3.2124 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2}{49 \sqrt{1-2 x}}+\frac{11}{21 (1-2 x)^{3/2}}+\frac{2}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

11/(21*(1 - 2*x)^(3/2)) - 2/(49*Sqrt[1 - 2*x]) + (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/49

_______________________________________________________________________________________

Rubi [A]  time = 0.0688882, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2}{49 \sqrt{1-2 x}}+\frac{11}{21 (1-2 x)^{3/2}}+\frac{2}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

11/(21*(1 - 2*x)^(3/2)) - 2/(49*Sqrt[1 - 2*x]) + (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/49

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.87967, size = 48, normalized size = 0.86 \[ \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{2}{49 \sqrt{- 2 x + 1}} + \frac{11}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 2/(49*sqrt(-2*x + 1)) + 11/(21
*(-2*x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.107154, size = 52, normalized size = 0.93 \[ \frac{84 x-6 \sqrt{21-42 x} (2 x-1) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+497}{1029 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(497 + 84*x - 6*Sqrt[21 - 42*x]*(-1 + 2*x)*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(10
29*(1 - 2*x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 38, normalized size = 0.7 \[{\frac{11}{21} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2}{49}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/21/(1-2*x)^(3/2)+2/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2/49/(1-2
*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50041, size = 69, normalized size = 1.23 \[ -\frac{1}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12 \, x + 71}{147 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) + 1/147*(12*x + 71)/(-2*x + 1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.21312, size = 105, normalized size = 1.88 \[ \frac{\sqrt{7}{\left (3 \, \sqrt{3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{7}{\left (12 \, x + 71\right )}\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(7)*(3*sqrt(3)*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sq
rt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(7)*(12*x + 71))/((2*x - 1)*sqrt(-2*x + 1
))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x + 3}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((5*x + 3)/((-2*x + 1)**(5/2)*(3*x + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.251697, size = 82, normalized size = 1.46 \[ -\frac{1}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{12 \, x + 71}{147 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 1/147*(12*x + 71)/((2*x - 1)*sqrt(-2*x + 1))