3.2061 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=61 \[ \frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (3 x+2)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

[Out]

64/(147*Sqrt[1 - 2*x]) + 1/(21*Sqrt[1 - 2*x]*(2 + 3*x)) - (64*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Rubi [A]  time = 0.0656541, antiderivative size = 61, 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 \[ \frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (3 x+2)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

64/(147*Sqrt[1 - 2*x]) + 1/(21*Sqrt[1 - 2*x]*(2 + 3*x)) - (64*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Rubi in Sympy [A]  time = 7.28895, size = 53, normalized size = 0.87 \[ - \frac{64 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1029} + \frac{64}{147 \sqrt{- 2 x + 1}} + \frac{1}{21 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-64*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1029 + 64/(147*sqrt(-2*x + 1)) + 1
/(21*sqrt(-2*x + 1)*(3*x + 2))

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Mathematica [A]  time = 0.106126, size = 56, normalized size = 0.92 \[ -\frac{\sqrt{1-2 x} (64 x+45)}{49 \left (6 x^2+x-2\right )}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(45 + 64*x))/(49*(-2 + x + 6*x^2)) - (64*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/(49*Sqrt[21])

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Maple [A]  time = 0.016, size = 45, normalized size = 0.7 \[{\frac{22}{49}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2}{147}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{64\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/49/(1-2*x)^(1/2)-2/147*(1-2*x)^(1/2)/(-4/3-2*x)-64/1029*arctanh(1/7*21^(1/2)*
(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50596, size = 88, normalized size = 1.44 \[ \frac{32}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (64 \, x + 45\right )}}{49 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 2/49*(64*x + 45)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))

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Fricas [A]  time = 0.226826, size = 96, normalized size = 1.57 \[ \frac{\sqrt{21}{\left (32 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (64 \, x + 45\right )}\right )}}{1029 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(21)*(32*(3*x + 2)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-
2*x + 1))/(3*x + 2)) + sqrt(21)*(64*x + 45))/((3*x + 2)*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)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.227776, size = 92, normalized size = 1.51 \[ \frac{32}{1029} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (64 \, x + 45\right )}}{49 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/1029*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 2/49*(64*x + 45)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))