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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = 0.00, size = 13, normalized size = 1.00 \[ \frac {\sqrt {x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[1 + x^4],x]

[Out]

Sqrt[1 + x^4]/2

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 9, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {x^{4} + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 9, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {x^{4} + 1} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\sqrt {x^{4}+1}}{2} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.31, size = 9, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {x^{4} + 1} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.04, size = 9, normalized size = 0.69 \[ \frac {\sqrt {x^4+1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4 + 1)^(1/2),x)

[Out]

(x^4 + 1)^(1/2)/2

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 8, normalized size = 0.62 \[ \frac {\sqrt {x^{4} + 1}}{2} \]

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

________________________________________________________________________________________