3.1822 \(\int \frac{\sqrt{1-2 x} (2+3 x)^4}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{5 (5 x+3)}+\frac{27}{175} \sqrt{1-2 x} (3 x+2)^3+\frac{12}{625} \sqrt{1-2 x} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (375 x+1256)}{3125}-\frac{262 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

[Out]

(12*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 + (27*Sqrt[1 - 2*x]*(2 + 3*x)^3)/175 - (Sqrt[
1 - 2*x]*(2 + 3*x)^4)/(5*(3 + 5*x)) - (3*Sqrt[1 - 2*x]*(1256 + 375*x))/3125 - (2
62*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.20314, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{5 (5 x+3)}+\frac{27}{175} \sqrt{1-2 x} (3 x+2)^3+\frac{12}{625} \sqrt{1-2 x} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (375 x+1256)}{3125}-\frac{262 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(12*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 + (27*Sqrt[1 - 2*x]*(2 + 3*x)^3)/175 - (Sqrt[
1 - 2*x]*(2 + 3*x)^4)/(5*(3 + 5*x)) - (3*Sqrt[1 - 2*x]*(1256 + 375*x))/3125 - (2
62*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.8669, size = 97, normalized size = 0.86 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{5 \left (5 x + 3\right )} + \frac{27 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{175} + \frac{12 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{625} - \frac{\sqrt{- 2 x + 1} \left (118125 x + 395640\right )}{328125} - \frac{262 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{171875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**4/(5*(5*x + 3)) + 27*sqrt(-2*x + 1)*(3*x + 2)**3/175
+ 12*sqrt(-2*x + 1)*(3*x + 2)**2/625 - sqrt(-2*x + 1)*(118125*x + 395640)/328125
- 262*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/171875

_______________________________________________________________________________________

Mathematica [A]  time = 0.107905, size = 68, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (101250 x^4+258525 x^3+206415 x^2-52485 x-63088\right )}{21875 (5 x+3)}-\frac{262 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-63088 - 52485*x + 206415*x^2 + 258525*x^3 + 101250*x^4))/(21875
*(3 + 5*x)) - (262*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 72, normalized size = 0.6 \[ -{\frac{81}{700} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{999}{1250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{4131}{2500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{24}{3125}\sqrt{1-2\,x}}+{\frac{2}{15625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{262\,\sqrt{55}}{171875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/700*(1-2*x)^(7/2)+999/1250*(1-2*x)^(5/2)-4131/2500*(1-2*x)^(3/2)+24/3125*(1-
2*x)^(1/2)+2/15625*(1-2*x)^(1/2)/(-6/5-2*x)-262/171875*arctanh(1/11*55^(1/2)*(1-
2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.52924, size = 120, normalized size = 1.06 \[ -\frac{81}{700} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{999}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{4131}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{131}{171875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{24}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/700*(-2*x + 1)^(7/2) + 999/1250*(-2*x + 1)^(5/2) - 4131/2500*(-2*x + 1)^(3/2
) + 131/171875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 24/3125*sqrt(-2*x + 1) - 1/3125*sqrt(-2*x + 1)/(5*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.211903, size = 107, normalized size = 0.95 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (101250 \, x^{4} + 258525 \, x^{3} + 206415 \, x^{2} - 52485 \, x - 63088\right )} \sqrt{-2 \, x + 1} + 917 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{1203125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1203125*sqrt(55)*(sqrt(55)*(101250*x^4 + 258525*x^3 + 206415*x^2 - 52485*x - 6
3088)*sqrt(-2*x + 1) + 917*(5*x + 3)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1)
)/(5*x + 3)))/(5*x + 3)

_______________________________________________________________________________________

Sympy [A]  time = 72.615, size = 211, normalized size = 1.87 \[ - \frac{81 \left (- 2 x + 1\right )^{\frac{7}{2}}}{700} + \frac{999 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1250} - \frac{4131 \left (- 2 x + 1\right )^{\frac{3}{2}}}{2500} + \frac{24 \sqrt{- 2 x + 1}}{3125} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{3125} + \frac{52 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81*(-2*x + 1)**(7/2)/700 + 999*(-2*x + 1)**(5/2)/1250 - 4131*(-2*x + 1)**(3/2)/
2500 + 24*sqrt(-2*x + 1)/3125 - 44*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x
+ 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(-2*
x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x >
-3/5)))/3125 + 52*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2
*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5
))/625

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217459, size = 143, normalized size = 1.27 \[ \frac{81}{700} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{999}{1250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{4131}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{131}{171875} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{24}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/700*(2*x - 1)^3*sqrt(-2*x + 1) + 999/1250*(2*x - 1)^2*sqrt(-2*x + 1) - 4131/2
500*(-2*x + 1)^(3/2) + 131/171875*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 24/3125*sqrt(-2*x + 1) - 1/3125*sqrt(-2*
x + 1)/(5*x + 3)