3.1851 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ \frac{(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac{137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac{137 \sqrt{1-2 x}}{10584 (3 x+2)}+\frac{137 \sqrt{1-2 x}}{1512 (3 x+2)^2}-\frac{137 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{5292 \sqrt{21}} \]

[Out]

(1 - 2*x)^(5/2)/(84*(2 + 3*x)^4) - (137*(1 - 2*x)^(3/2))/(756*(2 + 3*x)^3) + (13
7*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2) - (137*Sqrt[1 - 2*x])/(10584*(2 + 3*x)) - (1
37*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5292*Sqrt[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.101946, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac{137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac{137 \sqrt{1-2 x}}{10584 (3 x+2)}+\frac{137 \sqrt{1-2 x}}{1512 (3 x+2)^2}-\frac{137 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{5292 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(1 - 2*x)^(5/2)/(84*(2 + 3*x)^4) - (137*(1 - 2*x)^(3/2))/(756*(2 + 3*x)^3) + (13
7*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2) - (137*Sqrt[1 - 2*x])/(10584*(2 + 3*x)) - (1
37*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5292*Sqrt[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.3066, size = 94, normalized size = 0.87 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{84 \left (3 x + 2\right )^{4}} - \frac{137 \left (- 2 x + 1\right )^{\frac{3}{2}}}{756 \left (3 x + 2\right )^{3}} - \frac{137 \sqrt{- 2 x + 1}}{10584 \left (3 x + 2\right )} + \frac{137 \sqrt{- 2 x + 1}}{1512 \left (3 x + 2\right )^{2}} - \frac{137 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{111132} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)/(84*(3*x + 2)**4) - 137*(-2*x + 1)**(3/2)/(756*(3*x + 2)**3) -
137*sqrt(-2*x + 1)/(10584*(3*x + 2)) + 137*sqrt(-2*x + 1)/(1512*(3*x + 2)**2) -
137*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/111132

_______________________________________________________________________________________

Mathematica [A]  time = 0.110492, size = 63, normalized size = 0.58 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (3699 x^3-13245 x^2-7990 x+970\right )}{(3 x+2)^4}-274 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(970 - 7990*x - 13245*x^2 + 3699*x^3))/(2 + 3*x)^4 - 274*Sqr
t[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/222264

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 66, normalized size = 0.6 \[ -1296\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{137\, \left ( 1-2\,x \right ) ^{7/2}}{254016}}-{\frac{733\, \left ( 1-2\,x \right ) ^{5/2}}{326592}}+{\frac{1507\, \left ( 1-2\,x \right ) ^{3/2}}{139968}}-{\frac{959\,\sqrt{1-2\,x}}{139968}} \right ) }-{\frac{137\,\sqrt{21}}{111132}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1296*(-137/254016*(1-2*x)^(7/2)-733/326592*(1-2*x)^(5/2)+1507/139968*(1-2*x)^(3
/2)-959/139968*(1-2*x)^(1/2))/(-4-6*x)^4-137/111132*arctanh(1/7*21^(1/2)*(1-2*x)
^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.53593, size = 149, normalized size = 1.38 \[ \frac{137}{222264} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{3699 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 15393 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 73843 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 46991 \, \sqrt{-2 \, x + 1}}{5292 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/222264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/5292*(3699*(-2*x + 1)^(7/2) + 15393*(-2*x + 1)^(5/2) - 73843*(-2*x + 1
)^(3/2) + 46991*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x -
1)^2 + 8232*x - 1715)

_______________________________________________________________________________________

Fricas [A]  time = 0.215433, size = 140, normalized size = 1.3 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (3699 \, x^{3} - 13245 \, x^{2} - 7990 \, x + 970\right )} \sqrt{-2 \, x + 1} - 137 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{222264 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/222264*sqrt(21)*(sqrt(21)*(3699*x^3 - 13245*x^2 - 7990*x + 970)*sqrt(-2*x + 1
) - 137*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) + 21*sq
rt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213111, size = 135, normalized size = 1.25 \[ \frac{137}{222264} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3699 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 15393 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 73843 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 46991 \, \sqrt{-2 \, x + 1}}{84672 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/222264*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 1/84672*(3699*(2*x - 1)^3*sqrt(-2*x + 1) - 15393*(2*x - 1)^2*sqr
t(-2*x + 1) + 73843*(-2*x + 1)^(3/2) - 46991*sqrt(-2*x + 1))/(3*x + 2)^4