3.1858 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx\)

Optimal. Leaf size=82 \[ \frac{25}{42} (1-2 x)^{7/2}-\frac{31}{18} (1-2 x)^{5/2}+\frac{2}{81} (1-2 x)^{3/2}+\frac{14}{81} \sqrt{1-2 x}-\frac{14}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(14*Sqrt[1 - 2*x])/81 + (2*(1 - 2*x)^(3/2))/81 - (31*(1 - 2*x)^(5/2))/18 + (25*(
1 - 2*x)^(7/2))/42 - (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi [A]  time = 0.0985676, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{25}{42} (1-2 x)^{7/2}-\frac{31}{18} (1-2 x)^{5/2}+\frac{2}{81} (1-2 x)^{3/2}+\frac{14}{81} \sqrt{1-2 x}-\frac{14}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*Sqrt[1 - 2*x])/81 + (2*(1 - 2*x)^(3/2))/81 - (31*(1 - 2*x)^(5/2))/18 + (25*(
1 - 2*x)^(7/2))/42 - (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi in Sympy [A]  time = 10.1917, size = 71, normalized size = 0.87 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{7}{2}}}{42} - \frac{31 \left (- 2 x + 1\right )^{\frac{5}{2}}}{18} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} + \frac{14 \sqrt{- 2 x + 1}}{81} - \frac{14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

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),x)

[Out]

25*(-2*x + 1)**(7/2)/42 - 31*(-2*x + 1)**(5/2)/18 + 2*(-2*x + 1)**(3/2)/81 + 14*
sqrt(-2*x + 1)/81 - 14*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/243

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Mathematica [A]  time = 0.0799874, size = 56, normalized size = 0.68 \[ \frac{3 \sqrt{1-2 x} \left (-2700 x^3+144 x^2+1853 x-527\right )-98 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1701} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-527 + 1853*x + 144*x^2 - 2700*x^3) - 98*Sqrt[21]*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/1701

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Maple [A]  time = 0.008, size = 56, normalized size = 0.7 \[{\frac{2}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{31}{18} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{25}{42} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{14\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{14}{81}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/81*(1-2*x)^(3/2)-31/18*(1-2*x)^(5/2)+25/42*(1-2*x)^(7/2)-14/243*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)+14/81*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49161, size = 99, normalized size = 1.21 \[ \frac{25}{42} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{31}{18} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{7}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{14}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/42*(-2*x + 1)^(7/2) - 31/18*(-2*x + 1)^(5/2) + 2/81*(-2*x + 1)^(3/2) + 7/243*
sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 14/
81*sqrt(-2*x + 1)

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Fricas [A]  time = 0.227257, size = 92, normalized size = 1.12 \[ -\frac{1}{1701} \, \sqrt{3}{\left (\sqrt{3}{\left (2700 \, x^{3} - 144 \, x^{2} - 1853 \, x + 527\right )} \sqrt{-2 \, x + 1} - 49 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1701*sqrt(3)*(sqrt(3)*(2700*x^3 - 144*x^2 - 1853*x + 527)*sqrt(-2*x + 1) - 49
*sqrt(7)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 9.91931, size = 110, normalized size = 1.34 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{7}{2}}}{42} - \frac{31 \left (- 2 x + 1\right )^{\frac{5}{2}}}{18} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} + \frac{14 \sqrt{- 2 x + 1}}{81} + \frac{98 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(7/2)/42 - 31*(-2*x + 1)**(5/2)/18 + 2*(-2*x + 1)**(3/2)/81 + 14*
sqrt(-2*x + 1)/81 + 98*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 = 0.231448, size = 122, normalized size = 1.49 \[ -\frac{25}{42} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{31}{18} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{7}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{14}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/42*(2*x - 1)^3*sqrt(-2*x + 1) - 31/18*(2*x - 1)^2*sqrt(-2*x + 1) + 2/81*(-2*
x + 1)^(3/2) + 7/243*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) + 14/81*sqrt(-2*x + 1)