3.2103 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=61 \[ \frac{72}{605 \sqrt{1-2 x}}-\frac{1}{55 \sqrt{1-2 x} (5 x+3)}-\frac{72 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

[Out]

72/(605*Sqrt[1 - 2*x]) - 1/(55*Sqrt[1 - 2*x]*(3 + 5*x)) - (72*ArcTanh[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(121*Sqrt[55])

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Rubi [A]  time = 0.071361, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{72}{605 \sqrt{1-2 x}}-\frac{1}{55 \sqrt{1-2 x} (5 x+3)}-\frac{72 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

72/(605*Sqrt[1 - 2*x]) - 1/(55*Sqrt[1 - 2*x]*(3 + 5*x)) - (72*ArcTanh[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(121*Sqrt[55])

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Rubi in Sympy [A]  time = 6.8008, size = 53, normalized size = 0.87 \[ - \frac{72 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6655} + \frac{72}{605 \sqrt{- 2 x + 1}} - \frac{1}{55 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-72*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/6655 + 72/(605*sqrt(-2*x + 1)) -
1/(55*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.0991048, size = 56, normalized size = 0.92 \[ \frac{-\frac{55 \sqrt{1-2 x} (72 x+41)}{10 x^2+x-3}-72 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(41 + 72*x))/(-3 + x + 10*x^2) - 72*Sqrt[55]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/6655

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Maple [A]  time = 0.017, size = 45, normalized size = 0.7 \[{\frac{14}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{605}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{72\,\sqrt{55}}{6655}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/121/(1-2*x)^(1/2)+2/605*(1-2*x)^(1/2)/(-6/5-2*x)-72/6655*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.513, size = 88, normalized size = 1.44 \[ \frac{36}{6655} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (72 \, x + 41\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

36/6655*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 2/121*(72*x + 41)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.214842, size = 96, normalized size = 1.57 \[ \frac{\sqrt{55}{\left (36 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (72 \, x + 41\right )}\right )}}{6655 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6655*sqrt(55)*(36*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-
2*x + 1))/(5*x + 3)) + sqrt(55)*(72*x + 41))/((5*x + 3)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.218006, size = 92, normalized size = 1.51 \[ \frac{36}{6655} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (72 \, x + 41\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

36/6655*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 2/121*(72*x + 41)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))