3.1969 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=106 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)}-\frac{748 \sqrt{1-2 x}}{15 (5 x+3)}-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1562}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-748*Sqrt[1 - 2*x])/(15*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5*x)
) - (910*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + (1562*Sqrt[11/5]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi [A]  time = 0.193692, antiderivative size = 106, 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 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)}-\frac{748 \sqrt{1-2 x}}{15 (5 x+3)}-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1562}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-748*Sqrt[1 - 2*x])/(15*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5*x)
) - (910*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + (1562*Sqrt[11/5]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi in Sympy [A]  time = 20.8688, size = 87, normalized size = 0.82 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{748 \sqrt{- 2 x + 1}}{15 \left (5 x + 3\right )} - \frac{910 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} + \frac{1562 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

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)**2,x)

[Out]

7*(-2*x + 1)**(3/2)/(3*(3*x + 2)*(5*x + 3)) - 748*sqrt(-2*x + 1)/(15*(5*x + 3))
- 910*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/9 + 1562*sqrt(55)*atanh(sqrt(55)
*sqrt(-2*x + 1)/11)/25

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Mathematica [A]  time = 0.194505, size = 90, normalized size = 0.85 \[ \frac{1}{75} \left (4686 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{5 \sqrt{1-2 x} (2314 x+1461)}{(3 x+2) (5 x+3)}\right )-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-910*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + ((-5*Sqrt[1 - 2*x]*(1461 +
2314*x))/((2 + 3*x)*(3 + 5*x)) + 4686*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
])/75

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Maple [A]  time = 0.02, size = 70, normalized size = 0.7 \[{\frac{98}{9}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{910\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{25}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{1562\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

98/9*(1-2*x)^(1/2)/(-4/3-2*x)-910/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
+242/25*(1-2*x)^(1/2)/(-6/5-2*x)+1562/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55
^(1/2)

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Maxima [A]  time = 1.52375, size = 149, normalized size = 1.41 \[ -\frac{781}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{455}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (1157 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2618 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-781/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 455/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 4/15*(1157*(-2*x + 1)^(3/2) - 2618*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136
*x + 9)

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Fricas [A]  time = 0.21992, size = 188, normalized size = 1.77 \[ \frac{\sqrt{5} \sqrt{3}{\left (2343 \, \sqrt{11} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2275 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{5} \sqrt{3}{\left (2314 \, x + 1461\right )} \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/225*sqrt(5)*sqrt(3)*(2343*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)) + 2275*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)*sqrt(3)*(2314*x + 1461)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213573, size = 157, normalized size = 1.48 \[ -\frac{781}{25} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{455}{9} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (1157 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2618 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-781/25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 455/9*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 4/15*(1157*(-2*x + 1)^(3/2) - 2618*sqrt(-2*x + 1))/(15*
(2*x - 1)^2 + 136*x + 9)