3.1904 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=94 \[ -\frac{131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac{119 (1-2 x)^{3/2}}{3025}+\frac{357 \sqrt{1-2 x}}{1375}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

[Out]

(357*Sqrt[1 - 2*x])/1375 + (119*(1 - 2*x)^(3/2))/3025 - (1 - 2*x)^(5/2)/(550*(3
+ 5*x)^2) - (131*(1 - 2*x)^(5/2))/(6050*(3 + 5*x)) - (357*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(125*Sqrt[55])

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Rubi [A]  time = 0.111067, antiderivative size = 94, 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 \[ -\frac{131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac{119 (1-2 x)^{3/2}}{3025}+\frac{357 \sqrt{1-2 x}}{1375}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(357*Sqrt[1 - 2*x])/1375 + (119*(1 - 2*x)^(3/2))/3025 - (1 - 2*x)^(5/2)/(550*(3
+ 5*x)^2) - (131*(1 - 2*x)^(5/2))/(6050*(3 + 5*x)) - (357*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(125*Sqrt[55])

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Rubi in Sympy [A]  time = 11.426, size = 80, normalized size = 0.85 \[ - \frac{131 \left (- 2 x + 1\right )^{\frac{5}{2}}}{6050 \left (5 x + 3\right )} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{550 \left (5 x + 3\right )^{2}} + \frac{119 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3025} + \frac{357 \sqrt{- 2 x + 1}}{1375} - \frac{357 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-131*(-2*x + 1)**(5/2)/(6050*(5*x + 3)) - (-2*x + 1)**(5/2)/(550*(5*x + 3)**2) +
119*(-2*x + 1)**(3/2)/3025 + 357*sqrt(-2*x + 1)/1375 - 357*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/6875

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Mathematica [A]  time = 0.126539, size = 63, normalized size = 0.67 \[ \frac{\sqrt{1-2 x} \left (-600 x^3+1320 x^2+2105 x+656\right )}{250 (5 x+3)^2}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(656 + 2105*x + 1320*x^2 - 600*x^3))/(250*(3 + 5*x)^2) - (357*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[{\frac{6}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{174}{625}\sqrt{1-2\,x}}+{\frac{2}{25\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{127}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1419}{50}\sqrt{1-2\,x}} \right ) }-{\frac{357\,\sqrt{55}}{6875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(1-2*x)^(3/2)+174/625*(1-2*x)^(1/2)+2/25*(127/10*(1-2*x)^(3/2)-1419/50*(1-
2*x)^(1/2))/(-6-10*x)^2-357/6875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50632, size = 124, normalized size = 1.32 \[ \frac{6}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{357}{13750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{174}{625} \, \sqrt{-2 \, x + 1} + \frac{635 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1419 \, \sqrt{-2 \, x + 1}}{625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(
sqrt(55) + 5*sqrt(-2*x + 1))) + 174/625*sqrt(-2*x + 1) + 1/625*(635*(-2*x + 1)^(
3/2) - 1419*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.213614, size = 113, normalized size = 1.2 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (600 \, x^{3} - 1320 \, x^{2} - 2105 \, x - 656\right )} \sqrt{-2 \, x + 1} - 357 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{13750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13750*sqrt(55)*(sqrt(55)*(600*x^3 - 1320*x^2 - 2105*x - 656)*sqrt(-2*x + 1) -
357*(25*x^2 + 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3))
)/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213687, size = 116, normalized size = 1.23 \[ \frac{6}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{357}{13750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{174}{625} \, \sqrt{-2 \, x + 1} + \frac{635 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1419 \, \sqrt{-2 \, x + 1}}{2500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*
x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 174/625*sqrt(-2*x + 1) + 1/2500*(635*(-
2*x + 1)^(3/2) - 1419*sqrt(-2*x + 1))/(5*x + 3)^2