3.1847 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=69 \[ -\frac{1}{3} (1-2 x)^{5/2}-\frac{2}{27} (1-2 x)^{3/2}-\frac{14}{27} \sqrt{1-2 x}+\frac{14}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-14*Sqrt[1 - 2*x])/27 - (2*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(5/2)/3 + (14*Sqrt[7
/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi [A]  time = 0.0740745, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{3} (1-2 x)^{5/2}-\frac{2}{27} (1-2 x)^{3/2}-\frac{14}{27} \sqrt{1-2 x}+\frac{14}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-14*Sqrt[1 - 2*x])/27 - (2*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(5/2)/3 + (14*Sqrt[7
/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Rubi in Sympy [A]  time = 7.88838, size = 58, normalized size = 0.84 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{3} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{14 \sqrt{- 2 x + 1}}{27} + \frac{14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/3 - 2*(-2*x + 1)**(3/2)/27 - 14*sqrt(-2*x + 1)/27 + 14*sqrt(2
1)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/81

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Mathematica [A]  time = 0.060519, size = 51, normalized size = 0.74 \[ \frac{1}{81} \left (14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-3 \sqrt{1-2 x} \left (36 x^2-40 x+25\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*(25 - 40*x + 36*x^2) + 14*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 \[ -{\frac{2}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{14\,\sqrt{21}}{81}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{14}{27}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(1-2*x)^(3/2)-1/3*(1-2*x)^(5/2)+14/81*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*
21^(1/2)-14/27*(1-2*x)^(1/2)

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Maxima [A]  time = 1.5807, size = 86, normalized size = 1.25 \[ -\frac{1}{3} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7}{81} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{14}{27} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-2*x + 1)^(5/2) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*log(-(sqrt(21) - 3
*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/27*sqrt(-2*x + 1)

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Fricas [A]  time = 0.22607, size = 85, normalized size = 1.23 \[ -\frac{1}{81} \, \sqrt{3}{\left (\sqrt{3}{\left (36 \, x^{2} - 40 \, x + 25\right )} \sqrt{-2 \, x + 1} - 7 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/81*sqrt(3)*(sqrt(3)*(36*x^2 - 40*x + 25)*sqrt(-2*x + 1) - 7*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 = 14.2373, size = 99, normalized size = 1.43 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{3} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{14 \sqrt{- 2 x + 1}}{27} - \frac{98 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/3 - 2*(-2*x + 1)**(3/2)/27 - 14*sqrt(-2*x + 1)/27 - 98*Piecew
ise((-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 = 0.234531, size = 100, normalized size = 1.45 \[ -\frac{1}{3} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{2}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7}{81} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{14}{27} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*x - 1)^2*sqrt(-2*x + 1) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*ln(1/2*a
bs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/27*sqrt(-
2*x + 1)