3.1791 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{2+3 x} \, dx\)

Optimal. Leaf size=69 \[ \frac{5}{6} (1-2 x)^{5/2}-\frac{155}{54} (1-2 x)^{3/2}+\frac{2}{27} \sqrt{1-2 x}-\frac{2}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/27 - (155*(1 - 2*x)^(3/2))/54 + (5*(1 - 2*x)^(5/2))/6 - (2*Sqr
t[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi [A]  time = 0.0852035, 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 \[ \frac{5}{6} (1-2 x)^{5/2}-\frac{155}{54} (1-2 x)^{3/2}+\frac{2}{27} \sqrt{1-2 x}-\frac{2}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/27 - (155*(1 - 2*x)^(3/2))/54 + (5*(1 - 2*x)^(5/2))/6 - (2*Sqr
t[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi in Sympy [A]  time = 8.64901, size = 60, normalized size = 0.87 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{5}{2}}}{6} - \frac{155 \left (- 2 x + 1\right )^{\frac{3}{2}}}{54} + \frac{2 \sqrt{- 2 x + 1}}{27} - \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(5/2)/6 - 155*(-2*x + 1)**(3/2)/54 + 2*sqrt(-2*x + 1)/27 - 2*sqrt(
21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/81

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Mathematica [A]  time = 0.0626629, size = 51, normalized size = 0.74 \[ \frac{1}{81} \left (3 \sqrt{1-2 x} \left (90 x^2+65 x-53\right )-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-53 + 65*x + 90*x^2) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/81

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Maple [A]  time = 0.009, size = 47, normalized size = 0.7 \[ -{\frac{155}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{5}{6} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{21}}{81}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{2}{27}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-155/54*(1-2*x)^(3/2)+5/6*(1-2*x)^(5/2)-2/81*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
*21^(1/2)+2/27*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50432, size = 86, normalized size = 1.25 \[ \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{155}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{81} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{27} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*(-2*x + 1)^(5/2) - 155/54*(-2*x + 1)^(3/2) + 1/81*sqrt(21)*log(-(sqrt(21) -
3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/27*sqrt(-2*x + 1)

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Fricas [A]  time = 0.212784, size = 84, normalized size = 1.22 \[ \frac{1}{81} \, \sqrt{3}{\left (\sqrt{3}{\left (90 \, x^{2} + 65 \, x - 53\right )} \sqrt{-2 \, x + 1} + \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/81*sqrt(3)*(sqrt(3)*(90*x^2 + 65*x - 53)*sqrt(-2*x + 1) + sqrt(7)*log((sqrt(3)
*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 6.42481, size = 99, normalized size = 1.43 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{5}{2}}}{6} - \frac{155 \left (- 2 x + 1\right )^{\frac{3}{2}}}{54} + \frac{2 \sqrt{- 2 x + 1}}{27} + \frac{14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(5/2)/6 - 155*(-2*x + 1)**(3/2)/54 + 2*sqrt(-2*x + 1)/27 + 14*Piec
ewise((-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))/27

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GIAC/XCAS [A]  time = 0.21951, size = 100, normalized size = 1.45 \[ \frac{5}{6} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{155}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{81} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{27} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*(2*x - 1)^2*sqrt(-2*x + 1) - 155/54*(-2*x + 1)^(3/2) + 1/81*sqrt(21)*ln(1/2*
abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/27*sqrt(-
2*x + 1)