3.919 \(\int \frac{x^3}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\sqrt{x^4+1}}{2} \]

[Out]

Sqrt[1 + x^4]/2

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Rubi [A]  time = 0.0024358, 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} \[ \frac{\sqrt{x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[1 + x^4],x]

[Out]

Sqrt[1 + x^4]/2

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 \frac{x^3}{\sqrt{1+x^4}} \, dx &=\frac{\sqrt{1+x^4}}{2}\\ \end{align*}

Mathematica [A]  time = 0.0018307, size = 13, normalized size = 1. \[ \frac{\sqrt{x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[1 + x^4],x]

[Out]

Sqrt[1 + x^4]/2

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{2}\sqrt{{x}^{4}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.03734, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^4 + 1)

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Fricas [A]  time = 1.45237, size = 26, normalized size = 2. \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^4 + 1)

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Sympy [A]  time = 0.175, size = 8, normalized size = 0.62 \begin{align*} \frac{\sqrt{x^{4} + 1}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+1)**(1/2),x)

[Out]

sqrt(x**4 + 1)/2

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Giac [A]  time = 1.15676, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^4 + 1)