3.1804 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=93 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac{7}{9} \sqrt{1-2 x} (5 x+3)^2-\frac{2}{81} \sqrt{1-2 x} (170 x+211)-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(7*Sqrt[1 - 2*x]*(3 + 5*x)^2)/9 - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x)) - (2
*Sqrt[1 - 2*x]*(211 + 170*x))/81 - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sq
rt[21])

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Rubi [A]  time = 0.153111, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac{7}{9} \sqrt{1-2 x} (5 x+3)^2-\frac{2}{81} \sqrt{1-2 x} (170 x+211)-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*(3 + 5*x)^2)/9 - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x)) - (2
*Sqrt[1 - 2*x]*(211 + 170*x))/81 - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sq
rt[21])

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Rubi in Sympy [A]  time = 18.7327, size = 78, normalized size = 0.84 \[ \frac{7 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{9} - \frac{\sqrt{- 2 x + 1} \left (5100 x + 6330\right )}{1215} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{3 \left (3 x + 2\right )} - \frac{212 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1701} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*sqrt(-2*x + 1)*(5*x + 3)**2/9 - sqrt(-2*x + 1)*(5100*x + 6330)/1215 - sqrt(-2*
x + 1)*(5*x + 3)**3/(3*(3*x + 2)) - 212*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7
)/1701

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Mathematica [A]  time = 0.0990805, size = 63, normalized size = 0.68 \[ \frac{\sqrt{1-2 x} \left (1350 x^3+1725 x^2-110 x-439\right )}{81 (3 x+2)}-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-439 - 110*x + 1725*x^2 + 1350*x^3))/(81*(2 + 3*x)) - (212*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Maple [A]  time = 0.018, size = 63, normalized size = 0.7 \[{\frac{25}{18} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{725}{162} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{10}{27}\sqrt{1-2\,x}}-{\frac{2}{243}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{212\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(1-2*x)^(5/2)-725/162*(1-2*x)^(3/2)+10/27*(1-2*x)^(1/2)-2/243*(1-2*x)^(1/2
)/(-4/3-2*x)-212/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51705, size = 108, normalized size = 1.16 \[ \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{725}{162} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{106}{1701} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{10}{27} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(-2*x + 1)^(5/2) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 10/27*sqrt(-2*x + 1) +
1/81*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.212003, size = 100, normalized size = 1.08 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (1350 \, x^{3} + 1725 \, x^{2} - 110 \, x - 439\right )} \sqrt{-2 \, x + 1} + 106 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1701 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1701*sqrt(21)*(sqrt(21)*(1350*x^3 + 1725*x^2 - 110*x - 439)*sqrt(-2*x + 1) + 1
06*(3*x + 2)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(3*x + 2)

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Sympy [A]  time = 62.2676, size = 199, normalized size = 2.14 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{5}{2}}}{18} - \frac{725 \left (- 2 x + 1\right )^{\frac{3}{2}}}{162} + \frac{10 \sqrt{- 2 x + 1}}{27} + \frac{28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} + \frac{214 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(5/2)/18 - 725*(-2*x + 1)**(3/2)/162 + 10*sqrt(-2*x + 1)/27 + 28*
Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqrt(-
2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(
-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/81 + 214*Piecewise((-sqrt(21)*
acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*
sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/81

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GIAC/XCAS [A]  time = 0.21801, size = 122, normalized size = 1.31 \[ \frac{25}{18} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{725}{162} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{106}{1701} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{10}{27} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(2*x - 1)^2*sqrt(-2*x + 1) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*
ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 10/2
7*sqrt(-2*x + 1) + 1/81*sqrt(-2*x + 1)/(3*x + 2)