3.1860 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=94 \[ \frac{47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac{(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac{2873 (1-2 x)^{3/2}}{3969}+\frac{2873}{567} \sqrt{1-2 x}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(2873*Sqrt[1 - 2*x])/567 + (2873*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(5/2)/(126*(2
+ 3*x)^2) + (47*(1 - 2*x)^(5/2))/(294*(2 + 3*x)) - (2873*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(81*Sqrt[21])

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Rubi [A]  time = 0.111206, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac{(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac{2873 (1-2 x)^{3/2}}{3969}+\frac{2873}{567} \sqrt{1-2 x}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(2873*Sqrt[1 - 2*x])/567 + (2873*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(5/2)/(126*(2
+ 3*x)^2) + (47*(1 - 2*x)^(5/2))/(294*(2 + 3*x)) - (2873*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(81*Sqrt[21])

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Rubi in Sympy [A]  time = 12.4717, size = 80, normalized size = 0.85 \[ \frac{47 \left (- 2 x + 1\right )^{\frac{5}{2}}}{294 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{126 \left (3 x + 2\right )^{2}} + \frac{2873 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3969} + \frac{2873 \sqrt{- 2 x + 1}}{567} - \frac{2873 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1701} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

47*(-2*x + 1)**(5/2)/(294*(3*x + 2)) - (-2*x + 1)**(5/2)/(126*(3*x + 2)**2) + 28
73*(-2*x + 1)**(3/2)/3969 + 2873*sqrt(-2*x + 1)/567 - 2873*sqrt(21)*atanh(sqrt(2
1)*sqrt(-2*x + 1)/7)/1701

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Mathematica [A]  time = 0.113303, size = 63, normalized size = 0.67 \[ \frac{\sqrt{1-2 x} \left (-1800 x^3+5520 x^2+10195 x+3803\right )}{162 (3 x+2)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(3803 + 10195*x + 5520*x^2 - 1800*x^3))/(162*(2 + 3*x)^2) - (2873
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{130}{27}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{145}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1001}{54}\sqrt{1-2\,x}} \right ) }-{\frac{2873\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(1-2*x)^(3/2)+130/27*(1-2*x)^(1/2)+2/3*(-145/18*(1-2*x)^(3/2)+1001/54*(1-2
*x)^(1/2))/(-4-6*x)^2-2873/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48829, size = 124, normalized size = 1.32 \[ \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2873}{3402} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{130}{27} \, \sqrt{-2 \, x + 1} - \frac{435 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1001 \, \sqrt{-2 \, x + 1}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(
sqrt(21) + 3*sqrt(-2*x + 1))) + 130/27*sqrt(-2*x + 1) - 1/81*(435*(-2*x + 1)^(3/
2) - 1001*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.216735, size = 113, normalized size = 1.2 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1800 \, x^{3} - 5520 \, x^{2} - 10195 \, x - 3803\right )} \sqrt{-2 \, x + 1} - 2873 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3402 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3402*sqrt(21)*(sqrt(21)*(1800*x^3 - 5520*x^2 - 10195*x - 3803)*sqrt(-2*x + 1)
- 2873*(9*x^2 + 12*x + 4)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2
)))/(9*x^2 + 12*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254565, size = 116, normalized size = 1.23 \[ \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2873}{3402} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{130}{27} \, \sqrt{-2 \, x + 1} - \frac{435 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1001 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 130/27*sqrt(-2*x + 1) - 1/324*(435*(-2*x
+ 1)^(3/2) - 1001*sqrt(-2*x + 1))/(3*x + 2)^2