3.2026 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=77 \[ \frac{3 \sqrt{1-2 x}}{7 (3 x+2)}+\frac{72}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (72*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/7 - 10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.142999, 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 \[ \frac{3 \sqrt{1-2 x}}{7 (3 x+2)}+\frac{72}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (72*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/7 - 10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 14.2518, size = 65, normalized size = 0.84 \[ \frac{3 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )} + \frac{72 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{10 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(-2*x + 1)/(7*(3*x + 2)) + 72*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49
- 10*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.137786, size = 77, normalized size = 1. \[ \frac{3 \sqrt{1-2 x}}{7 (3 x+2)}+\frac{72}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (72*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/7 - 10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.016, size = 54, normalized size = 0.7 \[ -{\frac{2}{7}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{72\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{10\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*(1-2*x)^(1/2)/(-4/3-2*x)+72/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
-10/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51472, size = 120, normalized size = 1.56 \[ \frac{5}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{36}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{3 \, \sqrt{-2 \, x + 1}}{7 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 36/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 3/7*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.221918, size = 161, normalized size = 2.09 \[ \frac{\sqrt{11} \sqrt{7}{\left (35 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 36 \, \sqrt{11} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 3 \, \sqrt{11} \sqrt{7} \sqrt{-2 \, x + 1}\right )}}{539 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/539*sqrt(11)*sqrt(7)*(35*sqrt(7)*sqrt(5)*(3*x + 2)*log((sqrt(11)*(5*x - 8) + 1
1*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 36*sqrt(11)*sqrt(3)*(3*x + 2)*log((sqrt(7
)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + 3*sqrt(11)*sqrt(7)*sqrt(-2*
x + 1))/(3*x + 2)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.216304, size = 128, normalized size = 1.66 \[ \frac{5}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{36}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{3 \, \sqrt{-2 \, x + 1}}{7 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 36/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 3/7*sqrt(-2*x + 1)/(3*x + 2)