3.2156 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{217}{242 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

[Out]

49/(66*(1 - 2*x)^(3/2)) - 217/(242*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(121*Sqrt[55])

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Rubi [A]  time = 0.0874245, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{217}{242 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

49/(66*(1 - 2*x)^(3/2)) - 217/(242*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(121*Sqrt[55])

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Rubi in Sympy [A]  time = 11.3302, size = 48, normalized size = 0.89 \[ - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6655} - \frac{217}{242 \sqrt{- 2 x + 1}} + \frac{49}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/6655 - 217/(242*sqrt(-2*x + 1)) +
49/(66*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.094914, size = 46, normalized size = 0.85 \[ \frac{7 (93 x-8)}{363 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(-8 + 93*x))/(363*(1 - 2*x)^(3/2)) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1
21*Sqrt[55])

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Maple [A]  time = 0.015, size = 38, normalized size = 0.7 \[{\frac{49}{66} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{6655}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{217}{242}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/66/(1-2*x)^(3/2)-2/6655*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-217/242
/(1-2*x)^(1/2)

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Maxima [A]  time = 1.50084, size = 69, normalized size = 1.28 \[ \frac{1}{6655} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7 \,{\left (93 \, x - 8\right )}}{363 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6655*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + 7/363*(93*x - 8)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.222642, size = 97, normalized size = 1.8 \[ \frac{\sqrt{55}{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - 7 \, \sqrt{55}{\left (93 \, x - 8\right )}\right )}}{19965 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19965*sqrt(55)*(3*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-
2*x + 1))/(5*x + 3)) - 7*sqrt(55)*(93*x - 8))/((2*x - 1)*sqrt(-2*x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**2/((-2*x + 1)**(5/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.21117, size = 82, normalized size = 1.52 \[ \frac{1}{6655} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{7 \,{\left (93 \, x - 8\right )}}{363 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6655*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 7/363*(93*x - 8)/((2*x - 1)*sqrt(-2*x + 1))