3.2167 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{142}{6655 \sqrt{1-2 x}}-\frac{1231}{3630 \sqrt{1-2 x} (5 x+3)}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)}-\frac{142 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

[Out]

142/(6655*Sqrt[1 - 2*x]) + 49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)) - 1231/(3630*Sqrt[1
- 2*x]*(3 + 5*x)) - (142*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqrt[55])

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Rubi [A]  time = 0.111732, 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 \[ \frac{142}{6655 \sqrt{1-2 x}}-\frac{1231}{3630 \sqrt{1-2 x} (5 x+3)}+\frac{49}{66 (1-2 x)^{3/2} (5 x+3)}-\frac{142 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

142/(6655*Sqrt[1 - 2*x]) + 49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)) - 1231/(3630*Sqrt[1
- 2*x]*(3 + 5*x)) - (142*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqrt[55])

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Rubi in Sympy [A]  time = 9.71349, size = 65, normalized size = 0.8 \[ - \frac{142 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{73205} + \frac{142}{6655 \sqrt{- 2 x + 1}} + \frac{1231}{9075 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{1}{275 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-142*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/73205 + 142/(6655*sqrt(-2*x + 1)
) + 1231/(9075*(-2*x + 1)**(3/2)) - 1/(275*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.112496, size = 58, normalized size = 0.72 \[ \frac{-\frac{55 \left (852 x^2-2623 x-1866\right )}{(1-2 x)^{3/2} (5 x+3)}-426 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{219615} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(-1866 - 2623*x + 852*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 426*Sqrt[55]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/219615

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Maple [A]  time = 0.02, size = 54, normalized size = 0.7 \[{\frac{49}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{28}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{6655}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{142\,\sqrt{55}}{73205}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/363/(1-2*x)^(3/2)+28/1331/(1-2*x)^(1/2)+2/6655*(1-2*x)^(1/2)/(-6/5-2*x)-142/7
3205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49302, size = 100, normalized size = 1.23 \[ \frac{71}{73205} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (213 \,{\left (2 \, x - 1\right )}^{2} - 1771 \, x - 2079\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

71/73205*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 2/3993*(213*(2*x - 1)^2 - 1771*x - 2079)/(5*(-2*x + 1)^(5/2) - 11*(-2*x +
1)^(3/2))

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Fricas [A]  time = 0.220206, size = 111, normalized size = 1.37 \[ \frac{\sqrt{55}{\left (213 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (852 \, x^{2} - 2623 \, x - 1866\right )}\right )}}{219615 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/219615*sqrt(55)*(213*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) +
55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(852*x^2 - 2623*x - 1866))/((10*x^2 +
x - 3)*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((2+3*x)**2/(1-2*x)**(5/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221482, size = 104, normalized size = 1.28 \[ \frac{71}{73205} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7 \,{\left (24 \, x - 89\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

71/73205*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 7/3993*(24*x - 89)/((2*x - 1)*sqrt(-2*x + 1)) - 1/1331*sqrt(-2*x
+ 1)/(5*x + 3)