3.1992 \(\int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{67 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{\sqrt{1-2 x}}{42 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

Sqrt[1 - 2*x]/(42*(2 + 3*x)^2) - (67*Sqrt[1 - 2*x])/(294*(2 + 3*x)) - (67*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

_______________________________________________________________________________________

Rubi [A]  time = 0.0691723, antiderivative size = 68, 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 \[ -\frac{67 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{\sqrt{1-2 x}}{42 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(42*(2 + 3*x)^2) - (67*Sqrt[1 - 2*x])/(294*(2 + 3*x)) - (67*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.89843, size = 56, normalized size = 0.82 \[ - \frac{67 \sqrt{- 2 x + 1}}{294 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1}}{42 \left (3 x + 2\right )^{2}} - \frac{67 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3087} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67*sqrt(-2*x + 1)/(294*(3*x + 2)) + sqrt(-2*x + 1)/(42*(3*x + 2)**2) - 67*sqrt(
21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3087

_______________________________________________________________________________________

Mathematica [A]  time = 0.0951428, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{1-2 x} (201 x+127)}{294 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(127 + 201*x))/(294*(2 + 3*x)^2) - (67*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(147*Sqrt[21])

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 48, normalized size = 0.7 \[ -36\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{67\, \left ( 1-2\,x \right ) ^{3/2}}{1764}}+{\frac{65\,\sqrt{1-2\,x}}{756}} \right ) }-{\frac{67\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36*(-67/1764*(1-2*x)^(3/2)+65/756*(1-2*x)^(1/2))/(-4-6*x)^2-67/3087*arctanh(1/7
*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50663, size = 100, normalized size = 1.47 \[ \frac{67}{6174} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{201 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 455 \, \sqrt{-2 \, x + 1}}{147 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/6174*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 1/147*(201*(-2*x + 1)^(3/2) - 455*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7
)

_______________________________________________________________________________________

Fricas [A]  time = 0.23157, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (201 \, x + 127\right )} \sqrt{-2 \, x + 1} - 67 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{6174 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6174*sqrt(21)*(sqrt(21)*(201*x + 127)*sqrt(-2*x + 1) - 67*(9*x^2 + 12*x + 4)*
log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(9*x^2 + 12*x + 4)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220194, size = 92, normalized size = 1.35 \[ \frac{67}{6174} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{201 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 455 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/6174*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/588*(201*(-2*x + 1)^(3/2) - 455*sqrt(-2*x + 1))/(3*x + 2)^2