3.1957 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=92 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2)}+\frac{26}{15} \sqrt{1-2 x}+\frac{140}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(26*Sqrt[1 - 2*x])/15 + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)) + (140*Sqrt[7/3]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/5

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Rubi [A]  time = 0.189801, antiderivative size = 92, 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 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2)}+\frac{26}{15} \sqrt{1-2 x}+\frac{140}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(26*Sqrt[1 - 2*x])/15 + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)) + (140*Sqrt[7/3]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/5

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Rubi in Sympy [A]  time = 21.6008, size = 76, normalized size = 0.83 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )} + \frac{26 \sqrt{- 2 x + 1}}{15} + \frac{140 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(3*(3*x + 2)) + 26*sqrt(-2*x + 1)/15 + 140*sqrt(21)*atanh(sq
rt(21)*sqrt(-2*x + 1)/7)/9 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/25

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Mathematica [A]  time = 0.159934, size = 78, normalized size = 0.85 \[ \frac{1}{225} \left (\frac{15 \sqrt{1-2 x} (8 x+87)}{3 x+2}+3500 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2178 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((15*Sqrt[1 - 2*x]*(87 + 8*x))/(2 + 3*x) + 3500*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]] - 2178*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/225

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Maple [A]  time = 0.017, size = 63, normalized size = 0.7 \[{\frac{8}{45}\sqrt{1-2\,x}}-{\frac{98}{27}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{140\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{242\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/45*(1-2*x)^(1/2)-98/27*(1-2*x)^(1/2)/(-4/3-2*x)+140/9*arctanh(1/7*21^(1/2)*(1-
2*x)^(1/2))*21^(1/2)-242/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50163, size = 132, normalized size = 1.43 \[ \frac{121}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{70}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8}{45} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 70/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 8/45*sqrt(-2*x + 1) + 49/9*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.219872, size = 166, normalized size = 1.8 \[ \frac{\sqrt{5} \sqrt{3}{\left (363 \, \sqrt{11} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 350 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{5} \sqrt{3}{\left (8 \, x + 87\right )} \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/225*sqrt(5)*sqrt(3)*(363*sqrt(11)*sqrt(3)*(3*x + 2)*log((sqrt(5)*(5*x - 8) + 5
*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 350*sqrt(7)*sqrt(5)*(3*x + 2)*log((sqrt(3
)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(5)*sqrt(3)*(8*x + 87)*
sqrt(-2*x + 1))/(3*x + 2)

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Sympy [A]  time = 99.0665, size = 240, normalized size = 2.61 \[ \frac{8 \sqrt{- 2 x + 1}}{45} + \frac{1372 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{2842 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*sqrt(-2*x + 1)/45 + 1372*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 -
1)/4 + log(sqrt(21)*sqrt(-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)))/9 -
2842*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)/21, -2*x + 1 < 7/3))/9 + 2662*Piece
wise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(5
5)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/5

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GIAC/XCAS [A]  time = 0.217383, size = 140, normalized size = 1.52 \[ \frac{121}{25} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{70}{9} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8}{45} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 70/9*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) + 8/45*sqrt(-2*x + 1) + 49/9*sqrt(-2*x + 1)/(3*x + 2)