3.2082 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{125}{36} (1-2 x)^{3/2}+\frac{400}{9} \sqrt{1-2 x}+\frac{1331}{28 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

[Out]

1331/(28*Sqrt[1 - 2*x]) + (400*Sqrt[1 - 2*x])/9 - (125*(1 - 2*x)^(3/2))/36 + (2*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

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Rubi [A]  time = 0.105687, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{125}{36} (1-2 x)^{3/2}+\frac{400}{9} \sqrt{1-2 x}+\frac{1331}{28 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

1331/(28*Sqrt[1 - 2*x]) + (400*Sqrt[1 - 2*x])/9 - (125*(1 - 2*x)^(3/2))/36 + (2*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

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Rubi in Sympy [A]  time = 10.3349, size = 60, normalized size = 0.9 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{3}{2}}}{36} + \frac{400 \sqrt{- 2 x + 1}}{9} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1323} + \frac{1331}{28 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(3/2)/36 + 400*sqrt(-2*x + 1)/9 + 2*sqrt(21)*atanh(sqrt(21)*sqr
t(-2*x + 1)/7)/1323 + 1331/(28*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.134441, size = 51, normalized size = 0.76 \[ \frac{-875 x^2-4725 x+5576}{63 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(5576 - 4725*x - 875*x^2)/(63*Sqrt[1 - 2*x]) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(63*Sqrt[21])

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Maple [A]  time = 0.012, size = 47, normalized size = 0.7 \[ -{\frac{125}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{1323}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1331}{28}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{400}{9}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/36*(1-2*x)^(3/2)+2/1323*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1331/2
8/(1-2*x)^(1/2)+400/9*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49876, size = 86, normalized size = 1.28 \[ -\frac{125}{36} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{1323} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{400}{9} \, \sqrt{-2 \, x + 1} + \frac{1331}{28 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/36*(-2*x + 1)^(3/2) - 1/1323*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) + 400/9*sqrt(-2*x + 1) + 1331/28/sqrt(-2*x + 1)

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Fricas [A]  time = 0.232017, size = 86, normalized size = 1.28 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (875 \, x^{2} + 4725 \, x - 5576\right )} - \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1323 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1323*sqrt(21)*(sqrt(21)*(875*x^2 + 4725*x - 5576) - sqrt(-2*x + 1)*log((sqrt(
21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.230682, size = 90, normalized size = 1.34 \[ -\frac{125}{36} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{1323} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{400}{9} \, \sqrt{-2 \, x + 1} + \frac{1331}{28 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/36*(-2*x + 1)^(3/2) - 1/1323*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 400/9*sqrt(-2*x + 1) + 1331/28/sqrt(-2*x
+ 1)