3.2.34 \(\int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx\)

Optimal. Leaf size=18 \[ \frac {4 \left (x^4-x\right )^{5/4}}{15 x^5} \]

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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 {4 \left (x^4-x\right )^{5/4}}{15 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^4)^(1/4)/x^5,x]

[Out]

(4*(-x + x^4)^(5/4))/(15*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+x^4}}{x^5} \, dx &=\frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x \left (x^3-1\right )\right )^{5/4}}{15 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^4)^(1/4)/x^5,x]

[Out]

(4*(x*(-1 + x^3))^(5/4))/(15*x^5)

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IntegrateAlgebraic [A]  time = 0.18, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x + x^4)^(1/4)/x^5,x]

[Out]

(4*(-x + x^4)^(5/4))/(15*x^5)

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fricas [A]  time = 0.46, size = 19, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )}}{15 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/4)/x^5,x, algorithm="fricas")

[Out]

4/15*(x^4 - x)^(1/4)*(x^3 - 1)/x^4

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giac [A]  time = 0.51, size = 11, normalized size = 0.61 \begin {gather*} -\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^5,x, algorithm="giac")

[Out]

-4/15*(-1/x^3 + 1)^(5/4)

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maple [A]  time = 0.07, size = 20, normalized size = 1.11

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\)	\(20\)
gosper	\(\frac {4 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\)	\(24\)
risch	\(\frac {4 \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (x^{6}-2 x^{3}+1\right )}{15 x^{4} \left (x^{3}-1\right )}\)	\(32\)
meijerg	\(-\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {5}{4}}}{15 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}} x^{\frac {15}{4}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x)^(1/4)/x^5,x,method=_RETURNVERBOSE)

[Out]

4/15*(x^3-1)/x^4*(x^4-x)^(1/4)

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maxima [A]  time = 0.67, size = 25, normalized size = 1.39 \begin {gather*} \frac {4 \, {\left (x^{4} - x\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{15 \, x^{\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/4)/x^5,x, algorithm="maxima")

[Out]

4/15*(x^4 - x)*(x^2 + x + 1)^(1/4)*(x - 1)^(1/4)/x^(19/4)

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mupad [B]  time = 0.19, size = 31, normalized size = 1.72 \begin {gather*} -\frac {4\,{\left (x^4-x\right )}^{1/4}-4\,x^3\,{\left (x^4-x\right )}^{1/4}}{15\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - x)^(1/4)/x^5,x)

[Out]

-(4*(x^4 - x)^(1/4) - 4*x^3*(x^4 - x)^(1/4))/(15*x^4)

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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^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x)**(1/4)/x**5,x)

[Out]

Integral((x*(x - 1)*(x**2 + x + 1))**(1/4)/x**5, x)

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