3.1932 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=109 \[ \frac{143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac{(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac{211}{441} (1-2 x)^{5/2}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{1055}{81} \sqrt{1-2 x}-\frac{1055}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(1055*Sqrt[1 - 2*x])/81 + (1055*(1 - 2*x)^(3/2))/567 + (211*(1 - 2*x)^(5/2))/441
- (1 - 2*x)^(7/2)/(126*(2 + 3*x)^2) + (143*(1 - 2*x)^(7/2))/(882*(2 + 3*x)) - (
1055*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

_______________________________________________________________________________________

Rubi [A]  time = 0.137634, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac{(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac{211}{441} (1-2 x)^{5/2}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{1055}{81} \sqrt{1-2 x}-\frac{1055}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1055*Sqrt[1 - 2*x])/81 + (1055*(1 - 2*x)^(3/2))/567 + (211*(1 - 2*x)^(5/2))/441
- (1 - 2*x)^(7/2)/(126*(2 + 3*x)^2) + (143*(1 - 2*x)^(7/2))/(882*(2 + 3*x)) - (
1055*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.7367, size = 92, normalized size = 0.84 \[ \frac{143 \left (- 2 x + 1\right )^{\frac{7}{2}}}{882 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{126 \left (3 x + 2\right )^{2}} + \frac{211 \left (- 2 x + 1\right )^{\frac{5}{2}}}{441} + \frac{1055 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567} + \frac{1055 \sqrt{- 2 x + 1}}{81} - \frac{1055 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

143*(-2*x + 1)**(7/2)/(882*(3*x + 2)) - (-2*x + 1)**(7/2)/(126*(3*x + 2)**2) + 2
11*(-2*x + 1)**(5/2)/441 + 1055*(-2*x + 1)**(3/2)/567 + 1055*sqrt(-2*x + 1)/81 -
1055*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/243

_______________________________________________________________________________________

Mathematica [A]  time = 0.116225, size = 68, normalized size = 0.62 \[ \frac{1}{486} \left (\frac{3 \sqrt{1-2 x} \left (2160 x^4-3960 x^3+12828 x^2+25987 x+10007\right )}{(3 x+2)^2}-2110 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((3*Sqrt[1 - 2*x]*(10007 + 25987*x + 12828*x^2 - 3960*x^3 + 2160*x^4))/(2 + 3*x)
^2 - 2110*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/486

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 75, normalized size = 0.7 \[{\frac{10}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{130}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1006}{81}\sqrt{1-2\,x}}+{\frac{14}{9\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{149}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{343}{18}\sqrt{1-2\,x}} \right ) }-{\frac{1055\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(1-2*x)^(5/2)+130/81*(1-2*x)^(3/2)+1006/81*(1-2*x)^(1/2)+14/9*(-149/18*(1-
2*x)^(3/2)+343/18*(1-2*x)^(1/2))/(-4-6*x)^2-1055/243*arctanh(1/7*21^(1/2)*(1-2*x
)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49283, size = 136, normalized size = 1.25 \[ \frac{10}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{130}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1055}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1006}{81} \, \sqrt{-2 \, x + 1} - \frac{7 \,{\left (149 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(-2*x + 1)^(5/2) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt(21)*log(-(sqrt(
21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1006/81*sqrt(-2*x + 1)
- 7/81*(149*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

_______________________________________________________________________________________

Fricas [A]  time = 0.2221, size = 128, normalized size = 1.17 \[ \frac{\sqrt{3}{\left (1055 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt{-2 \, x + 1}\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/486*sqrt(3)*(1055*sqrt(7)*(9*x^2 + 12*x + 4)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7
)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(3)*(2160*x^4 - 3960*x^3 + 12828*x^2 + 25987*
x + 10007)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213631, size = 138, normalized size = 1.27 \[ \frac{10}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{130}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1055}{486} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1006}{81} \, \sqrt{-2 \, x + 1} - \frac{7 \,{\left (149 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/27*(2*x - 1)^2*sqrt(-2*x + 1) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt(21)*l
n(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1006/
81*sqrt(-2*x + 1) - 7/324*(149*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1))/(3*x + 2)^
2