3.2166 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{2 (17112 x+10309)}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) - (2*(10309 + 17112*x))/(19965*Sq
rt[1 - 2*x]*(3 + 5*x)) - (208*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6655*Sqrt[55])

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Rubi [A]  time = 0.116965, antiderivative size = 80, 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 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{2 (17112 x+10309)}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) - (2*(10309 + 17112*x))/(19965*Sq
rt[1 - 2*x]*(3 + 5*x)) - (208*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6655*Sqrt[55])

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Rubi in Sympy [A]  time = 12.0764, size = 68, normalized size = 0.85 \[ - \frac{208 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{366025} - \frac{34224 x + 20618}{19965 \sqrt{- 2 x + 1} \left (5 x + 3\right )} + \frac{7 \left (3 x + 2\right )^{2}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-208*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/366025 - (34224*x + 20618)/(1996
5*sqrt(-2*x + 1)*(5*x + 3)) + 7*(3*x + 2)**2/(33*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.117237, size = 58, normalized size = 0.72 \[ \frac{\frac{55 \left (106563 x^2+57832 x-3678\right )}{(1-2 x)^{3/2} (5 x+3)}-624 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1098075} \]

Antiderivative was successfully verified.

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

[Out]

((55*(-3678 + 57832*x + 106563*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 624*Sqrt[55]*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1098075

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Maple [A]  time = 0.022, size = 54, normalized size = 0.7 \[{\frac{343}{726} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{1421}{2662}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{33275}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{208\,\sqrt{55}}{366025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

343/726/(1-2*x)^(3/2)-1421/2662/(1-2*x)^(1/2)+2/33275*(1-2*x)^(1/2)/(-6/5-2*x)-2
08/366025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51114, size = 100, normalized size = 1.25 \[ \frac{104}{366025} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{106563 \,{\left (2 \, x - 1\right )}^{2} + 657580 \, x - 121275}{39930 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

104/366025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 1/39930*(106563*(2*x - 1)^2 + 657580*x - 121275)/(5*(-2*x + 1)^(5/2) - 1
1*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.221745, size = 112, normalized size = 1.4 \[ \frac{\sqrt{55}{\left (312 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (106563 \, x^{2} + 57832 \, x - 3678\right )}\right )}}{1098075 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1098075*sqrt(55)*(312*(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)*(106563*x^2 + 57832*x - 3678))/((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)**3/(1-2*x)**(5/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.214726, size = 104, normalized size = 1.3 \[ \frac{104}{366025} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{49 \,{\left (87 \, x - 5\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{6655 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

104/366025*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) - 49/3993*(87*x - 5)/((2*x - 1)*sqrt(-2*x + 1)) - 1/6655*sqrt(-2*
x + 1)/(5*x + 3)