3.2062 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=90 \[ -\frac{195 \sqrt{1-2 x}}{686 (3 x+2)}+\frac{65}{147 \sqrt{1-2 x} (3 x+2)}+\frac{1}{42 \sqrt{1-2 x} (3 x+2)^2}-\frac{65}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

1/(42*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 65/(147*Sqrt[1 - 2*x]*(2 + 3*x)) - (195*Sqrt[
1 - 2*x])/(686*(2 + 3*x)) - (65*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi [A]  time = 0.0930501, antiderivative size = 90, 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 \[ -\frac{195 \sqrt{1-2 x}}{686 (3 x+2)}+\frac{65}{147 \sqrt{1-2 x} (3 x+2)}+\frac{1}{42 \sqrt{1-2 x} (3 x+2)^2}-\frac{65}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

1/(42*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 65/(147*Sqrt[1 - 2*x]*(2 + 3*x)) - (195*Sqrt[
1 - 2*x])/(686*(2 + 3*x)) - (65*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi in Sympy [A]  time = 9.04876, size = 71, normalized size = 0.79 \[ - \frac{65 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{65}{343 \sqrt{- 2 x + 1}} - \frac{65}{294 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{1}{42 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-65*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 65/(343*sqrt(-2*x + 1)) - 6
5/(294*sqrt(-2*x + 1)*(3*x + 2)) + 1/(42*sqrt(-2*x + 1)*(3*x + 2)**2)

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Mathematica [A]  time = 0.128049, size = 58, normalized size = 0.64 \[ \frac{\frac{7 \left (1170 x^2+1105 x+233\right )}{\sqrt{1-2 x} (3 x+2)^2}-130 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4802} \]

Antiderivative was successfully verified.

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

[Out]

((7*(233 + 1105*x + 1170*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^2) - 130*Sqrt[21]*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/4802

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Maple [A]  time = 0.017, size = 57, normalized size = 0.6 \[{\frac{44}{343}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{36}{343\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{21}{4} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{427}{36}\sqrt{1-2\,x}} \right ) }-{\frac{65\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

44/343/(1-2*x)^(1/2)+36/343*(21/4*(1-2*x)^(3/2)-427/36*(1-2*x)^(1/2))/(-4-6*x)^2
-65/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50589, size = 112, normalized size = 1.24 \[ \frac{65}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{585 \,{\left (2 \, x - 1\right )}^{2} + 4550 \, x - 119}{343 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

65/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 1/343*(585*(2*x - 1)^2 + 4550*x - 119)/(9*(-2*x + 1)^(5/2) - 42*(-2*x + 1)^
(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]  time = 0.223657, size = 124, normalized size = 1.38 \[ \frac{\sqrt{7}{\left (65 \, \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1170 \, x^{2} + 1105 \, x + 233\right )}\right )}}{4802 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4802*sqrt(7)*(65*sqrt(3)*(9*x^2 + 12*x + 4)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x -
5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(1170*x^2 + 1105*x + 233))/
((9*x^2 + 12*x + 4)*sqrt(-2*x + 1))

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.226303, size = 104, normalized size = 1.16 \[ \frac{65}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{44}{343 \, \sqrt{-2 \, x + 1}} + \frac{27 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 61 \, \sqrt{-2 \, x + 1}}{196 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

65/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 44/343/sqrt(-2*x + 1) + 1/196*(27*(-2*x + 1)^(3/2) - 61*sqrt(-2*x +
1))/(3*x + 2)^2