\(\int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx\) [363]

Optimal result

Integrand size = 17, antiderivative size = 30 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2 \sqrt [4]{-x^2+x^4} \left (-5+x^2+4 x^4\right )}{45 x^5} \]

[Out]

2/45*(x^4-x^2)^(1/4)*(4*x^4+x^2-5)/x^5

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2 \left (x^4-x^2\right )^{5/4}}{9 x^7}+\frac {8 \left (x^4-x^2\right )^{5/4}}{45 x^5} \]

[In]

Int[(-x^2 + x^4)^(1/4)/x^6,x]

[Out]

(2*(-x^2 + x^4)^(5/4))/(9*x^7) + (8*(-x^2 + x^4)^(5/4))/(45*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])

Rule 2041

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*} \text {integral}& = \frac {2 \left (-x^2+x^4\right )^{5/4}}{9 x^7}+\frac {4}{9} \int \frac {\sqrt [4]{-x^2+x^4}}{x^4} \, dx \\ & = \frac {2 \left (-x^2+x^4\right )^{5/4}}{9 x^7}+\frac {8 \left (-x^2+x^4\right )^{5/4}}{45 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2 \sqrt [4]{x^2 \left (-1+x^2\right )} \left (-5+x^2+4 x^4\right )}{45 x^5} \]

[In]

Integrate[(-x^2 + x^4)^(1/4)/x^6,x]

[Out]

(2*(x^2*(-1 + x^2))^(1/4)*(-5 + x^2 + 4*x^4))/(45*x^5)

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90

method	result	size
trager	\(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (4 x^{4}+x^{2}-5\right )}{45 x^{5}}\)	\(27\)
pseudoelliptic	\(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (4 x^{4}+x^{2}-5\right )}{45 x^{5}}\)	\(27\)
gosper	\(\frac {2 \left (1+x \right ) \left (x -1\right ) \left (4 x^{2}+5\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{45 x^{5}}\)	\(30\)
risch	\(\frac {2 \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (4 x^{6}-3 x^{4}-6 x^{2}+5\right )}{45 x^{5} \left (x^{2}-1\right )}\)	\(41\)
meijerg	\(-\frac {2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{4}-\frac {1}{5} x^{2}+1\right ) \left (-x^{2}+1\right )^{\frac {1}{4}}}{9 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} x^{\frac {9}{2}}}\)	\(45\)

[In]

int((x^4-x^2)^(1/4)/x^6,x,method=_RETURNVERBOSE)

[Out]

2/45*(x^4-x^2)^(1/4)*(4*x^4+x^2-5)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2 \, {\left (4 \, x^{4} + x^{2} - 5\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{45 \, x^{5}} \]

[In]

integrate((x^4-x^2)^(1/4)/x^6,x, algorithm="fricas")

[Out]

2/45*(4*x^4 + x^2 - 5)*(x^4 - x^2)^(1/4)/x^5

Sympy [F]

\[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}{x^{6}}\, dx \]

[In]

integrate((x**4-x**2)**(1/4)/x**6,x)

[Out]

Integral((x**2*(x - 1)*(x + 1))**(1/4)/x**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2 \, {\left (4 \, x^{5} + x^{3} - 5 \, x\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{45 \, x^{\frac {11}{2}}} \]

[In]

integrate((x^4-x^2)^(1/4)/x^6,x, algorithm="maxima")

[Out]

2/45*(4*x^5 + x^3 - 5*x)*(x + 1)^(1/4)*(x - 1)^(1/4)/x^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=-\frac {2}{9} \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + \frac {2}{5} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} \]

[In]

integrate((x^4-x^2)^(1/4)/x^6,x, algorithm="giac")

[Out]

-2/9*(1/x^2 - 1)^2*(-1/x^2 + 1)^(1/4) + 2/5*(-1/x^2 + 1)^(5/4)

Mupad [B] (verification not implemented)

Time = 6.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx=\frac {2\,{\left (x^4-x^2\right )}^{1/4}\,\left (4\,x^4+x^2-5\right )}{45\,x^5} \]

[In]

int((x^4 - x^2)^(1/4)/x^6,x)

[Out]

(2*(x^4 - x^2)^(1/4)*(x^2 + 4*x^4 - 5))/(45*x^5)