3.2074 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2045 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{545 \sqrt{1-2 x}}{147 (3 x+2)^2}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{2045 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (545*Sqrt[1 - 2*x])/(147*(2 + 3*x)^2) - (20
45*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) - (2045*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(10
29*Sqrt[21])

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Rubi [A]  time = 0.113319, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2045 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{545 \sqrt{1-2 x}}{147 (3 x+2)^2}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{2045 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (545*Sqrt[1 - 2*x])/(147*(2 + 3*x)^2) - (20
45*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) - (2045*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(10
29*Sqrt[21])

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Rubi in Sympy [A]  time = 9.79878, size = 76, normalized size = 0.86 \[ - \frac{2045 \sqrt{- 2 x + 1}}{2058 \left (3 x + 2\right )} - \frac{545 \sqrt{- 2 x + 1}}{147 \left (3 x + 2\right )^{2}} - \frac{2045 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} + \frac{121}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2045*sqrt(-2*x + 1)/(2058*(3*x + 2)) - 545*sqrt(-2*x + 1)/(147*(3*x + 2)**2) -
2045*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21609 + 121/(14*sqrt(-2*x + 1)*(3
*x + 2)**2)

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Mathematica [A]  time = 0.142642, size = 58, normalized size = 0.66 \[ \frac{\frac{21 \left (12270 x^2+17305 x+6067\right )}{\sqrt{1-2 x} (3 x+2)^2}-4090 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{43218} \]

Antiderivative was successfully verified.

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

[Out]

((21*(6067 + 17305*x + 12270*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^2) - 4090*Sqrt[21]*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/43218

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Maple [A]  time = 0.019, size = 57, normalized size = 0.7 \[{\frac{242}{343}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{18}{343\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{133}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{917}{54}\sqrt{1-2\,x}} \right ) }-{\frac{2045\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

242/343/(1-2*x)^(1/2)+18/343*(-133/18*(1-2*x)^(3/2)+917/54*(1-2*x)^(1/2))/(-4-6*
x)^2-2045/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50191, size = 112, normalized size = 1.27 \[ \frac{2045}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6135 \,{\left (2 \, x - 1\right )}^{2} + 59150 \, x + 5999}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2045/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/1029*(6135*(2*x - 1)^2 + 59150*x + 5999)/(9*(-2*x + 1)^(5/2) - 42*(-2*
x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]  time = 0.227382, size = 116, normalized size = 1.32 \[ \frac{\sqrt{21}{\left (2045 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (12270 \, x^{2} + 17305 \, x + 6067\right )}\right )}}{43218 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43218*sqrt(21)*(2045*(9*x^2 + 12*x + 4)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5)
+ 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(12270*x^2 + 17305*x + 6067))/((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)**2/(1-2*x)**(3/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.26548, size = 104, normalized size = 1.18 \[ \frac{2045}{43218} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{343 \, \sqrt{-2 \, x + 1}} - \frac{57 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 131 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2045/43218*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 242/343/sqrt(-2*x + 1) - 1/588*(57*(-2*x + 1)^(3/2) - 131*sqrt(-
2*x + 1))/(3*x + 2)^2