3.1954 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{3+5 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3}{35} (1-2 x)^{7/2}+\frac{2}{125} (1-2 x)^{5/2}+\frac{22}{375} (1-2 x)^{3/2}+\frac{242}{625} \sqrt{1-2 x}-\frac{242}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(242*Sqrt[1 - 2*x])/625 + (22*(1 - 2*x)^(3/2))/375 + (2*(1 - 2*x)^(5/2))/125 - (
3*(1 - 2*x)^(7/2))/35 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi [A]  time = 0.0984079, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3}{35} (1-2 x)^{7/2}+\frac{2}{125} (1-2 x)^{5/2}+\frac{22}{375} (1-2 x)^{3/2}+\frac{242}{625} \sqrt{1-2 x}-\frac{242}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(242*Sqrt[1 - 2*x])/625 + (22*(1 - 2*x)^(3/2))/375 + (2*(1 - 2*x)^(5/2))/125 - (
3*(1 - 2*x)^(7/2))/35 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi in Sympy [A]  time = 9.3952, size = 71, normalized size = 0.87 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{125} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{242 \sqrt{- 2 x + 1}}{625} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)/35 + 2*(-2*x + 1)**(5/2)/125 + 22*(-2*x + 1)**(3/2)/375 + 2
42*sqrt(-2*x + 1)/625 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3125

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Mathematica [A]  time = 0.0613631, size = 56, normalized size = 0.68 \[ \frac{5 \sqrt{1-2 x} \left (9000 x^3-12660 x^2+4370 x+4937\right )-5082 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{65625} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(4937 + 4370*x - 12660*x^2 + 9000*x^3) - 5082*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/65625

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Maple [A]  time = 0.009, size = 56, normalized size = 0.7 \[{\frac{22}{375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{3}{35} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{242\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{625}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/375*(1-2*x)^(3/2)+2/125*(1-2*x)^(5/2)-3/35*(1-2*x)^(7/2)-242/3125*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/625*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49977, size = 99, normalized size = 1.21 \[ -\frac{3}{35} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/35*(-2*x + 1)^(7/2) + 2/125*(-2*x + 1)^(5/2) + 22/375*(-2*x + 1)^(3/2) + 121/
3125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 242/625*sqrt(-2*x + 1)

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Fricas [A]  time = 0.210469, size = 92, normalized size = 1.12 \[ \frac{1}{65625} \, \sqrt{5}{\left (\sqrt{5}{\left (9000 \, x^{3} - 12660 \, x^{2} + 4370 \, x + 4937\right )} \sqrt{-2 \, x + 1} + 2541 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/65625*sqrt(5)*(sqrt(5)*(9000*x^3 - 12660*x^2 + 4370*x + 4937)*sqrt(-2*x + 1) +
2541*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 = 9.95482, size = 110, normalized size = 1.34 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{125} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{242 \sqrt{- 2 x + 1}}{625} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)/35 + 2*(-2*x + 1)**(5/2)/125 + 22*(-2*x + 1)**(3/2)/375 + 2
42*sqrt(-2*x + 1)/625 + 2662*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/
11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x
+ 1 < 11/5))/625

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GIAC/XCAS [A]  time = 0.209679, size = 122, normalized size = 1.49 \[ \frac{3}{35} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{3125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/35*(2*x - 1)^3*sqrt(-2*x + 1) + 2/125*(2*x - 1)^2*sqrt(-2*x + 1) + 22/375*(-2*
x + 1)^(3/2) + 121/3125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) + 242/625*sqrt(-2*x + 1)