\(\int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx\) [164]

Optimal result

Integrand size = 21, antiderivative size = 19 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \]

[Out]

2/3*(x^6+x^4+1)^(3/4)/x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1604} \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (x^6+x^4+1\right )^{3/4}}{3 x^3} \]

[In]

Int[(-2 + x^6)/(x^4*(1 + x^4 + x^6)^(1/4)),x]

[Out]

(2*(1 + x^4 + x^6)^(3/4))/(3*x^3)

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 {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \left (1+x^4+x^6\right )^{3/4}}{3 x^3} \]

[In]

Integrate[(-2 + x^6)/(x^4*(1 + x^4 + x^6)^(1/4)),x]

[Out]

(2*(1 + x^4 + x^6)^(3/4))/(3*x^3)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
trager	\(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
risch	\(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
pseudoelliptic	\(\frac {2 \left (x^{6}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)

[In]

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

[Out]

2/3*(x^6+x^4+1)^(3/4)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \, {\left (x^{6} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

2/3*(x^6 + x^4 + 1)^(3/4)/x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2 \, {\left (x^{6} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

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

[Out]

2/3*(x^6 + x^4 + 1)^(3/4)/x^3

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^4+x^6}} \, dx=\frac {2\,{\left (x^6+x^4+1\right )}^{3/4}}{3\,x^3} \]

[In]

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

[Out]

(2*(x^4 + x^6 + 1)^(3/4))/(3*x^3)