3.2088 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=93 \[ -\frac{243}{560} (1-2 x)^{7/2}+\frac{5751 (1-2 x)^{5/2}}{1000}-\frac{17019}{500} (1-2 x)^{3/2}+\frac{806121 \sqrt{1-2 x}}{5000}+\frac{16807}{176 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]

[Out]

16807/(176*Sqrt[1 - 2*x]) + (806121*Sqrt[1 - 2*x])/5000 - (17019*(1 - 2*x)^(3/2)
)/500 + (5751*(1 - 2*x)^(5/2))/1000 - (243*(1 - 2*x)^(7/2))/560 - (2*ArcTanh[Sqr
t[5/11]*Sqrt[1 - 2*x]])/(6875*Sqrt[55])

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Rubi [A]  time = 0.173828, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{243}{560} (1-2 x)^{7/2}+\frac{5751 (1-2 x)^{5/2}}{1000}-\frac{17019}{500} (1-2 x)^{3/2}+\frac{806121 \sqrt{1-2 x}}{5000}+\frac{16807}{176 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

16807/(176*Sqrt[1 - 2*x]) + (806121*Sqrt[1 - 2*x])/5000 - (17019*(1 - 2*x)^(3/2)
)/500 + (5751*(1 - 2*x)^(5/2))/1000 - (243*(1 - 2*x)^(7/2))/560 - (2*ArcTanh[Sqr
t[5/11]*Sqrt[1 - 2*x]])/(6875*Sqrt[55])

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Rubi in Sympy [A]  time = 15.0141, size = 83, normalized size = 0.89 \[ - \frac{243 \left (- 2 x + 1\right )^{\frac{7}{2}}}{560} + \frac{5751 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1000} - \frac{17019 \left (- 2 x + 1\right )^{\frac{3}{2}}}{500} + \frac{806121 \sqrt{- 2 x + 1}}{5000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{378125} + \frac{16807}{176 \sqrt{- 2 x + 1}} \]

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),x)

[Out]

-243*(-2*x + 1)**(7/2)/560 + 5751*(-2*x + 1)**(5/2)/1000 - 17019*(-2*x + 1)**(3/
2)/500 + 806121*sqrt(-2*x + 1)/5000 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/1
1)/378125 + 16807/(176*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.146645, size = 61, normalized size = 0.66 \[ -\frac{334125 x^4+1545885 x^3+3732300 x^2+10459053 x-10972384}{48125 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(-10972384 + 10459053*x + 3732300*x^2 + 1545885*x^3 + 334125*x^4)/(48125*Sqrt[1
- 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6875*Sqrt[55])

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Maple [A]  time = 0.013, size = 65, normalized size = 0.7 \[ -{\frac{17019}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{5751}{1000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{243}{560} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{378125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{16807}{176}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{806121}{5000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17019/500*(1-2*x)^(3/2)+5751/1000*(1-2*x)^(5/2)-243/560*(1-2*x)^(7/2)-2/378125*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+16807/176/(1-2*x)^(1/2)+806121/500
0*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49929, size = 111, normalized size = 1.19 \[ -\frac{243}{560} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{5751}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{17019}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{378125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{806121}{5000} \, \sqrt{-2 \, x + 1} + \frac{16807}{176 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/560*(-2*x + 1)^(7/2) + 5751/1000*(-2*x + 1)^(5/2) - 17019/500*(-2*x + 1)^(3
/2) + 1/378125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 806121/5000*sqrt(-2*x + 1) + 16807/176/sqrt(-2*x + 1)

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Fricas [A]  time = 0.245988, size = 100, normalized size = 1.08 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (334125 \, x^{4} + 1545885 \, x^{3} + 3732300 \, x^{2} + 10459053 \, x - 10972384\right )} - 7 \, \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{2646875 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2646875*sqrt(55)*(sqrt(55)*(334125*x^4 + 1545885*x^3 + 3732300*x^2 + 10459053
*x - 10972384) - 7*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(
5*x + 3)))/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{5}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**5/((-2*x + 1)**(3/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.215486, size = 134, normalized size = 1.44 \[ \frac{243}{560} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{5751}{1000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{17019}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{378125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{806121}{5000} \, \sqrt{-2 \, x + 1} + \frac{16807}{176 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/560*(2*x - 1)^3*sqrt(-2*x + 1) + 5751/1000*(2*x - 1)^2*sqrt(-2*x + 1) - 1701
9/500*(-2*x + 1)^(3/2) + 1/378125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 806121/5000*sqrt(-2*x + 1) + 16807/176/s
qrt(-2*x + 1)