3.1897 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{11 \sqrt{1-2 x}}{5 (5 x+3)}-14 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{72}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-11*Sqrt[1 - 2*x])/(5*(3 + 5*x)) - 14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
] + (72*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi [A]  time = 0.126999, 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 \[ -\frac{11 \sqrt{1-2 x}}{5 (5 x+3)}-14 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{72}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-11*Sqrt[1 - 2*x])/(5*(3 + 5*x)) - 14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
] + (72*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi in Sympy [A]  time = 15.3008, size = 65, normalized size = 0.84 \[ - \frac{11 \sqrt{- 2 x + 1}}{5 \left (5 x + 3\right )} - \frac{14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} + \frac{72 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*sqrt(-2*x + 1)/(5*(5*x + 3)) - 14*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/
3 + 72*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/25

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Mathematica [A]  time = 0.185232, size = 76, normalized size = 0.99 \[ \frac{1}{25} \left (72 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{55 \sqrt{1-2 x}}{5 x+3}\right )-14 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + ((-55*Sqrt[1 - 2*x])/(3 + 5*x)
+ 72*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Maple [A]  time = 0.016, size = 54, normalized size = 0.7 \[ -{\frac{14\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{22}{25}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{72\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+22/25*(1-2*x)^(1/2)/(-6/5-2*x
)+72/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.58289, size = 120, normalized size = 1.56 \[ -\frac{36}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11 \, \sqrt{-2 \, x + 1}}{5 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + 7/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 11/5*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.234363, size = 161, normalized size = 2.09 \[ \frac{\sqrt{5} \sqrt{3}{\left (36 \, \sqrt{11} \sqrt{3}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 35 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - 11 \, \sqrt{5} \sqrt{3} \sqrt{-2 \, x + 1}\right )}}{75 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/75*sqrt(5)*sqrt(3)*(36*sqrt(11)*sqrt(3)*(5*x + 3)*log((sqrt(5)*(5*x - 8) - 5*s
qrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 35*sqrt(7)*sqrt(5)*(5*x + 3)*log((sqrt(3)*(
3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) - 11*sqrt(5)*sqrt(3)*sqrt(-2*x +
1))/(5*x + 3)

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Sympy [A]  time = 57.9494, size = 226, normalized size = 2.94 \[ - \frac{484 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{5} + 98 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) - \frac{814 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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(5
5)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5)))/5 + 98*Piecewise((-sq
rt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sq
rt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3)) - 814*Piecewise((-sqrt(55)*acoth(s
qrt(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))/5

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GIAC/XCAS [A]  time = 0.213204, size = 128, normalized size = 1.66 \[ -\frac{36}{25} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7}{3} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{11 \, \sqrt{-2 \, x + 1}}{5 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36/25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 7/3*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) - 11/5*sqrt(-2*x + 1)/(5*x + 3)