3.1893 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^3}{5 (5 x+3)}+\frac{27}{175} (1-2 x)^{3/2} (3 x+2)^2-\frac{6}{625} (1-2 x)^{3/2} (9 x+29)+\frac{192 \sqrt{1-2 x}}{3125}-\frac{192 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(192*Sqrt[1 - 2*x])/3125 + (27*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/175 - ((1 - 2*x)^(3/
2)*(2 + 3*x)^3)/(5*(3 + 5*x)) - (6*(1 - 2*x)^(3/2)*(29 + 9*x))/625 - (192*Sqrt[1
1/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.15843, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^3}{5 (5 x+3)}+\frac{27}{175} (1-2 x)^{3/2} (3 x+2)^2-\frac{6}{625} (1-2 x)^{3/2} (9 x+29)+\frac{192 \sqrt{1-2 x}}{3125}-\frac{192 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(192*Sqrt[1 - 2*x])/3125 + (27*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/175 - ((1 - 2*x)^(3/
2)*(2 + 3*x)^3)/(5*(3 + 5*x)) - (6*(1 - 2*x)^(3/2)*(29 + 9*x))/625 - (192*Sqrt[1
1/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 21.0747, size = 90, normalized size = 0.83 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{5 \left (5 x + 3\right )} + \frac{27 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{175} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5670 x + 18270\right )}{65625} + \frac{192 \sqrt{- 2 x + 1}}{3125} - \frac{192 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

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)**2,x)

[Out]

-(-2*x + 1)**(3/2)*(3*x + 2)**3/(5*(5*x + 3)) + 27*(-2*x + 1)**(3/2)*(3*x + 2)**
2/175 - (-2*x + 1)**(3/2)*(5670*x + 18270)/65625 + 192*sqrt(-2*x + 1)/3125 - 192
*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.10631, size = 68, normalized size = 0.63 \[ \frac{-\frac{5 \sqrt{1-2 x} \left (67500 x^4+62100 x^3-57165 x^2-27640 x+8738\right )}{5 x+3}-1344 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{109375} \]

Antiderivative was successfully verified.

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

[Out]

((-5*Sqrt[1 - 2*x]*(8738 - 27640*x - 57165*x^2 + 62100*x^3 + 67500*x^4))/(3 + 5*
x) - 1344*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/109375

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Maple [A]  time = 0.016, size = 72, normalized size = 0.7 \[{\frac{27}{350} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{351}{1250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{6}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{194}{3125}\sqrt{1-2\,x}}+{\frac{22}{15625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{192\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/350*(1-2*x)^(7/2)-351/1250*(1-2*x)^(5/2)+6/625*(1-2*x)^(3/2)+194/3125*(1-2*x)
^(1/2)+22/15625*(1-2*x)^(1/2)/(-6/5-2*x)-192/15625*arctanh(1/11*55^(1/2)*(1-2*x)
^(1/2))*55^(1/2)

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Maxima [A]  time = 1.53305, size = 120, normalized size = 1.11 \[ \frac{27}{350} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{351}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{6}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{96}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{194}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/350*(-2*x + 1)^(7/2) - 351/1250*(-2*x + 1)^(5/2) + 6/625*(-2*x + 1)^(3/2) + 9
6/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 194/3125*sqrt(-2*x + 1) - 11/3125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.213756, size = 116, normalized size = 1.07 \[ \frac{\sqrt{5}{\left (672 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (67500 \, x^{4} + 62100 \, x^{3} - 57165 \, x^{2} - 27640 \, x + 8738\right )} \sqrt{-2 \, x + 1}\right )}}{109375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/109375*sqrt(5)*(672*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqr
t(-2*x + 1))/(5*x + 3)) - sqrt(5)*(67500*x^4 + 62100*x^3 - 57165*x^2 - 27640*x +
8738)*sqrt(-2*x + 1))/(5*x + 3)

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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)*(2+3*x)**3/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216055, size = 143, normalized size = 1.32 \[ -\frac{27}{350} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{351}{1250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{6}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{96}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{194}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/350*(2*x - 1)^3*sqrt(-2*x + 1) - 351/1250*(2*x - 1)^2*sqrt(-2*x + 1) + 6/625
*(-2*x + 1)^(3/2) + 96/15625*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1)
)/(sqrt(55) + 5*sqrt(-2*x + 1))) + 194/3125*sqrt(-2*x + 1) - 11/3125*sqrt(-2*x +
1)/(5*x + 3)