3.1848 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{(1-2 x)^{5/2}}{21 (3 x+2)}+\frac{76}{189} (1-2 x)^{3/2}+\frac{76}{27} \sqrt{1-2 x}-\frac{76}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(76*Sqrt[1 - 2*x])/27 + (76*(1 - 2*x)^(3/2))/189 + (1 - 2*x)^(5/2)/(21*(2 + 3*x)
) - (76*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi [A]  time = 0.077049, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(1-2 x)^{5/2}}{21 (3 x+2)}+\frac{76}{189} (1-2 x)^{3/2}+\frac{76}{27} \sqrt{1-2 x}-\frac{76}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(76*Sqrt[1 - 2*x])/27 + (76*(1 - 2*x)^(3/2))/189 + (1 - 2*x)^(5/2)/(21*(2 + 3*x)
) - (76*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi in Sympy [A]  time = 8.73667, size = 61, normalized size = 0.8 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{21 \left (3 x + 2\right )} + \frac{76 \left (- 2 x + 1\right )^{\frac{3}{2}}}{189} + \frac{76 \sqrt{- 2 x + 1}}{27} - \frac{76 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)/(21*(3*x + 2)) + 76*(-2*x + 1)**(3/2)/189 + 76*sqrt(-2*x + 1)/
27 - 76*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/81

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Mathematica [A]  time = 0.0836496, size = 58, normalized size = 0.76 \[ \frac{1}{81} \left (\frac{3 \sqrt{1-2 x} \left (-60 x^2+212 x+175\right )}{3 x+2}-76 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((3*Sqrt[1 - 2*x]*(175 + 212*x - 60*x^2))/(2 + 3*x) - 76*Sqrt[21]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/81

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Maple [A]  time = 0.014, size = 54, normalized size = 0.7 \[{\frac{10}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{74}{27}\sqrt{1-2\,x}}-{\frac{14}{81}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{76\,\sqrt{21}}{81}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(1-2*x)^(3/2)+74/27*(1-2*x)^(1/2)-14/81*(1-2*x)^(1/2)/(-4/3-2*x)-76/81*arc
tanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.59866, size = 96, normalized size = 1.26 \[ \frac{10}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{38}{81} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{74}{27} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(-2*x + 1)^(3/2) + 38/81*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 74/27*sqrt(-2*x + 1) + 7/27*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.234974, size = 103, normalized size = 1.36 \[ \frac{\sqrt{3}{\left (38 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{3}{\left (60 \, x^{2} - 212 \, x - 175\right )} \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/81*sqrt(3)*(38*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x
+ 1))/(3*x + 2)) - sqrt(3)*(60*x^2 - 212*x - 175)*sqrt(-2*x + 1))/(3*x + 2)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242523, size = 100, normalized size = 1.32 \[ \frac{10}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{38}{81} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{74}{27} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(-2*x + 1)^(3/2) + 38/81*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1
))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 74/27*sqrt(-2*x + 1) + 7/27*sqrt(-2*x + 1)/(
3*x + 2)