3.1905 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=81 \[ -\frac{(1-2 x)^{5/2}}{110 (5 x+3)^2}-\frac{13 (1-2 x)^{3/2}}{110 (5 x+3)}-\frac{39}{275} \sqrt{1-2 x}+\frac{39 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

[Out]

(-39*Sqrt[1 - 2*x])/275 - (1 - 2*x)^(5/2)/(110*(3 + 5*x)^2) - (13*(1 - 2*x)^(3/2
))/(110*(3 + 5*x)) + (39*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi [A]  time = 0.07649, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(1-2 x)^{5/2}}{110 (5 x+3)^2}-\frac{13 (1-2 x)^{3/2}}{110 (5 x+3)}-\frac{39}{275} \sqrt{1-2 x}+\frac{39 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-39*Sqrt[1 - 2*x])/275 - (1 - 2*x)^(5/2)/(110*(3 + 5*x)^2) - (13*(1 - 2*x)^(3/2
))/(110*(3 + 5*x)) + (39*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi in Sympy [A]  time = 9.01845, size = 68, normalized size = 0.84 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{110 \left (5 x + 3\right )^{2}} - \frac{13 \left (- 2 x + 1\right )^{\frac{3}{2}}}{110 \left (5 x + 3\right )} - \frac{39 \sqrt{- 2 x + 1}}{275} + \frac{39 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/(110*(5*x + 3)**2) - 13*(-2*x + 1)**(3/2)/(110*(5*x + 3)) - 3
9*sqrt(-2*x + 1)/275 + 39*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1375

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Mathematica [A]  time = 0.0968499, size = 58, normalized size = 0.72 \[ \frac{39 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}}-\frac{\sqrt{1-2 x} \left (120 x^2+205 x+82\right )}{50 (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(82 + 205*x + 120*x^2))/(50*(3 + 5*x)^2) + (39*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Maple [A]  time = 0.015, size = 57, normalized size = 0.7 \[ -{\frac{12}{125}\sqrt{1-2\,x}}-{\frac{4}{5\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{61}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{693}{100}\sqrt{1-2\,x}} \right ) }+{\frac{39\,\sqrt{55}}{1375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12/125*(1-2*x)^(1/2)-4/5*(-61/20*(1-2*x)^(3/2)+693/100*(1-2*x)^(1/2))/(-6-10*x)
^2+39/1375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49858, size = 112, normalized size = 1.38 \[ -\frac{39}{2750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{12}{125} \, \sqrt{-2 \, x + 1} + \frac{305 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 693 \, \sqrt{-2 \, x + 1}}{125 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-39/2750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 12/125*sqrt(-2*x + 1) + 1/125*(305*(-2*x + 1)^(3/2) - 693*sqrt(-2*x + 1))/
(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.215126, size = 107, normalized size = 1.32 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (120 \, x^{2} + 205 \, x + 82\right )} \sqrt{-2 \, x + 1} - 39 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{2750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2750*sqrt(55)*(sqrt(55)*(120*x^2 + 205*x + 82)*sqrt(-2*x + 1) - 39*(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)/(3+5*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.211832, size = 104, normalized size = 1.28 \[ -\frac{39}{2750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{12}{125} \, \sqrt{-2 \, x + 1} + \frac{305 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 693 \, \sqrt{-2 \, x + 1}}{500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-39/2750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 12/125*sqrt(-2*x + 1) + 1/500*(305*(-2*x + 1)^(3/2) - 693*sqrt(-2
*x + 1))/(5*x + 3)^2