3.2037 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{5 \sqrt{1-2 x}}{11 (5 x+3)}-6 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{64}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-5*Sqrt[1 - 2*x])/(11*(3 + 5*x)) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
+ (64*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.141106, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 \sqrt{1-2 x}}{11 (5 x+3)}-6 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{64}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x])/(11*(3 + 5*x)) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
+ (64*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 14.3255, size = 65, normalized size = 0.84 \[ - \frac{5 \sqrt{- 2 x + 1}}{11 \left (5 x + 3\right )} - \frac{6 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} + \frac{64 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)/(11*(5*x + 3)) - 6*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7
+ 64*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.151109, size = 75, normalized size = 0.97 \[ -\frac{5 \sqrt{1-2 x}}{55 x+33}-6 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{64}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x])/(33 + 55*x) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] +
(64*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.017, size = 54, normalized size = 0.7 \[ -{\frac{6\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{2}{11}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{64\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/11*(1-2*x)^(1/2)/(-6/5-2*x)+
64/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50127, size = 120, normalized size = 1.56 \[ -\frac{32}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{3}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 3/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.22316, size = 161, normalized size = 2.09 \[ \frac{\sqrt{11} \sqrt{7}{\left (32 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 33 \, \sqrt{11} \sqrt{3}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - 5 \, \sqrt{11} \sqrt{7} \sqrt{-2 \, x + 1}\right )}}{847 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/847*sqrt(11)*sqrt(7)*(32*sqrt(7)*sqrt(5)*(5*x + 3)*log((sqrt(11)*(5*x - 8) - 1
1*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 33*sqrt(11)*sqrt(3)*(5*x + 3)*log((sqrt(7
)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - 5*sqrt(11)*sqrt(7)*sqrt(-2*
x + 1))/(5*x + 3)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.21248, size = 128, normalized size = 1.66 \[ -\frac{32}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{3}{7} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 3/7*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)