3.444 \(\int \left (2+5 x^4\right ) \sqrt{2 x+x^5} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{3} \left (x^5+2 x\right )^{3/2} \]

[Out]

(2*(2*x + x^5)^(3/2))/3

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Rubi [A]  time = 0.00718842, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2}{3} \left (x^5+2 x\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 5*x^4)*Sqrt[2*x + x^5],x]

[Out]

(2*(2*x + x^5)^(3/2))/3

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Rubi in Sympy [A]  time = 1.43104, size = 12, normalized size = 0.8 \[ \frac{2 \left (x^{5} + 2 x\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**4+2)*(x**5+2*x)**(1/2),x)

[Out]

2*(x**5 + 2*x)**(3/2)/3

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Mathematica [A]  time = 0.0158078, size = 15, normalized size = 1. \[ \frac{2}{3} \left (x \left (x^4+2\right )\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 5*x^4)*Sqrt[2*x + x^5],x]

[Out]

(2*(x*(2 + x^4))^(3/2))/3

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Maple [A]  time = 0.006, size = 18, normalized size = 1.2 \[{\frac{2\,x \left ({x}^{4}+2 \right ) }{3}\sqrt{{x}^{5}+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^4+2)*(x^5+2*x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.690947, size = 15, normalized size = 1. \[ \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 2*x)*(5*x^4 + 2),x, algorithm="maxima")

[Out]

2/3*(x^5 + 2*x)^(3/2)

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Fricas [A]  time = 0.258986, size = 15, normalized size = 1. \[ \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 2*x)*(5*x^4 + 2),x, algorithm="fricas")

[Out]

2/3*(x^5 + 2*x)^(3/2)

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Sympy [A]  time = 0.678819, size = 31, normalized size = 2.07 \[ \frac{2 x^{5} \sqrt{x^{5} + 2 x}}{3} + \frac{4 x \sqrt{x^{5} + 2 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**4+2)*(x**5+2*x)**(1/2),x)

[Out]

2*x**5*sqrt(x**5 + 2*x)/3 + 4*x*sqrt(x**5 + 2*x)/3

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GIAC/XCAS [A]  time = 0.272422, size = 15, normalized size = 1. \[ \frac{2}{3} \,{\left (x^{5} + 2 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 2*x)*(5*x^4 + 2),x, algorithm="giac")

[Out]

2/3*(x^5 + 2*x)^(3/2)