3.2090 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{9}{20} (1-2 x)^{3/2}+\frac{162}{25} \sqrt{1-2 x}+\frac{343}{44 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

[Out]

343/(44*Sqrt[1 - 2*x]) + (162*Sqrt[1 - 2*x])/25 - (9*(1 - 2*x)^(3/2))/20 - (2*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi [A]  time = 0.104242, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{9}{20} (1-2 x)^{3/2}+\frac{162}{25} \sqrt{1-2 x}+\frac{343}{44 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

343/(44*Sqrt[1 - 2*x]) + (162*Sqrt[1 - 2*x])/25 - (9*(1 - 2*x)^(3/2))/20 - (2*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi in Sympy [A]  time = 10.4715, size = 60, normalized size = 0.9 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{3}{2}}}{20} + \frac{162 \sqrt{- 2 x + 1}}{25} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15125} + \frac{343}{44 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(-2*x + 1)**(3/2)/20 + 162*sqrt(-2*x + 1)/25 - 2*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/15125 + 343/(44*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.10644, size = 51, normalized size = 0.76 \[ \frac{-495 x^2-3069 x+3802}{275 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(3802 - 3069*x - 495*x^2)/(275*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(275*Sqrt[55])

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Maple [A]  time = 0.013, size = 47, normalized size = 0.7 \[ -{\frac{9}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{343}{44}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{162}{25}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/20*(1-2*x)^(3/2)-2/15125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+343/44
/(1-2*x)^(1/2)+162/25*(1-2*x)^(1/2)

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Maxima [A]  time = 1.51147, size = 86, normalized size = 1.28 \[ -\frac{9}{20} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{162}{25} \, \sqrt{-2 \, x + 1} + \frac{343}{44 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/20*(-2*x + 1)^(3/2) + 1/15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) + 162/25*sqrt(-2*x + 1) + 343/44/sqrt(-2*x + 1)

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Fricas [A]  time = 0.22726, size = 86, normalized size = 1.28 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (495 \, x^{2} + 3069 \, x - 3802\right )} - \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{15125 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15125*sqrt(55)*(sqrt(55)*(495*x^2 + 3069*x - 3802) - sqrt(-2*x + 1)*log((sqrt
(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.244043, size = 90, normalized size = 1.34 \[ -\frac{9}{20} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{162}{25} \, \sqrt{-2 \, x + 1} + \frac{343}{44 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/20*(-2*x + 1)^(3/2) + 1/15125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 162/25*sqrt(-2*x + 1) + 343/44/sqrt(-2*x
+ 1)