3.2089 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{81}{200} (1-2 x)^{5/2}-\frac{963}{200} (1-2 x)^{3/2}+\frac{34371 \sqrt{1-2 x}}{1000}+\frac{2401}{88 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

[Out]

2401/(88*Sqrt[1 - 2*x]) + (34371*Sqrt[1 - 2*x])/1000 - (963*(1 - 2*x)^(3/2))/200
+ (81*(1 - 2*x)^(5/2))/200 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1375*Sqrt[5
5])

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Rubi [A]  time = 0.137858, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{81}{200} (1-2 x)^{5/2}-\frac{963}{200} (1-2 x)^{3/2}+\frac{34371 \sqrt{1-2 x}}{1000}+\frac{2401}{88 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

2401/(88*Sqrt[1 - 2*x]) + (34371*Sqrt[1 - 2*x])/1000 - (963*(1 - 2*x)^(3/2))/200
+ (81*(1 - 2*x)^(5/2))/200 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1375*Sqrt[5
5])

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Rubi in Sympy [A]  time = 12.6817, size = 71, normalized size = 0.89 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{5}{2}}}{200} - \frac{963 \left (- 2 x + 1\right )^{\frac{3}{2}}}{200} + \frac{34371 \sqrt{- 2 x + 1}}{1000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{75625} + \frac{2401}{88 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*(-2*x + 1)**(5/2)/200 - 963*(-2*x + 1)**(3/2)/200 + 34371*sqrt(-2*x + 1)/1000
- 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/75625 + 2401/(88*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.131266, size = 56, normalized size = 0.7 \[ \frac{-\frac{55 \left (4455 x^3+19800 x^2+71379 x-78712\right )}{\sqrt{1-2 x}}-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(-78712 + 71379*x + 19800*x^2 + 4455*x^3))/Sqrt[1 - 2*x] - 2*Sqrt[55]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/75625

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Maple [A]  time = 0.013, size = 56, normalized size = 0.7 \[ -{\frac{963}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{81}{200} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{75625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2401}{88}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{34371}{1000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-963/200*(1-2*x)^(3/2)+81/200*(1-2*x)^(5/2)-2/75625*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)+2401/88/(1-2*x)^(1/2)+34371/1000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.53585, size = 99, normalized size = 1.24 \[ \frac{81}{200} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{963}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{75625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{34371}{1000} \, \sqrt{-2 \, x + 1} + \frac{2401}{88 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/200*(-2*x + 1)^(5/2) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*log(-(sqrt
(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 34371/1000*sqrt(-2*x +
1) + 2401/88/sqrt(-2*x + 1)

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Fricas [A]  time = 0.227271, size = 93, normalized size = 1.16 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (4455 \, x^{3} + 19800 \, x^{2} + 71379 \, x - 78712\right )} - \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{75625 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/75625*sqrt(55)*(sqrt(55)*(4455*x^3 + 19800*x^2 + 71379*x - 78712) - 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. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**4/((-2*x + 1)**(3/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.212809, size = 112, normalized size = 1.4 \[ \frac{81}{200} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{963}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{75625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{34371}{1000} \, \sqrt{-2 \, x + 1} + \frac{2401}{88 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/200*(2*x - 1)^2*sqrt(-2*x + 1) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*
ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 343
71/1000*sqrt(-2*x + 1) + 2401/88/sqrt(-2*x + 1)