3.2021 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{27}{100} (1-2 x)^{5/2}+\frac{54}{25} (1-2 x)^{3/2}-\frac{3897}{500} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

[Out]

(-3897*Sqrt[1 - 2*x])/500 + (54*(1 - 2*x)^(3/2))/25 - (27*(1 - 2*x)^(5/2))/100 -
(2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

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Rubi [A]  time = 0.0811323, antiderivative size = 67, 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 \[ -\frac{27}{100} (1-2 x)^{5/2}+\frac{54}{25} (1-2 x)^{3/2}-\frac{3897}{500} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-3897*Sqrt[1 - 2*x])/500 + (54*(1 - 2*x)^(3/2))/25 - (27*(1 - 2*x)^(5/2))/100 -
(2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

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Rubi in Sympy [A]  time = 7.9821, size = 60, normalized size = 0.9 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{5}{2}}}{100} + \frac{54 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25} - \frac{3897 \sqrt{- 2 x + 1}}{500} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(5/2)/100 + 54*(-2*x + 1)**(3/2)/25 - 3897*sqrt(-2*x + 1)/500 -
2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/6875

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Mathematica [A]  time = 0.0717335, size = 51, normalized size = 0.76 \[ \frac{-495 \sqrt{1-2 x} \left (15 x^2+45 x+82\right )-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875} \]

Antiderivative was successfully verified.

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

[Out]

(-495*Sqrt[1 - 2*x]*(82 + 45*x + 15*x^2) - 2*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/6875

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Maple [A]  time = 0.009, size = 47, normalized size = 0.7 \[{\frac{54}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{27}{100} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{6875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{3897}{500}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

54/25*(1-2*x)^(3/2)-27/100*(1-2*x)^(5/2)-2/6875*arctanh(1/11*55^(1/2)*(1-2*x)^(1
/2))*55^(1/2)-3897/500*(1-2*x)^(1/2)

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Maxima [A]  time = 1.53089, size = 86, normalized size = 1.28 \[ -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{54}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{6875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3897}{500} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/100*(-2*x + 1)^(5/2) + 54/25*(-2*x + 1)^(3/2) + 1/6875*sqrt(55)*log(-(sqrt(5
5) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3897/500*sqrt(-2*x + 1)

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Fricas [A]  time = 0.27132, size = 78, normalized size = 1.16 \[ -\frac{1}{6875} \, \sqrt{55}{\left (9 \, \sqrt{55}{\left (15 \, x^{2} + 45 \, x + 82\right )} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6875*sqrt(55)*(9*sqrt(55)*(15*x^2 + 45*x + 82)*sqrt(-2*x + 1) - log((sqrt(55)
*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 5.04269, size = 102, normalized size = 1.52 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{5}{2}}}{100} + \frac{54 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25} - \frac{3897 \sqrt{- 2 x + 1}}{500} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(5/2)/100 + 54*(-2*x + 1)**(3/2)/25 - 3897*sqrt(-2*x + 1)/500 +
2*Piecewise((-sqrt(55)*acoth(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) > 5/1
1), (-sqrt(55)*atanh(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) < 5/11))/125

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GIAC/XCAS [A]  time = 0.223057, size = 100, normalized size = 1.49 \[ -\frac{27}{100} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{54}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{6875} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3897}{500} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/100*(2*x - 1)^2*sqrt(-2*x + 1) + 54/25*(-2*x + 1)^(3/2) + 1/6875*sqrt(55)*ln
(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3897/
500*sqrt(-2*x + 1)