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

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 = 0.0023835, 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, Rules used = {261} \[ \frac{1}{6} \left (x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^3 \sqrt{5+x^4} \, dx &=\frac{1}{6} \left (5+x^4\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0030452, 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 = 0.004, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{6} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 0.98323, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="maxima")

[Out]

1/6*(x^4 + 5)^(3/2)

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Fricas [A]  time = 1.43457, size = 28, normalized size = 2.15 \begin{align*} \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="fricas")

[Out]

1/6*(x^4 + 5)^(3/2)

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Sympy [B]  time = 0.277769, size = 24, normalized size = 1.85 \begin{align*} \frac{x^{4} \sqrt{x^{4} + 5}}{6} + \frac{5 \sqrt{x^{4} + 5}}{6} \end{align*}

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 [A]  time = 1.10859, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{6} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="giac")

[Out]

1/6*(x^4 + 5)^(3/2)