3.2.41 \(\int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 \left (x^4+x^2\right )^{5/4}}{5 x^5} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} -\frac {2 \left (x^4+x^2\right )^{5/4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2 + x^4)^(1/4)/x^4,x]

[Out]

(-2*(x^2 + x^4)^(5/4))/(5*x^5)

Rule 2014

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \left (x^2+1\right ) \sqrt [4]{x^4+x^2}}{5 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + x^4)^(1/4)/x^4,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \left (x^2+x^4\right )^{5/4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2 + x^4)^(1/4)/x^4,x]

[Out]

(-2*(x^2 + x^4)^(5/4))/(5*x^5)

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fricas [A]  time = 0.45, size = 19, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{5 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(x^4 + x^2)^(1/4)*(x^2 + 1)/x^3

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giac [A]  time = 0.30, size = 9, normalized size = 0.50 \begin {gather*} -\frac {2}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(1/x^2 + 1)^(5/4)

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maple [A]  time = 0.05, size = 13, normalized size = 0.72

method	result	size
meijerg	\(-\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5 x^{\frac {5}{2}}}\)	\(13\)
gosper	\(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\)	\(20\)
trager	\(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\)	\(20\)
risch	\(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{4}+2 x^{2}+1\right )}{5 x^{3} \left (x^{2}+1\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+x^2)^(1/4)/x^4,x,method=_RETURNVERBOSE)

[Out]

-2/5*(x^2+1)^(5/4)/x^(5/2)

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maxima [A]  time = 0.51, size = 17, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{5 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(x^3 + x)*(x^2 + 1)^(1/4)/x^(7/2)

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mupad [B]  time = 0.17, size = 29, normalized size = 1.61 \begin {gather*} -\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{5\,x}-\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{5\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + x^4)^(1/4)/x^4,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2*(x**2 + 1))**(1/4)/x**4, x)

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