3.4.22 \(\int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx\)

Optimal. Leaf size=28 \[ \frac {4 \sqrt [4]{x^4+x} \left (4 x^6-x^3-5\right )}{135 x^7} \]

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2016, 2014} \begin {gather*} \frac {16 \left (x^4+x\right )^{5/4}}{135 x^5}-\frac {4 \left (x^4+x\right )^{5/4}}{27 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + x^4)^(1/4)/x^8,x]

[Out]

(-4*(x + x^4)^(5/4))/(27*x^8) + (16*(x + x^4)^(5/4))/(135*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])

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx &=-\frac {4 \left (x+x^4\right )^{5/4}}{27 x^8}-\frac {4}{9} \int \frac {\sqrt [4]{x+x^4}}{x^5} \, dx\\ &=-\frac {4 \left (x+x^4\right )^{5/4}}{27 x^8}+\frac {16 \left (x+x^4\right )^{5/4}}{135 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} -\frac {4 \left (5-4 x^3\right ) \left (x^3+1\right ) \sqrt [4]{x^4+x}}{135 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + x^4)^(1/4)/x^8,x]

[Out]

(-4*(5 - 4*x^3)*(1 + x^3)*(x + x^4)^(1/4))/(135*x^7)

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IntegrateAlgebraic [A]  time = 0.22, size = 23, normalized size = 0.82 \begin {gather*} \frac {4 \left (-5+4 x^3\right ) \left (x+x^4\right )^{5/4}}{135 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x + x^4)^(1/4)/x^8,x]

[Out]

(4*(-5 + 4*x^3)*(x + x^4)^(5/4))/(135*x^8)

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fricas [A]  time = 0.46, size = 24, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (4 \, x^{6} - x^{3} - 5\right )} {\left (x^{4} + x\right )}^{\frac {1}{4}}}{135 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/4)/x^8,x, algorithm="fricas")

[Out]

4/135*(4*x^6 - x^3 - 5)*(x^4 + x)^(1/4)/x^7

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giac [A]  time = 0.26, size = 19, normalized size = 0.68 \begin {gather*} -\frac {4}{27} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {9}{4}} + \frac {4}{15} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/4)/x^8,x, algorithm="giac")

[Out]

-4/27*(1/x^3 + 1)^(9/4) + 4/15*(1/x^3 + 1)^(5/4)

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maple [A]  time = 0.07, size = 25, normalized size = 0.89

method	result	size
trager	\(\frac {4 \left (x^{4}+x \right )^{\frac {1}{4}} \left (4 x^{6}-x^{3}-5\right )}{135 x^{7}}\)	\(25\)
meijerg	\(-\frac {4 \left (-\frac {4}{5} x^{6}+\frac {1}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{27 x^{\frac {27}{4}}}\)	\(25\)
gosper	\(\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (4 x^{3}-5\right ) \left (x^{4}+x \right )^{\frac {1}{4}}}{135 x^{7}}\)	\(31\)
risch	\(\frac {4 \left (x \left (x^{3}+1\right )\right )^{\frac {1}{4}} \left (4 x^{9}+3 x^{6}-6 x^{3}-5\right )}{135 x^{7} \left (x^{3}+1\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+x)^(1/4)/x^8,x,method=_RETURNVERBOSE)

[Out]

4/135*(x^4+x)^(1/4)*(4*x^6-x^3-5)/x^7

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maxima [A]  time = 0.78, size = 34, normalized size = 1.21 \begin {gather*} \frac {4 \, {\left (4 \, x^{7} - x^{4} - 5 \, x\right )} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{135 \, x^{\frac {31}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/4)/x^8,x, algorithm="maxima")

[Out]

4/135*(4*x^7 - x^4 - 5*x)*(x^2 - x + 1)^(1/4)*(x + 1)^(1/4)/x^(31/4)

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mupad [B]  time = 0.34, size = 22, normalized size = 0.79 \begin {gather*} -\frac {4\,{\left (x^4+x\right )}^{1/4}\,\left (-4\,x^6+x^3+5\right )}{135\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^4)^(1/4)/x^8,x)

[Out]

-(4*(x + x^4)^(1/4)*(x^3 - 4*x^6 + 5))/(135*x^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+x)**(1/4)/x**8,x)

[Out]

Integral((x*(x + 1)*(x**2 - x + 1))**(1/4)/x**8, x)

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