3.1813 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{3+5 x} \, dx\)

Optimal. Leaf size=69 \[ \frac{9}{50} (1-2 x)^{5/2}-\frac{37}{50} (1-2 x)^{3/2}+\frac{2}{125} \sqrt{1-2 x}-\frac{2}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/125 - (37*(1 - 2*x)^(3/2))/50 + (9*(1 - 2*x)^(5/2))/50 - (2*Sq
rt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Rubi [A]  time = 0.0852675, antiderivative size = 69, 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 \[ \frac{9}{50} (1-2 x)^{5/2}-\frac{37}{50} (1-2 x)^{3/2}+\frac{2}{125} \sqrt{1-2 x}-\frac{2}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/125 - (37*(1 - 2*x)^(3/2))/50 + (9*(1 - 2*x)^(5/2))/50 - (2*Sq
rt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Rubi in Sympy [A]  time = 8.35039, size = 60, normalized size = 0.87 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{5}{2}}}{50} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}}}{50} + \frac{2 \sqrt{- 2 x + 1}}{125} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(5/2)/50 - 37*(-2*x + 1)**(3/2)/50 + 2*sqrt(-2*x + 1)/125 - 2*sqrt
(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/625

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Mathematica [A]  time = 0.0678694, size = 51, normalized size = 0.74 \[ \frac{1}{625} \left (5 \sqrt{1-2 x} \left (90 x^2+95 x-68\right )-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-68 + 95*x + 90*x^2) - 2*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/625

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Maple [A]  time = 0.011, size = 47, normalized size = 0.7 \[ -{\frac{37}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{9}{50} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37/50*(1-2*x)^(3/2)+9/50*(1-2*x)^(5/2)-2/625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2
))*55^(1/2)+2/125*(1-2*x)^(1/2)

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Maxima [A]  time = 1.52406, size = 86, normalized size = 1.25 \[ \frac{9}{50} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{37}{50} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/50*(-2*x + 1)^(5/2) - 37/50*(-2*x + 1)^(3/2) + 1/625*sqrt(55)*log(-(sqrt(55) -
5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.213615, size = 84, normalized size = 1.22 \[ \frac{1}{625} \, \sqrt{5}{\left (\sqrt{5}{\left (90 \, x^{2} + 95 \, x - 68\right )} \sqrt{-2 \, x + 1} + \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/625*sqrt(5)*(sqrt(5)*(90*x^2 + 95*x - 68)*sqrt(-2*x + 1) + 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 = 6.16729, size = 99, normalized size = 1.43 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{5}{2}}}{50} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}}}{50} + \frac{2 \sqrt{- 2 x + 1}}{125} + \frac{22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(5/2)/50 - 37*(-2*x + 1)**(3/2)/50 + 2*sqrt(-2*x + 1)/125 + 22*Pie
cewise((-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))/125

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GIAC/XCAS [A]  time = 0.228378, size = 100, normalized size = 1.45 \[ \frac{9}{50} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{37}{50} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/50*(2*x - 1)^2*sqrt(-2*x + 1) - 37/50*(-2*x + 1)^(3/2) + 1/625*sqrt(55)*ln(1/2
*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/125*sqr
t(-2*x + 1)