3.2002 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{25}{9} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{63 (3 x+2)}+\frac{46 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

(-25*Sqrt[1 - 2*x])/9 - Sqrt[1 - 2*x]/(63*(2 + 3*x)) + (46*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(21*Sqrt[21])

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Rubi [A]  time = 0.0845875, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{25}{9} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{63 (3 x+2)}+\frac{46 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-25*Sqrt[1 - 2*x])/9 - Sqrt[1 - 2*x]/(63*(2 + 3*x)) + (46*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(21*Sqrt[21])

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Rubi in Sympy [A]  time = 7.73569, size = 49, normalized size = 0.8 \[ - \frac{25 \sqrt{- 2 x + 1}}{9} - \frac{\sqrt{- 2 x + 1}}{63 \left (3 x + 2\right )} + \frac{46 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*sqrt(-2*x + 1)/9 - sqrt(-2*x + 1)/(63*(3*x + 2)) + 46*sqrt(21)*atanh(sqrt(21
)*sqrt(-2*x + 1)/7)/441

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Mathematica [A]  time = 0.096408, size = 51, normalized size = 0.84 \[ \frac{46 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-\frac{\sqrt{1-2 x} (175 x+117)}{63 x+42} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - 2*x]*(117 + 175*x))/(42 + 63*x)) + (46*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(21*Sqrt[21])

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Maple [A]  time = 0.014, size = 45, normalized size = 0.7 \[ -{\frac{25}{9}\sqrt{1-2\,x}}+{\frac{2}{189}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{46\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/9*(1-2*x)^(1/2)+2/189*(1-2*x)^(1/2)/(-4/3-2*x)+46/441*arctanh(1/7*21^(1/2)*(
1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49414, size = 84, normalized size = 1.38 \[ -\frac{23}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{25}{9} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{63 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 25/9*sqrt(-2*x + 1) - 1/63*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.213012, size = 86, normalized size = 1.41 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (175 \, x + 117\right )} \sqrt{-2 \, x + 1} - 23 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{441 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/441*sqrt(21)*(sqrt(21)*(175*x + 117)*sqrt(-2*x + 1) - 23*(3*x + 2)*log((sqrt(
21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(3*x + 2)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.228087, size = 88, normalized size = 1.44 \[ -\frac{23}{441} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25}{9} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{63 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23/441*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 25/9*sqrt(-2*x + 1) - 1/63*sqrt(-2*x + 1)/(3*x + 2)