3.2146 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{125}{12} \sqrt{1-2 x}-\frac{2178}{49 \sqrt{1-2 x}}+\frac{1331}{84 (1-2 x)^{3/2}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

1331/(84*(1 - 2*x)^(3/2)) - 2178/(49*Sqrt[1 - 2*x]) - (125*Sqrt[1 - 2*x])/12 + (
2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi [A]  time = 0.0986792, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{125}{12} \sqrt{1-2 x}-\frac{2178}{49 \sqrt{1-2 x}}+\frac{1331}{84 (1-2 x)^{3/2}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

1331/(84*(1 - 2*x)^(3/2)) - 2178/(49*Sqrt[1 - 2*x]) - (125*Sqrt[1 - 2*x])/12 + (
2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi in Sympy [A]  time = 11.931, size = 60, normalized size = 0.9 \[ - \frac{125 \sqrt{- 2 x + 1}}{12} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3087} - \frac{2178}{49 \sqrt{- 2 x + 1}} + \frac{1331}{84 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*sqrt(-2*x + 1)/12 + 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3087 - 2178
/(49*sqrt(-2*x + 1)) + 1331/(84*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.155796, size = 51, normalized size = 0.76 \[ \frac{2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \left (6125 x^2-19193 x+5736\right )}{(1-2 x)^{3/2}}}{3087} \]

Antiderivative was successfully verified.

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

[Out]

((-21*(5736 - 19193*x + 6125*x^2))/(1 - 2*x)^(3/2) + 2*Sqrt[21]*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/3087

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Maple [A]  time = 0.016, size = 47, normalized size = 0.7 \[{\frac{1331}{84} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2178}{49}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{125}{12}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/84/(1-2*x)^(3/2)+2/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2178/4
9/(1-2*x)^(1/2)-125/12*(1-2*x)^(1/2)

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Maxima [A]  time = 1.47948, size = 81, normalized size = 1.21 \[ -\frac{1}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{125}{12} \, \sqrt{-2 \, x + 1} + \frac{121 \,{\left (432 \, x - 139\right )}}{588 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 125/12*sqrt(-2*x + 1) + 121/588*(432*x - 139)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.224706, size = 101, normalized size = 1.51 \[ \frac{\sqrt{21}{\left ({\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (6125 \, x^{2} - 19193 \, x + 5736\right )}\right )}}{3087 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3087*sqrt(21)*((2*x - 1)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x
+ 1))/(3*x + 2)) + sqrt(21)*(6125*x^2 - 19193*x + 5736))/((2*x - 1)*sqrt(-2*x +
1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (5 x + 3\right )^{3}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((5*x + 3)**3/((-2*x + 1)**(5/2)*(3*x + 2)), x)

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GIAC/XCAS [A]  time = 0.217101, size = 95, normalized size = 1.42 \[ -\frac{1}{3087} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{125}{12} \, \sqrt{-2 \, x + 1} - \frac{121 \,{\left (432 \, x - 139\right )}}{588 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3087*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 125/12*sqrt(-2*x + 1) - 121/588*(432*x - 139)/((2*x - 1)*sqrt(-2*x
+ 1))