\(\int \frac {\sqrt {-x+x^4}}{x^6} \, dx\) [136]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (-x+x^4\right )^{3/2}}{9 x^6} \]

[Out]

2/9*(x^4-x)^(3/2)/x^6

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} \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (x^4-x\right )^{3/2}}{9 x^6} \]

[In]

Int[Sqrt[-x + x^4]/x^6,x]

[Out]

(2*(-x + x^4)^(3/2))/(9*x^6)

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 {2 \left (-x+x^4\right )^{3/2}}{9 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \left (x \left (-1+x^3\right )\right )^{3/2}}{9 x^6} \]

[In]

Integrate[Sqrt[-x + x^4]/x^6,x]

[Out]

(2*(x*(-1 + x^3))^(3/2))/(9*x^6)

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
trager	\(\frac {2 \left (x^{3}-1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\)	\(20\)
pseudoelliptic	\(\frac {2 \left (x^{3}-1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\)	\(20\)
gosper	\(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\)	\(24\)
risch	\(\frac {\frac {2}{9} x^{6}-\frac {4}{9} x^{3}+\frac {2}{9}}{x^{4} \sqrt {x \left (x^{3}-1\right )}}\)	\(25\)
default	\(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\)	\(30\)
elliptic	\(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\)	\(30\)
meijerg	\(-\frac {2 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\)	\(33\)

[In]

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

[Out]

2/9/x^5*(x^3-1)*(x^4-x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \, \sqrt {x^{4} - x} {\left (x^{3} - 1\right )}}{9 \, x^{5}} \]

[In]

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

[Out]

2/9*sqrt(x^4 - x)*(x^3 - 1)/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2 \, {\left (x^{4} - x\right )} \sqrt {x^{2} + x + 1} \sqrt {x - 1}}{9 \, x^{\frac {11}{2}}} \]

[In]

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

[Out]

2/9*(x^4 - x)*sqrt(x^2 + x + 1)*sqrt(x - 1)/x^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

2/9*(-1/x^3 + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 5.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-x+x^4}}{x^6} \, dx=\frac {2\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \]

[In]

int((x^4 - x)^(1/2)/x^6,x)

[Out]

(2*(x^4 - x)^(1/2)*(x^3 - 1))/(9*x^5)