3.2038 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=106 \[ -\frac{340 \sqrt{1-2 x}}{77 (5 x+3)}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-340*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x))
- (426*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.201302, antiderivative size = 106, 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 \[ -\frac{340 \sqrt{1-2 x}}{77 (5 x+3)}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-340*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x))
- (426*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 20.8829, size = 87, normalized size = 0.82 \[ - \frac{204 \sqrt{- 2 x + 1}}{77 \left (3 x + 2\right )} - \frac{5 \sqrt{- 2 x + 1}}{11 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{426 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} + \frac{650 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-204*sqrt(-2*x + 1)/(77*(3*x + 2)) - 5*sqrt(-2*x + 1)/(11*(3*x + 2)*(5*x + 3)) -
426*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 + 650*sqrt(55)*atanh(sqrt(55)*
sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.177719, size = 91, normalized size = 0.86 \[ -\frac{\sqrt{1-2 x} (1020 x+647)}{77 (3 x+2) (5 x+3)}-\frac{426}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{650}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(647 + 1020*x))/(77*(2 + 3*x)*(3 + 5*x)) - (426*Sqrt[3/7]*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
])/11

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Maple [A]  time = 0.019, size = 70, normalized size = 0.7 \[{\frac{6}{7}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{426\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{10}{11}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{650\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/7*(1-2*x)^(1/2)/(-4/3-2*x)-426/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
+10/11*(1-2*x)^(1/2)/(-6/5-2*x)+650/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^
(1/2)

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Maxima [A]  time = 1.51178, size = 149, normalized size = 1.41 \[ -\frac{325}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{213}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1157 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-325/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 213/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 1157*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 13
6*x + 9)

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Fricas [A]  time = 0.225291, size = 188, normalized size = 1.77 \[ \frac{\sqrt{11} \sqrt{7}{\left (2275 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2343 \, \sqrt{11} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (1020 \, x + 647\right )} \sqrt{-2 \, x + 1}\right )}}{5929 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5929*sqrt(11)*sqrt(7)*(2275*sqrt(7)*sqrt(5)*(15*x^2 + 19*x + 6)*log((sqrt(11)*
(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 2343*sqrt(11)*sqrt(3)*(15*x^
2 + 19*x + 6)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sq
rt(11)*sqrt(7)*(1020*x + 647)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215047, size = 157, normalized size = 1.48 \[ -\frac{325}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{213}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1157 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-325/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 213/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 1157*sqrt(-2*x + 1))/(15
*(2*x - 1)^2 + 136*x + 9)