3.1898 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac{68 \sqrt{1-2 x}}{3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-68*Sqrt[1 - 2*x])/(3*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)) -
134*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 138*Sqrt[11/5]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.193019, antiderivative size = 102, 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 \[ -\frac{68 \sqrt{1-2 x}}{3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-68*Sqrt[1 - 2*x])/(3*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)) -
134*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 138*Sqrt[11/5]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 22.0718, size = 87, normalized size = 0.85 \[ - \frac{68 \sqrt{- 2 x + 1}}{3 \left (5 x + 3\right )} + \frac{7 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{134 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} + \frac{138 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-68*sqrt(-2*x + 1)/(3*(5*x + 3)) + 7*sqrt(-2*x + 1)/(3*(3*x + 2)*(5*x + 3)) - 13
4*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3 + 138*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/5

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Mathematica [A]  time = 0.157702, size = 85, normalized size = 0.83 \[ -\frac{\sqrt{1-2 x} (68 x+43)}{(3 x+2) (5 x+3)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - 2*x]*(43 + 68*x))/((2 + 3*x)*(3 + 5*x))) - 134*Sqrt[7/3]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]] + 138*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 70, normalized size = 0.7 \[{\frac{14}{3}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{134\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{22}{5}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{138\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/3*(1-2*x)^(1/2)/(-4/3-2*x)-134/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
+22/5*(1-2*x)^(1/2)/(-6/5-2*x)+138/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/
2)

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Maxima [A]  time = 1.50816, size = 149, normalized size = 1.46 \[ -\frac{69}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{67}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (34 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-69/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 67/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 4*(34*(-2*x + 1)^(3/2) - 77*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)

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Fricas [A]  time = 0.235378, size = 188, normalized size = 1.84 \[ \frac{\sqrt{5} \sqrt{3}{\left (69 \, \sqrt{11} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 67 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{5} \sqrt{3}{\left (68 \, x + 43\right )} \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*sqrt(5)*sqrt(3)*(69*sqrt(11)*sqrt(3)*(15*x^2 + 19*x + 6)*log((sqrt(5)*(5*x
- 8) - 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 67*sqrt(7)*sqrt(5)*(15*x^2 + 19*x
+ 6)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(5)*sq
rt(3)*(68*x + 43)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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Sympy [A]  time = 119.229, size = 321, normalized size = 3.15 \[ - 196 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right ) - 484 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right ) + 924 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) - 1540 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-196*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*s
qrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*
sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 484*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))) + 924*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqr
t(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/
21, -2*x + 1 < 7/3)) - 1540*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/1
1)/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 = 0.214257, size = 157, normalized size = 1.54 \[ -\frac{69}{5} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{67}{3} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (34 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-69/5*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 67/3*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 4*(34*(-2*x + 1)^(3/2) - 77*sqrt(-2*x + 1))/(15*(2*x - 1)^
2 + 136*x + 9)