3.1924 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac{41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{205 \sqrt{1-2 x}}{444528 (3 x+2)}+\frac{205 \sqrt{1-2 x}}{190512 (3 x+2)^2}-\frac{205 \sqrt{1-2 x}}{13608 (3 x+2)^3}+\frac{205 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

[Out]

(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) - (41*(1 - 2*x)^(5/2))/(378*(2 + 3*x)^5) + (20
5*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) - (205*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3)
+ (205*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (205*Sqrt[1 - 2*x])/(444528*(2 + 3*
x)) + (205*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.158577, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac{41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac{205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{205 \sqrt{1-2 x}}{444528 (3 x+2)}+\frac{205 \sqrt{1-2 x}}{190512 (3 x+2)^2}-\frac{205 \sqrt{1-2 x}}{13608 (3 x+2)^3}+\frac{205 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) - (41*(1 - 2*x)^(5/2))/(378*(2 + 3*x)^5) + (20
5*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) - (205*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3)
+ (205*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (205*Sqrt[1 - 2*x])/(444528*(2 + 3*
x)) + (205*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.9173, size = 131, normalized size = 0.89 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{126 \left (3 x + 2\right )^{6}} - \frac{41 \left (- 2 x + 1\right )^{\frac{5}{2}}}{378 \left (3 x + 2\right )^{5}} + \frac{205 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4536 \left (3 x + 2\right )^{4}} + \frac{205 \sqrt{- 2 x + 1}}{444528 \left (3 x + 2\right )} + \frac{205 \sqrt{- 2 x + 1}}{190512 \left (3 x + 2\right )^{2}} - \frac{205 \sqrt{- 2 x + 1}}{13608 \left (3 x + 2\right )^{3}} + \frac{205 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{4667544} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**7,x)

[Out]

(-2*x + 1)**(7/2)/(126*(3*x + 2)**6) - 41*(-2*x + 1)**(5/2)/(378*(3*x + 2)**5) +
205*(-2*x + 1)**(3/2)/(4536*(3*x + 2)**4) + 205*sqrt(-2*x + 1)/(444528*(3*x + 2
)) + 205*sqrt(-2*x + 1)/(190512*(3*x + 2)**2) - 205*sqrt(-2*x + 1)/(13608*(3*x +
2)**3) + 205*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/4667544

_______________________________________________________________________________________

Mathematica [A]  time = 0.122311, size = 73, normalized size = 0.49 \[ \frac{\frac{21 \sqrt{1-2 x} \left (49815 x^5+204795 x^4-824526 x^3-176850 x^2+154312 x-51904\right )}{(3 x+2)^6}+410 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9335088} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

((21*Sqrt[1 - 2*x]*(-51904 + 154312*x - 176850*x^2 - 824526*x^3 + 204795*x^4 + 4
9815*x^5))/(2 + 3*x)^6 + 410*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9335088

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 84, normalized size = 0.6 \[ -46656\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ({\frac{205\, \left ( 1-2\,x \right ) ^{11/2}}{42674688}}-{\frac{3485\, \left ( 1-2\,x \right ) ^{9/2}}{54867456}}-{\frac{439\, \left ( 1-2\,x \right ) ^{7/2}}{3919104}}+{\frac{451\, \left ( 1-2\,x \right ) ^{5/2}}{559872}}-{\frac{24395\, \left ( 1-2\,x \right ) ^{3/2}}{30233088}}+{\frac{10045\,\sqrt{1-2\,x}}{30233088}} \right ) }+{\frac{205\,\sqrt{21}}{4667544}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^7,x)

[Out]

-46656*(205/42674688*(1-2*x)^(11/2)-3485/54867456*(1-2*x)^(9/2)-439/3919104*(1-2
*x)^(7/2)+451/559872*(1-2*x)^(5/2)-24395/30233088*(1-2*x)^(3/2)+10045/30233088*(
1-2*x)^(1/2))/(-4-6*x)^6+205/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)

_______________________________________________________________________________________

Maxima [A]  time = 1.49485, size = 197, normalized size = 1.33 \[ -\frac{205}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49815 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 658665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 1161594 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 8353422 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 8367485 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 3445435 \, \sqrt{-2 \, x + 1}}{222264 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

-205/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/222264*(49815*(-2*x + 1)^(11/2) - 658665*(-2*x + 1)^(9/2) - 1161594*
(-2*x + 1)^(7/2) + 8353422*(-2*x + 1)^(5/2) - 8367485*(-2*x + 1)^(3/2) + 3445435
*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 1852
20*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

_______________________________________________________________________________________

Fricas [A]  time = 0.236599, size = 181, normalized size = 1.22 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt{-2 \, x + 1} + 205 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{9335088 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/9335088*sqrt(21)*(sqrt(21)*(49815*x^5 + 204795*x^4 - 824526*x^3 - 176850*x^2 +
154312*x - 51904)*sqrt(-2*x + 1) + 205*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^
3 + 2160*x^2 + 576*x + 64)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2
)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

_______________________________________________________________________________________

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)**(5/2)*(3+5*x)/(2+3*x)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240267, size = 178, normalized size = 1.2 \[ -\frac{205}{9335088} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{49815 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 658665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 1161594 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 8353422 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 8367485 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3445435 \, \sqrt{-2 \, x + 1}}{14224896 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

-205/9335088*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 1/14224896*(49815*(2*x - 1)^5*sqrt(-2*x + 1) + 658665*(2*x - 1
)^4*sqrt(-2*x + 1) - 1161594*(2*x - 1)^3*sqrt(-2*x + 1) - 8353422*(2*x - 1)^2*sq
rt(-2*x + 1) + 8367485*(-2*x + 1)^(3/2) - 3445435*sqrt(-2*x + 1))/(3*x + 2)^6