\(\int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx\) [134]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5} \]

[Out]

4/15*(x^4-x)^(5/4)/x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2039} \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (x^4-x\right )^{5/4}}{15 x^5} \]

[In]

Int[(-x + x^4)^(1/4)/x^5,x]

[Out]

(4*(-x + x^4)^(5/4))/(15*x^5)

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x+x^4\right )^{5/4}}{15 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{5/4}}{15 x^5} \]

[In]

Integrate[(-x + x^4)^(1/4)/x^5,x]

[Out]

(4*(x*(-1 + x^3))^(5/4))/(15*x^5)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\)	\(20\)
pseudoelliptic	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\)	\(20\)
gosper	\(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{15 x^{4}}\)	\(24\)
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	\(-\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {5}{4}}}{15 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} x^{\frac {15}{4}}}\)	\(33\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )}}{15 \, x^{4}} \]

[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

Sympy [F]

\[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\int \frac {\sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{5}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4 \, {\left (x^{4} - x\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{15 \, x^{\frac {19}{4}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=\frac {4}{15} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} \]

[In]

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

[Out]

4/15*(-1/x^3 + 1)^(5/4)

Mupad [B] (verification not implemented)

Time = 5.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt [4]{-x+x^4}}{x^5} \, dx=-\frac {4\,{\left (x^4-x\right )}^{1/4}-4\,x^3\,{\left (x^4-x\right )}^{1/4}}{15\,x^4} \]

[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)