3.1811 \(\int \frac{\sqrt{1-2 x} (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=95 \[ \frac{9}{40} (1-2 x)^{9/2}-\frac{2889 (1-2 x)^{7/2}}{1400}+\frac{34371 (1-2 x)^{5/2}}{5000}-\frac{45473 (1-2 x)^{3/2}}{5000}+\frac{2 \sqrt{1-2 x}}{3125}-\frac{2 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(2*Sqrt[1 - 2*x])/3125 - (45473*(1 - 2*x)^(3/2))/5000 + (34371*(1 - 2*x)^(5/2))/
5000 - (2889*(1 - 2*x)^(7/2))/1400 + (9*(1 - 2*x)^(9/2))/40 - (2*Sqrt[11/5]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.0994315, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{9}{40} (1-2 x)^{9/2}-\frac{2889 (1-2 x)^{7/2}}{1400}+\frac{34371 (1-2 x)^{5/2}}{5000}-\frac{45473 (1-2 x)^{3/2}}{5000}+\frac{2 \sqrt{1-2 x}}{3125}-\frac{2 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/3125 - (45473*(1 - 2*x)^(3/2))/5000 + (34371*(1 - 2*x)^(5/2))/
5000 - (2889*(1 - 2*x)^(7/2))/1400 + (9*(1 - 2*x)^(9/2))/40 - (2*Sqrt[11/5]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 10.2274, size = 83, normalized size = 0.87 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{9}{2}}}{40} - \frac{2889 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1400} + \frac{34371 \left (- 2 x + 1\right )^{\frac{5}{2}}}{5000} - \frac{45473 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5000} + \frac{2 \sqrt{- 2 x + 1}}{3125} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(9/2)/40 - 2889*(-2*x + 1)**(7/2)/1400 + 34371*(-2*x + 1)**(5/2)/5
000 - 45473*(-2*x + 1)**(3/2)/5000 + 2*sqrt(-2*x + 1)/3125 - 2*sqrt(55)*atanh(sq
rt(55)*sqrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.093379, size = 61, normalized size = 0.64 \[ \frac{5 \sqrt{1-2 x} \left (78750 x^4+203625 x^3+177930 x^2+27865 x-88776\right )-14 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{109375} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-88776 + 27865*x + 177930*x^2 + 203625*x^3 + 78750*x^4) - 14*S
qrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/109375

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Maple [A]  time = 0.01, size = 65, normalized size = 0.7 \[ -{\frac{45473}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{34371}{5000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2889}{1400} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{9}{40} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{3125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-45473/5000*(1-2*x)^(3/2)+34371/5000*(1-2*x)^(5/2)-2889/1400*(1-2*x)^(7/2)+9/40*
(1-2*x)^(9/2)-2/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/3125*(1-2*
x)^(1/2)

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Maxima [A]  time = 1.49521, size = 111, normalized size = 1.17 \[ \frac{9}{40} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{2889}{1400} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{34371}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{45473}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/40*(-2*x + 1)^(9/2) - 2889/1400*(-2*x + 1)^(7/2) + 34371/5000*(-2*x + 1)^(5/2)
- 45473/5000*(-2*x + 1)^(3/2) + 1/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.211416, size = 99, normalized size = 1.04 \[ \frac{1}{109375} \, \sqrt{5}{\left (\sqrt{5}{\left (78750 \, x^{4} + 203625 \, x^{3} + 177930 \, x^{2} + 27865 \, x - 88776\right )} \sqrt{-2 \, x + 1} + 7 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/109375*sqrt(5)*(sqrt(5)*(78750*x^4 + 203625*x^3 + 177930*x^2 + 27865*x - 88776
)*sqrt(-2*x + 1) + 7*sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1)
)/(5*x + 3)))

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Sympy [A]  time = 7.56549, size = 122, normalized size = 1.28 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{9}{2}}}{40} - \frac{2889 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1400} + \frac{34371 \left (- 2 x + 1\right )^{\frac{5}{2}}}{5000} - \frac{45473 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5000} + \frac{2 \sqrt{- 2 x + 1}}{3125} + \frac{22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(9/2)/40 - 2889*(-2*x + 1)**(7/2)/1400 + 34371*(-2*x + 1)**(5/2)/5
000 - 45473*(-2*x + 1)**(3/2)/5000 + 2*sqrt(-2*x + 1)/3125 + 22*Piecewise((-sqrt
(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sq
rt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/3125

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GIAC/XCAS [A]  time = 0.241419, size = 143, normalized size = 1.51 \[ \frac{9}{40} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{2889}{1400} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{34371}{5000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{45473}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/40*(2*x - 1)^4*sqrt(-2*x + 1) + 2889/1400*(2*x - 1)^3*sqrt(-2*x + 1) + 34371/5
000*(2*x - 1)^2*sqrt(-2*x + 1) - 45473/5000*(-2*x + 1)^(3/2) + 1/15625*sqrt(55)*
ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3
125*sqrt(-2*x + 1)