3.2072 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=54 \[ \frac{25}{6} \sqrt{1-2 x}+\frac{121}{14 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

121/(14*Sqrt[1 - 2*x]) + (25*Sqrt[1 - 2*x])/6 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(21*Sqrt[21])

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Rubi [A]  time = 0.0888791, 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 \[ \frac{25}{6} \sqrt{1-2 x}+\frac{121}{14 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(14*Sqrt[1 - 2*x]) + (25*Sqrt[1 - 2*x])/6 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(21*Sqrt[21])

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Rubi in Sympy [A]  time = 9.8332, size = 48, normalized size = 0.89 \[ \frac{25 \sqrt{- 2 x + 1}}{6} - \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} + \frac{121}{14 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*sqrt(-2*x + 1)/6 - 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/441 + 121/(14*
sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0958874, size = 50, normalized size = 0.93 \[ \frac{\sqrt{1-2 x} (175 x-269)}{42 x-21}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-269 + 175*x))/(-21 + 42*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/(21*Sqrt[21])

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Maple [A]  time = 0.015, size = 38, normalized size = 0.7 \[ -{\frac{2\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{121}{14}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{25}{6}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+121/14/(1-2*x)^(1/2)+25/6*(1
-2*x)^(1/2)

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Maxima [A]  time = 1.50665, size = 74, normalized size = 1.37 \[ \frac{1}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{25}{6} \, \sqrt{-2 \, x + 1} + \frac{121}{14 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))
+ 25/6*sqrt(-2*x + 1) + 121/14/sqrt(-2*x + 1)

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Fricas [A]  time = 0.238919, size = 80, normalized size = 1.48 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (175 \, x - 269\right )} - \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{441 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/441*sqrt(21)*(sqrt(21)*(175*x - 269) - sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5)
+ 21*sqrt(-2*x + 1))/(3*x + 2)))/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214678, size = 78, normalized size = 1.44 \[ \frac{1}{441} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{25}{6} \, \sqrt{-2 \, x + 1} + \frac{121}{14 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/441*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 25/6*sqrt(-2*x + 1) + 121/14/sqrt(-2*x + 1)