845 \sqrt{1-2 x}}{1078 (5 x+3)}+\frac{162 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-35845*Sqrt[1 - 2*x])/(1078*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 +
5*x)) + (162*Sqrt[1 - 2*x])/(49*(2 + 3*x)*(3 + 5*x)) - (22479*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.8159, size = 107, normalized size = 0.8 \[ - \frac{21507 \sqrt{- 2 x + 1}}{1078 \left (3 x + 2\right )} - \frac{309 \sqrt{- 2 x + 1}}{154 \left (3 x + 2\right )^{2}} - \frac{5 \sqrt{- 2 x + 1}}{11 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{22479 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} + \frac{4900 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-21507*sqrt(-2*x + 1)/(1078*(3*x + 2)) - 309*sqrt(-2*x + 1)/(154*(3*x + 2)**2) -
5*sqrt(-2*x + 1)/(11*(3*x + 2)**2*(5*x + 3)) - 22479*sqrt(21)*atanh(sqrt(21)*sq
rt(-2*x + 1)/7)/343 + 4900*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121

_______________________________________________________________________________________

Mathematica [A]  time = 0.190927, size = 96, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (322605 x^2+419448 x+136021\right )}{1078 (3 x+2)^2 (5 x+3)}-\frac{22479}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{4900}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(136021 + 419448*x + 322605*x^2))/(1078*(2 + 3*x)^2*(3 + 5*x)) -
(22479*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqrt[5/11]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 82, normalized size = 0.6 \[ 486\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{143\, \left ( 1-2\,x \right ) ^{3/2}}{882}}-{\frac{145\,\sqrt{1-2\,x}}{378}} \right ) }-{\frac{22479\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{50}{11}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{4900\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

486*(143/882*(1-2*x)^(3/2)-145/378*(1-2*x)^(1/2))/(-4-6*x)^2-22479/343*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+50/11*(1-2*x)^(1/2)/(-6/5-2*x)+4900/121*arct
anh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49444, size = 173, normalized size = 1.3 \[ -\frac{2450}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22479}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{322605 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 1484106 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1705585 \, \sqrt{-2 \, x + 1}}{539 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2450/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 22479/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 1/539*(322605*(-2*x + 1)^(5/2) - 1484106*(-2*x + 1)^(3/2) + 170558
5*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

_______________________________________________________________________________________

Fricas [A]  time = 0.220337, size = 215, normalized size = 1.62 \[ \frac{\sqrt{11} \sqrt{7}{\left (240100 \, \sqrt{7} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 247269 \, \sqrt{11} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (322605 \, x^{2} + 419448 \, x + 136021\right )} \sqrt{-2 \, x + 1}\right )}}{83006 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/83006*sqrt(11)*sqrt(7)*(240100*sqrt(7)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12)*l
og((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 247269*sqrt(11)
*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-
2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(322605*x^2 + 419448*x + 136021)*sqrt(-2
*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

_______________________________________________________________________________________

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)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220785, size = 166, normalized size = 1.25 \[ -\frac{2450}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22479}{686} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{125 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} + \frac{9 \,{\left (429 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1015 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2450/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) + 22479/686*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 125/11*sqrt(-2*x + 1)/(5*x + 3) + 9/196*(429*(-2*
x + 1)^(3/2) - 1015*sqrt(-2*x + 1))/(3*x + 2)^2