3.2136 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{130}{1029 \sqrt{1-2 x}}-\frac{365}{294 \sqrt{1-2 x} (3 x+2)}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)}+\frac{130 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{343 \sqrt{21}} \]

[Out]

-130/(1029*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)) - 365/(294*Sqrt[1
- 2*x]*(2 + 3*x)) + (130*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(343*Sqrt[21])

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Rubi [A]  time = 0.11372, antiderivative size = 81, 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 \[ -\frac{130}{1029 \sqrt{1-2 x}}-\frac{365}{294 \sqrt{1-2 x} (3 x+2)}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)}+\frac{130 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{343 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-130/(1029*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)) - 365/(294*Sqrt[1
- 2*x]*(2 + 3*x)) + (130*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(343*Sqrt[21])

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Rubi in Sympy [A]  time = 10.1901, size = 70, normalized size = 0.86 \[ \frac{130 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7203} - \frac{130}{1029 \sqrt{- 2 x + 1}} - \frac{365}{294 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{121}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

130*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7203 - 130/(1029*sqrt(-2*x + 1)) -
365/(294*sqrt(-2*x + 1)*(3*x + 2)) + 121/(42*(-2*x + 1)**(3/2)*(3*x + 2))

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Mathematica [A]  time = 0.11575, size = 58, normalized size = 0.72 \[ \frac{\frac{7 \left (780 x^2+2685 x+1427\right )}{(1-2 x)^{3/2} (3 x+2)}+130 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203} \]

Antiderivative was successfully verified.

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

[Out]

((7*(1427 + 2685*x + 780*x^2))/((1 - 2*x)^(3/2)*(2 + 3*x)) + 130*Sqrt[21]*ArcTan
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/7203

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Maple [A]  time = 0.02, size = 54, normalized size = 0.7 \[{\frac{121}{147} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{44}{343}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{1029}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{130\,\sqrt{21}}{7203}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/147/(1-2*x)^(3/2)-44/343/(1-2*x)^(1/2)+2/1029*(1-2*x)^(1/2)/(-4/3-2*x)+130/7
203*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48775, size = 100, normalized size = 1.23 \[ -\frac{65}{7203} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (195 \,{\left (2 \, x - 1\right )}^{2} + 3465 \, x + 1232\right )}}{1029 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-65/7203*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 2/1029*(195*(2*x - 1)^2 + 3465*x + 1232)/(3*(-2*x + 1)^(5/2) - 7*(-2*x + 1
)^(3/2))

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Fricas [A]  time = 0.223683, size = 112, normalized size = 1.38 \[ \frac{\sqrt{21}{\left (195 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{21}{\left (780 \, x^{2} + 2685 \, x + 1427\right )}\right )}}{21609 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21609*sqrt(21)*(195*(6*x^2 + x - 2)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) - 2
1*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(21)*(780*x^2 + 2685*x + 1427))/((6*x^2 + 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)**2/(1-2*x)**(5/2)/(2+3*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217961, size = 104, normalized size = 1.28 \[ -\frac{65}{7203} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{11 \,{\left (24 \, x + 65\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-65/7203*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 11/1029*(24*x + 65)/((2*x - 1)*sqrt(-2*x + 1)) - 1/343*sqrt(-2*x +
1)/(3*x + 2)