3.1828 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{10 \sqrt{1-2 x}}{5 x+3}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)}-138 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+134 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-10*Sqrt[1 - 2*x])/(3 + 5*x) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)) - 138*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 134*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Rubi [A]  time = 0.174733, 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 \[ -\frac{10 \sqrt{1-2 x}}{5 x+3}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)}-138 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+134 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x])/(3 + 5*x) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)) - 138*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 134*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Rubi in Sympy [A]  time = 21.5309, size = 83, normalized size = 0.86 \[ - \frac{10 \sqrt{- 2 x + 1}}{5 x + 3} + \frac{\sqrt{- 2 x + 1}}{\left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{138 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} + \frac{134 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*sqrt(-2*x + 1)/(5*x + 3) + sqrt(-2*x + 1)/((3*x + 2)*(5*x + 3)) - 138*sqrt(2
1)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7 + 134*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x +
1)/11)/11

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Mathematica [A]  time = 0.148007, size = 85, normalized size = 0.88 \[ -\frac{\sqrt{1-2 x} (30 x+19)}{(3 x+2) (5 x+3)}-138 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+134 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - 2*x]*(19 + 30*x))/((2 + 3*x)*(3 + 5*x))) - 138*Sqrt[3/7]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]] + 134*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.017, size = 70, normalized size = 0.7 \[ 2\,{\frac{\sqrt{1-2\,x}}{-4/3-2\,x}}-{\frac{138\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+2\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+{\frac{134\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1-2*x)^(1/2)/(-4/3-2*x)-138/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2*
(1-2*x)^(1/2)/(-6/5-2*x)+134/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51994, size = 149, normalized size = 1.54 \[ -\frac{67}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{69}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 34 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + 69/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 4*(15*(-2*x + 1)^(3/2) - 34*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)

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Fricas [A]  time = 0.222161, size = 188, normalized size = 1.94 \[ \frac{\sqrt{11} \sqrt{7}{\left (67 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 69 \, \sqrt{11} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (30 \, x + 19\right )} \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/77*sqrt(11)*sqrt(7)*(67*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)) + 69*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)) - sqrt(11)
*sqrt(7)*(30*x + 19)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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Sympy [A]  time = 42.2571, size = 321, normalized size = 3.31 \[ - 84 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right ) - 220 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right ) + 408 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) - 680 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-84*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sq
rt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 220*Piecewise((sqrt(55)*
(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4
- 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)
))/605, (x <= 1/2) & (x > -3/5))) + 408*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt
(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2
1, -2*x + 1 < 7/3)) - 680*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)
/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1
< 11/5))

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GIAC/XCAS [A]  time = 0.230463, size = 157, normalized size = 1.62 \[ -\frac{67}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{69}{7} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 34 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 69/7*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) + 4*(15*(-2*x + 1)^(3/2) - 34*sqrt(-2*x + 1))/(15*(2*x - 1)
^2 + 136*x + 9)