3.1859 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=89 \[ -\frac{(1-2 x)^{5/2}}{63 (3 x+2)}-\frac{5}{9} (1-2 x)^{5/2}-\frac{146}{567} (1-2 x)^{3/2}-\frac{146}{81} \sqrt{1-2 x}+\frac{146}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-146*Sqrt[1 - 2*x])/81 - (146*(1 - 2*x)^(3/2))/567 - (5*(1 - 2*x)^(5/2))/9 - (1
- 2*x)^(5/2)/(63*(2 + 3*x)) + (146*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
81

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Rubi [A]  time = 0.10659, antiderivative size = 89, 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 \[ -\frac{(1-2 x)^{5/2}}{63 (3 x+2)}-\frac{5}{9} (1-2 x)^{5/2}-\frac{146}{567} (1-2 x)^{3/2}-\frac{146}{81} \sqrt{1-2 x}+\frac{146}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-146*Sqrt[1 - 2*x])/81 - (146*(1 - 2*x)^(3/2))/567 - (5*(1 - 2*x)^(5/2))/9 - (1
- 2*x)^(5/2)/(63*(2 + 3*x)) + (146*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
81

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Rubi in Sympy [A]  time = 11.9506, size = 73, normalized size = 0.82 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{63 \left (3 x + 2\right )} - \frac{146 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567} - \frac{146 \sqrt{- 2 x + 1}}{81} + \frac{146 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

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)**2,x)

[Out]

-5*(-2*x + 1)**(5/2)/9 - (-2*x + 1)**(5/2)/(63*(3*x + 2)) - 146*(-2*x + 1)**(3/2
)/567 - 146*sqrt(-2*x + 1)/81 + 146*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/24
3

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Mathematica [A]  time = 0.0995483, size = 63, normalized size = 0.71 \[ \frac{1}{243} \left (146 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{3 \sqrt{1-2 x} \left (540 x^3-300 x^2+187 x+425\right )}{3 x+2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-3*Sqrt[1 - 2*x]*(425 + 187*x - 300*x^2 + 540*x^3))/(2 + 3*x) + 146*Sqrt[21]*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243

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Maple [A]  time = 0.017, size = 63, normalized size = 0.7 \[ -{\frac{5}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{20}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{16}{9}\sqrt{1-2\,x}}+{\frac{14}{243}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{146\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(1-2*x)^(5/2)-20/81*(1-2*x)^(3/2)-16/9*(1-2*x)^(1/2)+14/243*(1-2*x)^(1/2)/(
-4/3-2*x)+146/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.47897, size = 108, normalized size = 1.21 \[ -\frac{5}{9} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{73}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{16}{9} \, \sqrt{-2 \, x + 1} - \frac{7 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(-2*x + 1)^(5/2) - 20/81*(-2*x + 1)^(3/2) - 73/243*sqrt(21)*log(-(sqrt(21)
- 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 16/9*sqrt(-2*x + 1) - 7/81*
sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.222277, size = 109, normalized size = 1.22 \[ \frac{\sqrt{3}{\left (73 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{3}{\left (540 \, x^{3} - 300 \, x^{2} + 187 \, x + 425\right )} \sqrt{-2 \, x + 1}\right )}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/243*sqrt(3)*(73*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x
+ 1))/(3*x + 2)) - sqrt(3)*(540*x^3 - 300*x^2 + 187*x + 425)*sqrt(-2*x + 1))/(3
*x + 2)

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Sympy [A]  time = 158.729, size = 201, normalized size = 2.26 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9} - \frac{20 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{16 \sqrt{- 2 x + 1}}{9} - \frac{196 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} - \frac{1036 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(5/2)/9 - 20*(-2*x + 1)**(3/2)/81 - 16*sqrt(-2*x + 1)/9 - 196*Pie
cewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqrt(-2*x
+ 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(-2*
x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/81 - 1036*Piecewise((-sqrt(21)*ac
oth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sq
rt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/81

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GIAC/XCAS [A]  time = 0.220847, size = 122, normalized size = 1.37 \[ -\frac{5}{9} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{73}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{16}{9} \, \sqrt{-2 \, x + 1} - \frac{7 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(2*x - 1)^2*sqrt(-2*x + 1) - 20/81*(-2*x + 1)^(3/2) - 73/243*sqrt(21)*ln(1/
2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 16/9*sqrt
(-2*x + 1) - 7/81*sqrt(-2*x + 1)/(3*x + 2)