3.1812 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{27}{140} (1-2 x)^{7/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{2}{625} \sqrt{1-2 x}-\frac{2}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2))/125 -
(27*(1 - 2*x)^(7/2))/140 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi [A]  time = 0.0887274, antiderivative size = 82, 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 \[ -\frac{27}{140} (1-2 x)^{7/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{2}{625} \sqrt{1-2 x}-\frac{2}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2))/125 -
(27*(1 - 2*x)^(7/2))/140 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi in Sympy [A]  time = 9.34449, size = 71, normalized size = 0.87 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{7}{2}}}{140} + \frac{162 \left (- 2 x + 1\right )^{\frac{5}{2}}}{125} - \frac{1299 \left (- 2 x + 1\right )^{\frac{3}{2}}}{500} + \frac{2 \sqrt{- 2 x + 1}}{625} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(7/2)/140 + 162*(-2*x + 1)**(5/2)/125 - 1299*(-2*x + 1)**(3/2)/5
00 + 2*sqrt(-2*x + 1)/625 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3125

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Mathematica [A]  time = 0.0690639, size = 56, normalized size = 0.68 \[ \frac{5 \sqrt{1-2 x} \left (6750 x^3+12555 x^2+5115 x-6526\right )-14 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{21875} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-6526 + 5115*x + 12555*x^2 + 6750*x^3) - 14*Sqrt[55]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/21875

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Maple [A]  time = 0.01, size = 56, normalized size = 0.7 \[ -{\frac{1299}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{162}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{27}{140} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{625}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1299/500*(1-2*x)^(3/2)+162/125*(1-2*x)^(5/2)-27/140*(1-2*x)^(7/2)-2/3125*arctan
h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/625*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49774, size = 99, normalized size = 1.21 \[ -\frac{27}{140} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{162}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1299}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/140*(-2*x + 1)^(7/2) + 162/125*(-2*x + 1)^(5/2) - 1299/500*(-2*x + 1)^(3/2)
+ 1/3125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 2/625*sqrt(-2*x + 1)

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Fricas [A]  time = 0.213524, size = 92, normalized size = 1.12 \[ \frac{1}{21875} \, \sqrt{5}{\left (\sqrt{5}{\left (6750 \, x^{3} + 12555 \, x^{2} + 5115 \, x - 6526\right )} \sqrt{-2 \, x + 1} + 7 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21875*sqrt(5)*(sqrt(5)*(6750*x^3 + 12555*x^2 + 5115*x - 6526)*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 = 6.94827, size = 110, normalized size = 1.34 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{7}{2}}}{140} + \frac{162 \left (- 2 x + 1\right )^{\frac{5}{2}}}{125} - \frac{1299 \left (- 2 x + 1\right )^{\frac{3}{2}}}{500} + \frac{2 \sqrt{- 2 x + 1}}{625} + \frac{22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(7/2)/140 + 162*(-2*x + 1)**(5/2)/125 - 1299*(-2*x + 1)**(3/2)/5
00 + 2*sqrt(-2*x + 1)/625 + 22*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 = 0.247271, size = 122, normalized size = 1.49 \[ \frac{27}{140} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{162}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1299}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{3125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/140*(2*x - 1)^3*sqrt(-2*x + 1) + 162/125*(2*x - 1)^2*sqrt(-2*x + 1) - 1299/50
0*(-2*x + 1)^(3/2) + 1/3125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))
/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/625*sqrt(-2*x + 1)