3.944 \(\int \frac {x}{(1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac {x^2}{2 \sqrt {x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {264} \[ \frac {x^2}{2 \sqrt {x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {x^2}{2 \sqrt {x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

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fricas [B]  time = 0.84, size = 25, normalized size = 1.56 \[ \frac {x^{4} + \sqrt {x^{4} + 1} x^{2} + 1}{2 \, {\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 12, normalized size = 0.75 \[ \frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \[ \frac {x^{2}}{2 \sqrt {x^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.28, size = 12, normalized size = 0.75 \[ \frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

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mupad [B]  time = 1.05, size = 12, normalized size = 0.75 \[ \frac {x^2}{2\,\sqrt {x^4+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 + 1)^(3/2),x)

[Out]

x^2/(2*(x^4 + 1)^(1/2))

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sympy [A]  time = 0.66, size = 12, normalized size = 0.75 \[ \frac {x^{2}}{2 \sqrt {x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(2*sqrt(x**4 + 1))

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