3.2125 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

20/(147*(1 - 2*x)^(3/2)) + 60/(343*Sqrt[1 - 2*x]) + 1/(21*(1 - 2*x)^(3/2)*(2 + 3
*x)) - (60*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

_______________________________________________________________________________________

Rubi [A]  time = 0.0871042, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

20/(147*(1 - 2*x)^(3/2)) + 60/(343*Sqrt[1 - 2*x]) + 1/(21*(1 - 2*x)^(3/2)*(2 + 3
*x)) - (60*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.37323, size = 65, normalized size = 0.86 \[ - \frac{60 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{60}{343 \sqrt{- 2 x + 1}} + \frac{20}{147 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{1}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-60*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 60/(343*sqrt(-2*x + 1)) + 2
0/(147*(-2*x + 1)**(3/2)) + 1/(21*(-2*x + 1)**(3/2)*(3*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.116484, size = 58, normalized size = 0.76 \[ \frac{\frac{7 \left (-1080 x^2+240 x+689\right )}{(1-2 x)^{3/2} (3 x+2)}-180 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203} \]

Antiderivative was successfully verified.

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

[Out]

((7*(689 + 240*x - 1080*x^2))/((1 - 2*x)^(3/2)*(2 + 3*x)) - 180*Sqrt[21]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/7203

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 54, normalized size = 0.7 \[{\frac{22}{147} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{62}{343}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{60\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/147/(1-2*x)^(3/2)+62/343/(1-2*x)^(1/2)-2/343*(1-2*x)^(1/2)/(-4/3-2*x)-60/2401
*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.4917, size = 100, normalized size = 1.32 \[ \frac{30}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (270 \,{\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 2/1029*(270*(2*x - 1)^2 + 840*x - 959)/(3*(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(
3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.2222, size = 119, normalized size = 1.57 \[ \frac{\sqrt{7}{\left (90 \, \sqrt{3}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1080 \, x^{2} - 240 \, x - 689\right )}\right )}}{7203 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7203*sqrt(7)*(90*sqrt(3)*(6*x^2 + x - 2)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5)
+ 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(1080*x^2 - 240*x - 689))/((6*
x^2 + x - 2)*sqrt(-2*x + 1))

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.244749, size = 104, normalized size = 1.37 \[ \frac{30}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (93 \, x - 85\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{3 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 4/1029*(93*x - 85)/((2*x - 1)*sqrt(-2*x + 1)) + 3/343*sqrt(-2*x + 1
)/(3*x + 2)