3.1849 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=81 \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]

[Out]

(-71*Sqrt[1 - 2*x])/63 + (1 - 2*x)^(5/2)/(42*(2 + 3*x)^2) - (71*(1 - 2*x)^(3/2))
/(126*(2 + 3*x)) + (71*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])

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Rubi [A]  time = 0.0737455, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-71*Sqrt[1 - 2*x])/63 + (1 - 2*x)^(5/2)/(42*(2 + 3*x)^2) - (71*(1 - 2*x)^(3/2))
/(126*(2 + 3*x)) + (71*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])

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Rubi in Sympy [A]  time = 9.07737, size = 68, normalized size = 0.84 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{42 \left (3 x + 2\right )^{2}} - \frac{71 \left (- 2 x + 1\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )} - \frac{71 \sqrt{- 2 x + 1}}{63} + \frac{71 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{189} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)/(42*(3*x + 2)**2) - 71*(-2*x + 1)**(3/2)/(126*(3*x + 2)) - 71*
sqrt(-2*x + 1)/63 + 71*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/189

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Mathematica [A]  time = 0.100154, size = 58, normalized size = 0.72 \[ \frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (120 x^2+235 x+101\right )}{18 (3 x+2)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(101 + 235*x + 120*x^2))/(18*(2 + 3*x)^2) + (71*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(9*Sqrt[21])

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Maple [A]  time = 0.016, size = 57, normalized size = 0.7 \[ -{\frac{20}{27}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{25}{4} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{71\,\sqrt{21}}{189}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20/27*(1-2*x)^(1/2)-4/3*(-25/4*(1-2*x)^(3/2)+511/36*(1-2*x)^(1/2))/(-4-6*x)^2+7
1/189*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.53401, size = 112, normalized size = 1.38 \[ -\frac{71}{378} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{27 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-71/378*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 20/27*sqrt(-2*x + 1) + 1/27*(225*(-2*x + 1)^(3/2) - 511*sqrt(-2*x + 1))/(9*
(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.236758, size = 107, normalized size = 1.32 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (120 \, x^{2} + 235 \, x + 101\right )} \sqrt{-2 \, x + 1} - 71 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{378 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/378*sqrt(21)*(sqrt(21)*(120*x^2 + 235*x + 101)*sqrt(-2*x + 1) - 71*(9*x^2 + 1
2*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+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242581, size = 104, normalized size = 1.28 \[ -\frac{71}{378} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{108 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-71/378*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 20/27*sqrt(-2*x + 1) + 1/108*(225*(-2*x + 1)^(3/2) - 511*sqrt(-2*x
+ 1))/(3*x + 2)^2