3.1872 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=118 \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-31*Sqrt[1 - 2*x]*(3 + 5*x)^2)/3 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(6*(2 + 3*x)^2
) + (6*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x) + (Sqrt[1 - 2*x]*(367 + 1715*x))/54
+ (887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi [A]  time = 0.203561, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-31*Sqrt[1 - 2*x]*(3 + 5*x)^2)/3 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(6*(2 + 3*x)^2
) + (6*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x) + (Sqrt[1 - 2*x]*(367 + 1715*x))/54
+ (887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi in Sympy [A]  time = 20.1884, size = 97, normalized size = 0.82 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (192375 x + 56925\right )}{17010} - \frac{6 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{6 \left (3 x + 2\right )^{2}} - \frac{887 \sqrt{- 2 x + 1}}{189} + \frac{887 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{567} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*(192375*x + 56925)/17010 - 6*(-2*x + 1)**(3/2)*(5*x + 3)**2/(7
*(3*x + 2)) - (-2*x + 1)**(3/2)*(5*x + 3)**3/(6*(3*x + 2)**2) - 887*sqrt(-2*x +
1)/189 + 887*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/567

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Mathematica [A]  time = 0.107982, size = 63, normalized size = 0.53 \[ \frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (1800 x^4+570 x^2+2965 x+1367\right )}{54 (3 x+2)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(1367 + 2965*x + 570*x^2 + 1800*x^4))/(54*(2 + 3*x)^2) + (887*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ -{\frac{25}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{370}{81}\sqrt{1-2\,x}}-{\frac{2}{9\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{215}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{497}{18}\sqrt{1-2\,x}} \right ) }+{\frac{887\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(1-2*x)^(5/2)-50/81*(1-2*x)^(3/2)-370/81*(1-2*x)^(1/2)-2/9*(-215/18*(1-2*
x)^(3/2)+497/18*(1-2*x)^(1/2))/(-4-6*x)^2+887/567*arctanh(1/7*21^(1/2)*(1-2*x)^(
1/2))*21^(1/2)

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Maxima [A]  time = 1.49383, size = 136, normalized size = 1.15 \[ -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(-2*x + 1)^(5/2) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt(21)*log(-(sqrt(
21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 370/81*sqrt(-2*x + 1) +
1/81*(215*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.211789, size = 113, normalized size = 0.96 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1800 \, x^{4} + 570 \, x^{2} + 2965 \, x + 1367\right )} \sqrt{-2 \, x + 1} - 887 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1134 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1134*sqrt(21)*(sqrt(21)*(1800*x^4 + 570*x^2 + 2965*x + 1367)*sqrt(-2*x + 1) -
887*(9*x^2 + 12*x + 4)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))
/(9*x^2 + 12*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223522, size = 138, normalized size = 1.17 \[ -\frac{25}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(2*x - 1)^2*sqrt(-2*x + 1) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt(21)*l
n(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 370/8
1*sqrt(-2*x + 1) + 1/324*(215*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(3*x + 2)^2