3.804 \(\int x^3 \sqrt{5+x^4} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{6} \left (x^4+5\right )^{3/2} \]

[Out]

(5 + x^4)^(3/2)/6

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Rubi [A]  time = 0.00685628, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{6} \left (x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

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Rubi in Sympy [A]  time = 1.62052, size = 8, normalized size = 0.62 \[ \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(x**4+5)**(1/2),x)

[Out]

(x**4 + 5)**(3/2)/6

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Mathematica [A]  time = 0.00618847, size = 13, normalized size = 1. \[ \frac{1}{6} \left (x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

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Maple [A]  time = 0.006, size = 10, normalized size = 0.8 \[{\frac{1}{6} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(x^4+5)^(1/2),x)

[Out]

1/6*(x^4+5)^(3/2)

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Maxima [A]  time = 1.43292, size = 12, normalized size = 0.92 \[ \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*x^3,x, algorithm="maxima")

[Out]

1/6*(x^4 + 5)^(3/2)

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Fricas [A]  time = 0.228481, size = 12, normalized size = 0.92 \[ \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*x^3,x, algorithm="fricas")

[Out]

1/6*(x^4 + 5)^(3/2)

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Sympy [A]  time = 0.474179, size = 24, normalized size = 1.85 \[ \frac{x^{4} \sqrt{x^{4} + 5}}{6} + \frac{5 \sqrt{x^{4} + 5}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(x**4+5)**(1/2),x)

[Out]

x**4*sqrt(x**4 + 5)/6 + 5*sqrt(x**4 + 5)/6

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GIAC/XCAS [A]  time = 0.213963, size = 12, normalized size = 0.92 \[ \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*x^3,x, algorithm="giac")

[Out]

1/6*(x^4 + 5)^(3/2)