\(\int \frac {\sqrt [4]{-1+x^4-x^5} (-4+x^5)}{x^6} \, dx\) [201]

Optimal result

Integrand size = 23, antiderivative size = 21 \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=-\frac {4 \left (-1+x^4-x^5\right )^{5/4}}{5 x^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1604} \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=-\frac {4 \left (-x^5+x^4-1\right )^{5/4}}{5 x^5} \]

[In]

Int[((-1 + x^4 - x^5)^(1/4)*(-4 + x^5))/x^6,x]

[Out]

(-4*(-1 + x^4 - x^5)^(5/4))/(5*x^5)

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (-1+x^4-x^5\right )^{5/4}}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=-\frac {4 \left (-1+x^4-x^5\right )^{5/4}}{5 x^5} \]

[In]

Integrate[((-1 + x^4 - x^5)^(1/4)*(-4 + x^5))/x^6,x]

[Out]

(-4*(-1 + x^4 - x^5)^(5/4))/(5*x^5)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(-\frac {4 \left (-x^{5}+x^{4}-1\right )^{\frac {5}{4}}}{5 x^{5}}\)	\(18\)
pseudoelliptic	\(-\frac {4 \left (-x^{5}+x^{4}-1\right )^{\frac {5}{4}}}{5 x^{5}}\)	\(18\)
trager	\(\frac {4 \left (x^{5}-x^{4}+1\right ) \left (-x^{5}+x^{4}-1\right )^{\frac {1}{4}}}{5 x^{5}}\)	\(28\)
risch	\(-\frac {4 \left (x^{10}-2 x^{9}+x^{8}+2 x^{5}-2 x^{4}+1\right )}{5 \left (-x^{5}+x^{4}-1\right )^{\frac {3}{4}} x^{5}}\)	\(41\)

[In]

int((-x^5+x^4-1)^(1/4)*(x^5-4)/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=\frac {4 \, {\left (x^{5} - x^{4} + 1\right )} {\left (-x^{5} + x^{4} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]

[In]

integrate((-x^5+x^4-1)^(1/4)*(x^5-4)/x^6,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=\int \frac {\left (x^{5} - 4\right ) \sqrt [4]{- x^{5} + x^{4} - 1}}{x^{6}}\, dx \]

[In]

integrate((-x**5+x**4-1)**(1/4)*(x**5-4)/x**6,x)

[Out]

Integral((x**5 - 4)*(-x**5 + x**4 - 1)**(1/4)/x**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=\frac {4 \, {\left (x^{5} - x^{4} + 1\right )} {\left (-x^{5} + x^{4} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]

[In]

integrate((-x^5+x^4-1)^(1/4)*(x^5-4)/x^6,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=\int { \frac {{\left (x^{5} - 4\right )} {\left (-x^{5} + x^{4} - 1\right )}^{\frac {1}{4}}}{x^{6}} \,d x } \]

[In]

integrate((-x^5+x^4-1)^(1/4)*(x^5-4)/x^6,x, algorithm="giac")

[Out]

integrate((x^5 - 4)*(-x^5 + x^4 - 1)^(1/4)/x^6, x)

Mupad [B] (verification not implemented)

Time = 5.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [4]{-1+x^4-x^5} \left (-4+x^5\right )}{x^6} \, dx=-\frac {4\,{\left (-x^5+x^4-1\right )}^{5/4}}{5\,x^5} \]

[In]

int(((x^5 - 4)*(x^4 - x^5 - 1)^(1/4))/x^6,x)

[Out]

-(4*(x^4 - x^5 - 1)^(5/4))/(5*x^5)