3.1903 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{48 \sqrt{1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{693}{625} \sqrt{1-2 x} (3 x+2)^2+\frac{63 \sqrt{1-2 x} (125 x+92)}{6250}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

[Out]

(693*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(10*(3 + 5*x
)^2) - (48*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(25*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(92 + 1
25*x))/6250 - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Rubi [A]  time = 0.2003, antiderivative size = 120, 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 \[ -\frac{48 \sqrt{1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{693}{625} \sqrt{1-2 x} (3 x+2)^2+\frac{63 \sqrt{1-2 x} (125 x+92)}{6250}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(693*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(10*(3 + 5*x
)^2) - (48*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(25*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(92 + 1
25*x))/6250 - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Rubi in Sympy [A]  time = 20.5463, size = 97, normalized size = 0.81 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{10 \left (5 x + 3\right )^{2}} - \frac{48 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{275 \left (5 x + 3\right )} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (104895 x + 62370\right )}{206250} + \frac{5943 \sqrt{- 2 x + 1}}{34375} - \frac{5943 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{171875} \]

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)**3,x)

[Out]

-(-2*x + 1)**(3/2)*(3*x + 2)**3/(10*(5*x + 3)**2) - 48*(-2*x + 1)**(3/2)*(3*x +
2)**2/(275*(5*x + 3)) + (-2*x + 1)**(3/2)*(104895*x + 62370)/206250 + 5943*sqrt(
-2*x + 1)/34375 - 5943*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/171875

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Mathematica [A]  time = 0.141885, size = 68, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (-27000 x^4-14400 x^3+37530 x^2+36295 x+8644\right )}{6250 (5 x+3)^2}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(8644 + 36295*x + 37530*x^2 - 14400*x^3 - 27000*x^4))/(6250*(3 +
5*x)^2) - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Maple [A]  time = 0.016, size = 75, normalized size = 0.6 \[ -{\frac{27}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{18}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{558}{3125}\sqrt{1-2\,x}}+{\frac{2}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{193}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{429}{10}\sqrt{1-2\,x}} \right ) }-{\frac{5943\,\sqrt{55}}{171875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/625*(1-2*x)^(5/2)+18/625*(1-2*x)^(3/2)+558/3125*(1-2*x)^(1/2)+2/125*(193/10*
(1-2*x)^(3/2)-429/10*(1-2*x)^(1/2))/(-6-10*x)^2-5943/171875*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5025, size = 136, normalized size = 1.13 \[ -\frac{27}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{18}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5943}{343750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{558}{3125} \, \sqrt{-2 \, x + 1} + \frac{193 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 429 \, \sqrt{-2 \, x + 1}}{625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/625*(-2*x + 1)^(5/2) + 18/625*(-2*x + 1)^(3/2) + 5943/343750*sqrt(55)*log(-(
sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 558/3125*sqrt(-2*x
+ 1) + 1/625*(193*(-2*x + 1)^(3/2) - 429*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*
x + 11)

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Fricas [A]  time = 0.212464, size = 120, normalized size = 1. \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (27000 \, x^{4} + 14400 \, x^{3} - 37530 \, x^{2} - 36295 \, x - 8644\right )} \sqrt{-2 \, x + 1} - 5943 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{343750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/343750*sqrt(55)*(sqrt(55)*(27000*x^4 + 14400*x^3 - 37530*x^2 - 36295*x - 8644
)*sqrt(-2*x + 1) - 5943*(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/(3+5*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214068, size = 138, normalized size = 1.15 \[ -\frac{27}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{18}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5943}{343750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{558}{3125} \, \sqrt{-2 \, x + 1} + \frac{193 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 429 \, \sqrt{-2 \, x + 1}}{2500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/625*(2*x - 1)^2*sqrt(-2*x + 1) + 18/625*(-2*x + 1)^(3/2) + 5943/343750*sqrt(
55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) +
558/3125*sqrt(-2*x + 1) + 1/2500*(193*(-2*x + 1)^(3/2) - 429*sqrt(-2*x + 1))/(5
*x + 3)^2