3.1919 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=89 \[ \frac{(1-2 x)^{7/2}}{21 (3 x+2)}+\frac{16}{63} (1-2 x)^{5/2}+\frac{80}{81} (1-2 x)^{3/2}+\frac{560}{81} \sqrt{1-2 x}-\frac{560}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(560*Sqrt[1 - 2*x])/81 + (80*(1 - 2*x)^(3/2))/81 + (16*(1 - 2*x)^(5/2))/63 + (1
- 2*x)^(7/2)/(21*(2 + 3*x)) - (560*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/8
1

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Rubi [A]  time = 0.099307, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(1-2 x)^{7/2}}{21 (3 x+2)}+\frac{16}{63} (1-2 x)^{5/2}+\frac{80}{81} (1-2 x)^{3/2}+\frac{560}{81} \sqrt{1-2 x}-\frac{560}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(560*Sqrt[1 - 2*x])/81 + (80*(1 - 2*x)^(3/2))/81 + (16*(1 - 2*x)^(5/2))/63 + (1
- 2*x)^(7/2)/(21*(2 + 3*x)) - (560*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/8
1

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Rubi in Sympy [A]  time = 9.62198, size = 73, normalized size = 0.82 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{21 \left (3 x + 2\right )} + \frac{16 \left (- 2 x + 1\right )^{\frac{5}{2}}}{63} + \frac{80 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} + \frac{560 \sqrt{- 2 x + 1}}{81} - \frac{560 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(7/2)/(21*(3*x + 2)) + 16*(-2*x + 1)**(5/2)/63 + 80*(-2*x + 1)**(3/2
)/81 + 560*sqrt(-2*x + 1)/81 - 560*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/243

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Mathematica [A]  time = 0.088985, size = 63, normalized size = 0.71 \[ \frac{1}{243} \left (\frac{3 \sqrt{1-2 x} \left (216 x^3-516 x^2+1474 x+1325\right )}{3 x+2}-560 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((3*Sqrt[1 - 2*x]*(1325 + 1474*x - 516*x^2 + 216*x^3))/(2 + 3*x) - 560*Sqrt[21]*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243

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Maple [A]  time = 0.016, size = 63, normalized size = 0.7 \[{\frac{2}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{74}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{182}{27}\sqrt{1-2\,x}}-{\frac{98}{243}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{560\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(1-2*x)^(5/2)+74/81*(1-2*x)^(3/2)+182/27*(1-2*x)^(1/2)-98/243*(1-2*x)^(1/2)/
(-4/3-2*x)-560/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50181, size = 108, normalized size = 1.21 \[ \frac{2}{9} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{74}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{280}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{182}{27} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(-2*x + 1)^(5/2) + 74/81*(-2*x + 1)^(3/2) + 280/243*sqrt(21)*log(-(sqrt(21)
- 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 182/27*sqrt(-2*x + 1) + 49/
81*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.21846, size = 108, normalized size = 1.21 \[ \frac{\sqrt{3}{\left (280 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (216 \, x^{3} - 516 \, x^{2} + 1474 \, x + 1325\right )} \sqrt{-2 \, x + 1}\right )}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/243*sqrt(3)*(280*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*
x + 1))/(3*x + 2)) + sqrt(3)*(216*x^3 - 516*x^2 + 1474*x + 1325)*sqrt(-2*x + 1))
/(3*x + 2)

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Sympy [A]  time = 153.727, size = 199, normalized size = 2.24 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9} + \frac{74 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} + \frac{182 \sqrt{- 2 x + 1}}{27} + \frac{1372 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} + \frac{4018 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)/9 + 74*(-2*x + 1)**(3/2)/81 + 182*sqrt(-2*x + 1)/27 + 1372*P
iecewise((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 + 4018*Piecewise((-sqrt(21)*
acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*
sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/81

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GIAC/XCAS [A]  time = 0.219286, size = 122, normalized size = 1.37 \[ \frac{2}{9} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{74}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{280}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{182}{27} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(2*x - 1)^2*sqrt(-2*x + 1) + 74/81*(-2*x + 1)^(3/2) + 280/243*sqrt(21)*ln(1/
2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 182/27*sq
rt(-2*x + 1) + 49/81*sqrt(-2*x + 1)/(3*x + 2)