3.443 \(\int \left (1+x^4\right ) \sqrt{5 x+x^5} \, dx\)

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

[Out]

(2*(5*x + x^5)^(3/2))/15

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Rubi [A]  time = 0.00727577, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2}{15} \left (x^5+5 x\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)*Sqrt[5*x + x^5],x]

[Out]

(2*(5*x + x^5)^(3/2))/15

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Rubi in Sympy [A]  time = 1.27445, size = 12, normalized size = 0.8 \[ \frac{2 \left (x^{5} + 5 x\right )^{\frac{3}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x**5 + 5*x)**(3/2)/15

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Mathematica [A]  time = 0.0149563, size = 15, normalized size = 1. \[ \frac{2}{15} \left (x \left (x^4+5\right )\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)*Sqrt[5*x + x^5],x]

[Out]

(2*(x*(5 + x^4))^(3/2))/15

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Maple [A]  time = 0.007, size = 18, normalized size = 1.2 \[{\frac{2\,x \left ({x}^{4}+5 \right ) }{15}\sqrt{{x}^{5}+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)*(x^5+5*x)^(1/2),x)

[Out]

2/15*x*(x^4+5)*(x^5+5*x)^(1/2)

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Maxima [A]  time = 0.691572, size = 15, normalized size = 1. \[ \frac{2}{15} \,{\left (x^{5} + 5 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 5*x)*(x^4 + 1),x, algorithm="maxima")

[Out]

2/15*(x^5 + 5*x)^(3/2)

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Fricas [A]  time = 0.261134, size = 15, normalized size = 1. \[ \frac{2}{15} \,{\left (x^{5} + 5 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 5*x)*(x^4 + 1),x, algorithm="fricas")

[Out]

2/15*(x^5 + 5*x)^(3/2)

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Sympy [A]  time = 0.684449, size = 31, normalized size = 2.07 \[ \frac{2 x^{5} \sqrt{x^{5} + 5 x}}{15} + \frac{2 x \sqrt{x^{5} + 5 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**5*sqrt(x**5 + 5*x)/15 + 2*x*sqrt(x**5 + 5*x)/3

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GIAC/XCAS [A]  time = 0.271071, size = 15, normalized size = 1. \[ \frac{2}{15} \,{\left (x^{5} + 5 \, x\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^5 + 5*x)*(x^4 + 1),x, algorithm="giac")

[Out]

2/15*(x^5 + 5*x)^(3/2)