3.1892 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(258*Sqrt[1 - 2*x])/15625 - (2*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/875 + (11*(1 - 2*x)^
(3/2)*(2 + 3*x)^3)/75 - ((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(5*(3 + 5*x)) - ((1 - 2*x)
^(3/2)*(5678 + 3663*x))/9375 - (258*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/15625

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Rubi [A]  time = 0.20574, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(258*Sqrt[1 - 2*x])/15625 - (2*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/875 + (11*(1 - 2*x)^
(3/2)*(2 + 3*x)^3)/75 - ((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(5*(3 + 5*x)) - ((1 - 2*x)
^(3/2)*(5678 + 3663*x))/9375 - (258*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/15625

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Rubi in Sympy [A]  time = 27.9593, size = 109, normalized size = 0.85 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}}{5 \left (5 x + 3\right )} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{75} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{875} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (1153845 x + 1788570\right )}{2953125} + \frac{258 \sqrt{- 2 x + 1}}{15625} - \frac{258 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x)**2,x)

[Out]

-(-2*x + 1)**(3/2)*(3*x + 2)**4/(5*(5*x + 3)) + 11*(-2*x + 1)**(3/2)*(3*x + 2)**
3/75 - 2*(-2*x + 1)**(3/2)*(3*x + 2)**2/875 - (-2*x + 1)**(3/2)*(1153845*x + 178
8570)/2953125 + 258*sqrt(-2*x + 1)/15625 - 258*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x
+ 1)/11)/78125

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Mathematica [A]  time = 0.128429, size = 78, normalized size = 0.61 \[ -\frac{5 \sqrt{1-2 x} \left (787500 x^5+1395000 x^4+157275 x^3-924335 x^2-143235 x+161312\right )+1806 \sqrt{55} (5 x+3) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{546875 (5 x+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

-(5*Sqrt[1 - 2*x]*(161312 - 143235*x - 924335*x^2 + 157275*x^3 + 1395000*x^4 + 7
87500*x^5) + 1806*Sqrt[55]*(3 + 5*x)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(546875*
(3 + 5*x))

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Maple [A]  time = 0.016, size = 81, normalized size = 0.6 \[ -{\frac{9}{100} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{999}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{12393}{12500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{8}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{52}{3125}\sqrt{1-2\,x}}+{\frac{22}{78125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{258\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x)

[Out]

-9/100*(1-2*x)^(9/2)+999/1750*(1-2*x)^(7/2)-12393/12500*(1-2*x)^(5/2)+8/3125*(1-
2*x)^(3/2)+52/3125*(1-2*x)^(1/2)+22/78125*(1-2*x)^(1/2)/(-6/5-2*x)-258/78125*arc
tanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49627, size = 132, normalized size = 1.03 \[ -\frac{9}{100} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{999}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{12393}{12500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="maxima")

[Out]

-9/100*(-2*x + 1)^(9/2) + 999/1750*(-2*x + 1)^(7/2) - 12393/12500*(-2*x + 1)^(5/
2) + 8/3125*(-2*x + 1)^(3/2) + 129/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 52/3125*sqrt(-2*x + 1) - 11/15625*sqrt(-2*
x + 1)/(5*x + 3)

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Fricas [A]  time = 0.21365, size = 123, normalized size = 0.96 \[ \frac{\sqrt{5}{\left (903 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt{-2 \, x + 1}\right )}}{546875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="fricas")

[Out]

1/546875*sqrt(5)*(903*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqr
t(-2*x + 1))/(5*x + 3)) - sqrt(5)*(787500*x^5 + 1395000*x^4 + 157275*x^3 - 92433
5*x^2 - 143235*x + 161312)*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)**4/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216547, size = 165, normalized size = 1.29 \[ -\frac{9}{100} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{999}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{12393}{12500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="giac")

[Out]

-9/100*(2*x - 1)^4*sqrt(-2*x + 1) - 999/1750*(2*x - 1)^3*sqrt(-2*x + 1) - 12393/
12500*(2*x - 1)^2*sqrt(-2*x + 1) + 8/3125*(-2*x + 1)^(3/2) + 129/78125*sqrt(55)*
ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 52/
3125*sqrt(-2*x + 1) - 11/15625*sqrt(-2*x + 1)/(5*x + 3)