3.2073 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1091 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)}+\frac{134 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)) - (1091*Sqrt[1 - 2*x])/(294*(2 + 3*x)) + (134*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi [A]  time = 0.0943732, antiderivative size = 68, 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 \[ -\frac{1091 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)}+\frac{134 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)) - (1091*Sqrt[1 - 2*x])/(294*(2 + 3*x)) + (134*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi in Sympy [A]  time = 8.18631, size = 56, normalized size = 0.82 \[ - \frac{1091 \sqrt{- 2 x + 1}}{294 \left (3 x + 2\right )} + \frac{134 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3087} + \frac{121}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1091*sqrt(-2*x + 1)/(294*(3*x + 2)) + 134*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1
)/7)/3087 + 121/(14*sqrt(-2*x + 1)*(3*x + 2))

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Mathematica [A]  time = 0.108928, size = 56, normalized size = 0.82 \[ \frac{134 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} (1091 x+725)}{6 x^2+x-2}}{3087} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(725 + 1091*x))/(-2 + x + 6*x^2) + 134*Sqrt[21]*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/3087

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Maple [A]  time = 0.019, size = 45, normalized size = 0.7 \[{\frac{121}{49}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{441}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{134\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/49/(1-2*x)^(1/2)+2/441*(1-2*x)^(1/2)/(-4/3-2*x)+134/3087*arctanh(1/7*21^(1/2
)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50351, size = 88, normalized size = 1.29 \[ -\frac{67}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1091 \, x + 725\right )}}{147 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 2/147*(1091*x + 725)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))

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Fricas [A]  time = 0.230505, size = 96, normalized size = 1.41 \[ \frac{\sqrt{21}{\left (67 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (1091 \, x + 725\right )}\right )}}{3087 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3087*sqrt(21)*(67*(3*x + 2)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-
2*x + 1))/(3*x + 2)) + sqrt(21)*(1091*x + 725))/((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)**2/(1-2*x)**(3/2)/(2+3*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221127, size = 92, normalized size = 1.35 \[ -\frac{67}{3087} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (1091 \, x + 725\right )}}{147 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/3087*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 2/147*(1091*x + 725)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))