3.1793 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=81 \[ \frac{139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac{(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac{863}{441} \sqrt{1-2 x}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

[Out]

(863*Sqrt[1 - 2*x])/441 - (1 - 2*x)^(3/2)/(126*(2 + 3*x)^2) + (139*(1 - 2*x)^(3/
2))/(882*(2 + 3*x)) - (863*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.0934741, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac{(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac{863}{441} \sqrt{1-2 x}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(863*Sqrt[1 - 2*x])/441 - (1 - 2*x)^(3/2)/(126*(2 + 3*x)^2) + (139*(1 - 2*x)^(3/
2))/(882*(2 + 3*x)) - (863*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.2312, size = 68, normalized size = 0.84 \[ \frac{139 \left (- 2 x + 1\right )^{\frac{3}{2}}}{882 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )^{2}} + \frac{863 \sqrt{- 2 x + 1}}{441} - \frac{863 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1323} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

139*(-2*x + 1)**(3/2)/(882*(3*x + 2)) - (-2*x + 1)**(3/2)/(126*(3*x + 2)**2) + 8
63*sqrt(-2*x + 1)/441 - 863*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1323

_______________________________________________________________________________________

Mathematica [A]  time = 0.096089, size = 58, normalized size = 0.72 \[ \frac{\sqrt{1-2 x} \left (2100 x^2+2941 x+1025\right )}{126 (3 x+2)^2}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(1025 + 2941*x + 2100*x^2))/(126*(2 + 3*x)^2) - (863*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 57, normalized size = 0.7 \[{\frac{50}{27}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{47}{14} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{139}{18}\sqrt{1-2\,x}} \right ) }-{\frac{863\,\sqrt{21}}{1323}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/27*(1-2*x)^(1/2)+2/3*(-47/14*(1-2*x)^(3/2)+139/18*(1-2*x)^(1/2))/(-4-6*x)^2-8
63/1323*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51689, size = 112, normalized size = 1.38 \[ \frac{863}{2646} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{50}{27} \, \sqrt{-2 \, x + 1} - \frac{423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 973 \, \sqrt{-2 \, x + 1}}{189 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

863/2646*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 50/27*sqrt(-2*x + 1) - 1/189*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x + 1))/(
9*(2*x - 1)^2 + 84*x + 7)

_______________________________________________________________________________________

Fricas [A]  time = 0.221341, size = 107, normalized size = 1.32 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt{-2 \, x + 1} + 863 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{2646 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2646*sqrt(21)*(sqrt(21)*(2100*x^2 + 2941*x + 1025)*sqrt(-2*x + 1) + 863*(9*x^2
+ 12*x + 4)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(9*x^2 + 1
2*x + 4)

_______________________________________________________________________________________

Sympy [A]  time = 125.215, size = 323, normalized size = 3.99 \[ \frac{50 \sqrt{- 2 x + 1}}{27} + \frac{32 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{3} + \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{27} + \frac{130 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50*sqrt(-2*x + 1)/27 + 32*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 -
1)/4 + log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 +
1)) - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/3 +
56*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)
*sqrt(-2*x + 1)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt
(21)*sqrt(-2*x + 1)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*
(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/27 + 130*Pi
ecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(
21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210964, size = 104, normalized size = 1.28 \[ \frac{863}{2646} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{50}{27} \, \sqrt{-2 \, x + 1} - \frac{423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 973 \, \sqrt{-2 \, x + 1}}{756 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

863/2646*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 50/27*sqrt(-2*x + 1) - 1/756*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x
+ 1))/(3*x + 2)^2