3.432 \(\int \sqrt{x+x^{5/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

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Rubi [A]  time = 0.00959661, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x + x^(5/2)],x]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

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Rubi in Sympy [A]  time = 1.02098, size = 17, normalized size = 0.85 \[ \frac{4 \left (x^{\frac{5}{2}} + x\right )^{\frac{3}{2}}}{9 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(x**(5/2) + x)**(3/2)/(9*x**(3/2))

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Mathematica [A]  time = 0.0193071, size = 29, normalized size = 1.45 \[ \left (\frac{4 x}{9}+\frac{4}{9 \sqrt{x}}\right ) \sqrt{x \left (x^{3/2}+1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x + x^(5/2)],x]

[Out]

(4/(9*Sqrt[x]) + (4*x)/9)*Sqrt[x*(1 + x^(3/2))]

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Maple [A]  time = 0.013, size = 18, normalized size = 0.9 \[{\frac{4}{9}\sqrt{x+{x}^{{\frac{5}{2}}}} \left ( 1+{x}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+x^(5/2))^(1/2),x)

[Out]

4/9*(x+x^(5/2))^(1/2)/x^(1/2)*(1+x^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{\frac{5}{2}} + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(5/2) + x),x, algorithm="maxima")

[Out]

integrate(sqrt(x^(5/2) + x), x)

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Fricas [A]  time = 0.253817, size = 26, normalized size = 1.3 \[ \frac{4 \, \sqrt{x^{\frac{5}{2}} + x}{\left (x^{2} + \sqrt{x}\right )}}{9 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(5/2) + x),x, algorithm="fricas")

[Out]

4/9*sqrt(x^(5/2) + x)*(x^2 + sqrt(x))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{\frac{5}{2}} + x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**(5/2) + x), x)

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GIAC/XCAS [A]  time = 0.21918, size = 15, normalized size = 0.75 \[ \frac{4}{9} \,{\left (x^{\frac{3}{2}} + 1\right )}^{\frac{3}{2}} - \frac{4}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^(5/2) + x),x, algorithm="giac")

[Out]

4/9*(x^(3/2) + 1)^(3/2) - 4/9