3.1881 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=95 \[ -\frac{3}{20} (1-2 x)^{9/2}+\frac{162}{175} (1-2 x)^{7/2}-\frac{3897 (1-2 x)^{5/2}}{2500}+\frac{2 (1-2 x)^{3/2}}{1875}+\frac{22 \sqrt{1-2 x}}{3125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(22*Sqrt[1 - 2*x])/3125 + (2*(1 - 2*x)^(3/2))/1875 - (3897*(1 - 2*x)^(5/2))/2500
+ (162*(1 - 2*x)^(7/2))/175 - (3*(1 - 2*x)^(9/2))/20 - (22*Sqrt[11/5]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.104002, antiderivative size = 95, 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 \[ -\frac{3}{20} (1-2 x)^{9/2}+\frac{162}{175} (1-2 x)^{7/2}-\frac{3897 (1-2 x)^{5/2}}{2500}+\frac{2 (1-2 x)^{3/2}}{1875}+\frac{22 \sqrt{1-2 x}}{3125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[1 - 2*x])/3125 + (2*(1 - 2*x)^(3/2))/1875 - (3897*(1 - 2*x)^(5/2))/2500
+ (162*(1 - 2*x)^(7/2))/175 - (3*(1 - 2*x)^(9/2))/20 - (22*Sqrt[11/5]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 11.0173, size = 83, normalized size = 0.87 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{9}{2}}}{20} + \frac{162 \left (- 2 x + 1\right )^{\frac{7}{2}}}{175} - \frac{3897 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1875} + \frac{22 \sqrt{- 2 x + 1}}{3125} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(9/2)/20 + 162*(-2*x + 1)**(7/2)/175 - 3897*(-2*x + 1)**(5/2)/250
0 + 2*(-2*x + 1)**(3/2)/1875 + 22*sqrt(-2*x + 1)/3125 - 22*sqrt(55)*atanh(sqrt(5
5)*sqrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.0813522, size = 61, normalized size = 0.64 \[ \frac{-5 \sqrt{1-2 x} \left (157500 x^4+171000 x^3-83565 x^2-123295 x+50858\right )-462 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{328125} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(50858 - 123295*x - 83565*x^2 + 171000*x^3 + 157500*x^4) - 462
*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/328125

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Maple [A]  time = 0.008, size = 65, normalized size = 0.7 \[{\frac{2}{1875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3897}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{162}{175} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{3}{20} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{22\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{3125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1875*(1-2*x)^(3/2)-3897/2500*(1-2*x)^(5/2)+162/175*(1-2*x)^(7/2)-3/20*(1-2*x)^
(9/2)-22/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/3125*(1-2*x)^(1/
2)

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Maxima [A]  time = 1.49428, size = 111, normalized size = 1.17 \[ -\frac{3}{20} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{162}{175} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{3897}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/20*(-2*x + 1)^(9/2) + 162/175*(-2*x + 1)^(7/2) - 3897/2500*(-2*x + 1)^(5/2) +
2/1875*(-2*x + 1)^(3/2) + 11/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/
(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/3125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.215449, size = 99, normalized size = 1.04 \[ -\frac{1}{328125} \, \sqrt{5}{\left (\sqrt{5}{\left (157500 \, x^{4} + 171000 \, x^{3} - 83565 \, x^{2} - 123295 \, x + 50858\right )} \sqrt{-2 \, x + 1} - 231 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/328125*sqrt(5)*(sqrt(5)*(157500*x^4 + 171000*x^3 - 83565*x^2 - 123295*x + 508
58)*sqrt(-2*x + 1) - 231*sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x
+ 1))/(5*x + 3)))

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Sympy [A]  time = 13.2415, size = 122, normalized size = 1.28 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{9}{2}}}{20} + \frac{162 \left (- 2 x + 1\right )^{\frac{7}{2}}}{175} - \frac{3897 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1875} + \frac{22 \sqrt{- 2 x + 1}}{3125} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(9/2)/20 + 162*(-2*x + 1)**(7/2)/175 - 3897*(-2*x + 1)**(5/2)/250
0 + 2*(-2*x + 1)**(3/2)/1875 + 22*sqrt(-2*x + 1)/3125 + 242*Piecewise((-sqrt(55)
*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(5
5)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/3125

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GIAC/XCAS [A]  time = 0.213089, size = 143, normalized size = 1.51 \[ -\frac{3}{20} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{162}{175} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{3897}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/20*(2*x - 1)^4*sqrt(-2*x + 1) - 162/175*(2*x - 1)^3*sqrt(-2*x + 1) - 3897/250
0*(2*x - 1)^2*sqrt(-2*x + 1) + 2/1875*(-2*x + 1)^(3/2) + 11/15625*sqrt(55)*ln(1/
2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/3125*
sqrt(-2*x + 1)