3.1850 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ \frac{(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac{52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac{52 \sqrt{1-2 x}}{189 (3 x+2)}-\frac{104 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

[Out]

(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) - (52*(1 - 2*x)^(3/2))/(189*(2 + 3*x)^2) + (52*
Sqrt[1 - 2*x])/(189*(2 + 3*x)) - (104*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqr
t[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.0802553, antiderivative size = 88, 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 \[ \frac{(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac{52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac{52 \sqrt{1-2 x}}{189 (3 x+2)}-\frac{104 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) - (52*(1 - 2*x)^(3/2))/(189*(2 + 3*x)^2) + (52*
Sqrt[1 - 2*x])/(189*(2 + 3*x)) - (104*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqr
t[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.93894, size = 75, normalized size = 0.85 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{63 \left (3 x + 2\right )^{3}} - \frac{52 \left (- 2 x + 1\right )^{\frac{3}{2}}}{189 \left (3 x + 2\right )^{2}} + \frac{52 \sqrt{- 2 x + 1}}{189 \left (3 x + 2\right )} - \frac{104 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3969} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)/(2+3*x)**4,x)

[Out]

(-2*x + 1)**(5/2)/(63*(3*x + 2)**3) - 52*(-2*x + 1)**(3/2)/(189*(3*x + 2)**2) +
52*sqrt(-2*x + 1)/(189*(3*x + 2)) - 104*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7
)/3969

_______________________________________________________________________________________

Mathematica [A]  time = 0.0907821, size = 58, normalized size = 0.66 \[ \frac{\frac{21 \sqrt{1-2 x} \left (792 x^2+664 x+107\right )}{(3 x+2)^3}-104 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

((21*Sqrt[1 - 2*x]*(107 + 664*x + 792*x^2))/(2 + 3*x)^3 - 104*Sqrt[21]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/3969

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 57, normalized size = 0.7 \[ 216\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{22\, \left ( 1-2\,x \right ) ^{5/2}}{567}}+{\frac{104\, \left ( 1-2\,x \right ) ^{3/2}}{729}}-{\frac{91\,\sqrt{1-2\,x}}{729}} \right ) }-{\frac{104\,\sqrt{21}}{3969}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x)

[Out]

216*(-22/567*(1-2*x)^(5/2)+104/729*(1-2*x)^(3/2)-91/729*(1-2*x)^(1/2))/(-4-6*x)^
3-104/3969*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50705, size = 124, normalized size = 1.41 \[ \frac{52}{3969} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8 \,{\left (198 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 728 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 637 \, \sqrt{-2 \, x + 1}\right )}}{189 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

52/3969*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 8/189*(198*(-2*x + 1)^(5/2) - 728*(-2*x + 1)^(3/2) + 637*sqrt(-2*x + 1))/(2
7*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

_______________________________________________________________________________________

Fricas [A]  time = 0.21725, size = 120, normalized size = 1.36 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (792 \, x^{2} + 664 \, x + 107\right )} \sqrt{-2 \, x + 1} + 52 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3969 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

1/3969*sqrt(21)*(sqrt(21)*(792*x^2 + 664*x + 107)*sqrt(-2*x + 1) + 52*(27*x^3 +
54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(27*
x^3 + 54*x^2 + 36*x + 8)

_______________________________________________________________________________________

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)/(2+3*x)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21188, size = 113, normalized size = 1.28 \[ \frac{52}{3969} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{198 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 728 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 637 \, \sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^4,x, algorithm="giac")

[Out]

52/3969*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/189*(198*(2*x - 1)^2*sqrt(-2*x + 1) - 728*(-2*x + 1)^(3/2) + 637*
sqrt(-2*x + 1))/(3*x + 2)^3