3.2099 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=120 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^3}{605 (5 x+3)}+\frac{10836 \sqrt{1-2 x} (3 x+2)^2}{15125}+\frac{504 \sqrt{1-2 x} (1500 x+4499)}{75625}-\frac{336 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

[Out]

(10836*Sqrt[1 - 2*x]*(2 + 3*x)^2)/15125 - (36*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*(3
+ 5*x)) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (504*Sqrt[1 - 2*x]*(44
99 + 1500*x))/75625 - (336*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(75625*Sqrt[55])

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Rubi [A]  time = 0.22589, 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 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^3}{605 (5 x+3)}+\frac{10836 \sqrt{1-2 x} (3 x+2)^2}{15125}+\frac{504 \sqrt{1-2 x} (1500 x+4499)}{75625}-\frac{336 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(10836*Sqrt[1 - 2*x]*(2 + 3*x)^2)/15125 - (36*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*(3
+ 5*x)) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (504*Sqrt[1 - 2*x]*(44
99 + 1500*x))/75625 - (336*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(75625*Sqrt[55])

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Rubi in Sympy [A]  time = 25.5445, size = 104, normalized size = 0.87 \[ - \frac{36 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{605 \left (5 x + 3\right )} + \frac{10836 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{15125} + \frac{\sqrt{- 2 x + 1} \left (11340000 x + 34012440\right )}{1134375} - \frac{336 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{4159375} + \frac{7 \left (3 x + 2\right )^{4}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36*sqrt(-2*x + 1)*(3*x + 2)**3/(605*(5*x + 3)) + 10836*sqrt(-2*x + 1)*(3*x + 2)
**2/15125 + sqrt(-2*x + 1)*(11340000*x + 34012440)/1134375 - 336*sqrt(55)*atanh(
sqrt(55)*sqrt(-2*x + 1)/11)/4159375 + 7*(3*x + 2)**4/(11*sqrt(-2*x + 1)*(5*x + 3
))

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Mathematica [A]  time = 0.137113, size = 71, normalized size = 0.59 \[ \frac{\frac{55 \sqrt{1-2 x} \left (735075 x^4+3789720 x^3+14309460 x^2-6264264 x-8186648\right )}{10 x^2+x-3}-336 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-8186648 - 6264264*x + 14309460*x^2 + 3789720*x^3 + 735075*x
^4))/(-3 + x + 10*x^2) - 336*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/4159375

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Maple [A]  time = 0.02, size = 72, normalized size = 0.6 \[{\frac{243}{1000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2943}{1000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{107109}{5000}\sqrt{1-2\,x}}+{\frac{16807}{968}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{378125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{336\,\sqrt{55}}{4159375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(1-2*x)^(5/2)-2943/1000*(1-2*x)^(3/2)+107109/5000*(1-2*x)^(1/2)+16807/9
68/(1-2*x)^(1/2)+2/378125*(1-2*x)^(1/2)/(-6/5-2*x)-336/4159375*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51051, size = 124, normalized size = 1.03 \[ \frac{243}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{2943}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{168}{4159375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{107109}{5000} \, \sqrt{-2 \, x + 1} - \frac{52521891 \, x + 31513117}{302500 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(-2*x + 1)^(5/2) - 2943/1000*(-2*x + 1)^(3/2) + 168/4159375*sqrt(55)*lo
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 107109/5000*sq
rt(-2*x + 1) - 1/302500*(52521891*x + 31513117)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2
*x + 1))

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Fricas [A]  time = 0.253281, size = 117, normalized size = 0.98 \[ \frac{\sqrt{55}{\left (168 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (735075 \, x^{4} + 3789720 \, x^{3} + 14309460 \, x^{2} - 6264264 \, x - 8186648\right )}\right )}}{4159375 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4159375*sqrt(55)*(168*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sq
rt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(735075*x^4 + 3789720*x^3 + 14309460*x^2 - 6
264264*x - 8186648))/((5*x + 3)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221179, size = 138, normalized size = 1.15 \[ \frac{243}{1000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{2943}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{168}{4159375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{107109}{5000} \, \sqrt{-2 \, x + 1} - \frac{52521891 \, x + 31513117}{302500 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1000*(2*x - 1)^2*sqrt(-2*x + 1) - 2943/1000*(-2*x + 1)^(3/2) + 168/4159375*s
qrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 107109/5000*sqrt(-2*x + 1) - 1/302500*(52521891*x + 31513117)/(5*(-2*x + 1)
^(3/2) - 11*sqrt(-2*x + 1))