3.313 \(\int \frac{(2-3 x)^3 \sqrt{x}}{(1+x)^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{54 x^{5/2}}{5}+72 x^{3/2}-\frac{125 \sqrt{x}}{x+1}-450 \sqrt{x}+575 \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

-450*Sqrt[x] + 72*x^(3/2) - (54*x^(5/2))/5 - (125*Sqrt[x])/(1 + x) + 575*ArcTan[
Sqrt[x]]

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Rubi [A]  time = 0.0809778, antiderivative size = 58, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{\sqrt{x} (2-3 x)^3}{x+1}-\frac{21}{5} \sqrt{x} (2-3 x)^2-\frac{3}{5} (917-171 x) \sqrt{x}+575 \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(917 - 171*x)*Sqrt[x])/5 - (21*(2 - 3*x)^2*Sqrt[x])/5 - ((2 - 3*x)^3*Sqrt[x]
)/(1 + x) + 575*ArcTan[Sqrt[x]]

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Rubi in Sympy [A]  time = 11.6609, size = 54, normalized size = 1.23 \[ - \frac{8 \sqrt{x} \left (- \frac{1539 x}{8} + \frac{8253}{8}\right )}{15} - \frac{\sqrt{x} \left (- 3 x + 2\right )^{3}}{x + 1} - \frac{21 \sqrt{x} \left (- 3 x + 2\right )^{2}}{5} + 575 \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*sqrt(x)*(-1539*x/8 + 8253/8)/15 - sqrt(x)*(-3*x + 2)**3/(x + 1) - 21*sqrt(x)*
(-3*x + 2)**2/5 + 575*atan(sqrt(x))

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Mathematica [A]  time = 0.0367129, size = 38, normalized size = 0.86 \[ 575 \tan ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \left (54 x^3-306 x^2+1890 x+2875\right )}{5 (x+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(2875 + 1890*x - 306*x^2 + 54*x^3))/(5*(1 + x)) + 575*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.018, size = 33, normalized size = 0.8 \[ 72\,{x}^{3/2}-{\frac{54}{5}{x}^{{\frac{5}{2}}}}+575\,\arctan \left ( \sqrt{x} \right ) -450\,\sqrt{x}-125\,{\frac{\sqrt{x}}{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

72*x^(3/2)-54/5*x^(5/2)+575*arctan(x^(1/2))-450*x^(1/2)-125*x^(1/2)/(1+x)

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Maxima [A]  time = 1.57159, size = 43, normalized size = 0.98 \[ -\frac{54}{5} \, x^{\frac{5}{2}} + 72 \, x^{\frac{3}{2}} - 450 \, \sqrt{x} - \frac{125 \, \sqrt{x}}{x + 1} + 575 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54/5*x^(5/2) + 72*x^(3/2) - 450*sqrt(x) - 125*sqrt(x)/(x + 1) + 575*arctan(sqrt
(x))

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Fricas [A]  time = 0.212857, size = 50, normalized size = 1.14 \[ \frac{2875 \,{\left (x + 1\right )} \arctan \left (\sqrt{x}\right ) -{\left (54 \, x^{3} - 306 \, x^{2} + 1890 \, x + 2875\right )} \sqrt{x}}{5 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2875*(x + 1)*arctan(sqrt(x)) - (54*x^3 - 306*x^2 + 1890*x + 2875)*sqrt(x))/
(x + 1)

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Sympy [A]  time = 26.8666, size = 39, normalized size = 0.89 \[ - \frac{54 x^{\frac{5}{2}}}{5} + 72 x^{\frac{3}{2}} - 450 \sqrt{x} - \frac{125 \sqrt{x}}{x + 1} + 575 \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54*x**(5/2)/5 + 72*x**(3/2) - 450*sqrt(x) - 125*sqrt(x)/(x + 1) + 575*atan(sqrt
(x))

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GIAC/XCAS [A]  time = 0.269752, size = 43, normalized size = 0.98 \[ -\frac{54}{5} \, x^{\frac{5}{2}} + 72 \, x^{\frac{3}{2}} - 450 \, \sqrt{x} - \frac{125 \, \sqrt{x}}{x + 1} + 575 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54/5*x^(5/2) + 72*x^(3/2) - 450*sqrt(x) - 125*sqrt(x)/(x + 1) + 575*arctan(sqrt
(x))